Handbook of Graph Theory

DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN HANDBOOK OF GRAPH THEORY EDITED BY JONATHAN ...

Author: Jonathan L. Gross | Jay Yellen

543 downloads 2465 Views 42MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN

HANDBOOK OF

GRAPH THEORY EDITED BY

JONATHAN L. GROSS JAY YELLEN

CRC PR E S S Boca Raton London New York Washington, D.C.

© 2004 CRC Press LLC

DISCRETE MATHEMATICS and ITS APPLICATIONS Series Editor

Kenneth H. Rosen, Ph.D. AT&T Laboratories Middletown, New Jersey

Charles J. Colbourn and Jeffrey H. Dinitz, The CRC Handbook of Combinatorial Designs Charalambos A. Charalambides, Enumerative Combinatorics Steven Furino, Ying Miao, and Jianxing Yin, Frames and Resolvable Designs: Uses, Constructions, and Existence Randy Goldberg and Lance Riek, A Practical Handbook of Speech Coders Jacob E. Goodman and Joseph O’Rourke, Handbook of Discrete and Computational Geometry Jonathan L. Gross and Jay Yellen, Graph Theory and Its Applications Jonathan L. Gross and Jay Yellen, Handbook of Graph Theory Darrel R. Hankerson, Greg A. Harris, and Peter D. Johnson, Introduction to Information Theory and Data Compression Daryl D. Harms, Miroslav Kraetzl, Charles J. Colbourn, and John S. Devitt, Network Reliability: Experiments with a Symbolic Algebra Environment David M. Jackson and Terry I. Visentin, An Atlas of Smaller Maps in Orientable and Nonorientable Surfaces Richard E. Klima, Ernest Stitzinger, and Neil P. Sigmon, Abstract Algebra Applications with Maple Patrick Knupp and Kambiz Salari, Verification of Computer Codes in Computational Science and Engineering Donald L. Kreher and Douglas R. Stinson, Combinatorial Algorithms: Generation Enumeration and Search Charles C. Lindner and Christopher A. Rodgers, Design Theory Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, Handbook of Applied Cryptography Richard A. Mollin, Algebraic Number Theory Richard A. Mollin, Fundamental Number Theory with Applications Richard A. Mollin, An Introduction to Crytography Richard A. Mollin, Quadratics

© 2004 CRC Press LLC

Continued Titles Richard A. Mollin, RSA and Public-Key Cryptography Kenneth H. Rosen, Handbook of Discrete and Combinatorial Mathematics Douglas R. Shier and K.T. Wallenius, Applied Mathematical Modeling: A Multidisciplinary Approach Douglas R. Stinson, Cryptography: Theory and Practice, Second Edition Roberto Togneri and Christopher J. deSilva, Fundamentals of Information Theory and Coding Design Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography

© 2004 CRC Press LLC

8522 disclaimer.fm Page 1 Tuesday, November 4, 2003 12:31 PM

Library of Congress Cataloging-in-Publication Data Handbook of graph theory / editors-in-chief, Jonathan L. Gross, Jay Yellen. p. cm. — (Discrete mathematics and its applications) Includes bibliographical references and index. ISBN 1-58488-090-2 (alk. paper) 1. Graph theory—Handbooks, manuals, etc. I. Gross, Jonathan L. II. Yellen, Jay. QA166.H36 2003 511'.5—dc22

2003065270

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microÞlming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. All rights reserved. Authorization to photocopy items for internal or personal use, or the personal or internal use of speciÞc clients, may be granted by CRC Press LLC, provided that $1.50 per page photocopied is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923 USA. The fee code for users of the Transactional Reporting Service is ISBN 1-58488-090-2/04/$0.00+$1.50. The fee is subject to change without notice. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. SpeciÞc permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identiÞcation and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com © 2004 CRC Press LLC No claim to original U.S. Government works International Standard Book Number 1-58488-090-2 Library of Congress Card Number 2003065270 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

© 2004 CRC Press LLC

PREFACE

! " "

# $

% & Format

' (

) $ * ( ) + (

" $ ,- + . /

! ) ( . 0 '

) ' !

1 ( " Terminology and Notations

2 3 4 . ) ( 5

0 ( 6 ' 7 ' ) 8 / 7 " ) ) *

© 2004 CRC Press LLC

' 2(

9 ( ) : Æ 6 7 * ( ;

6 ) 2 ) . Feedback

" (

(

Acknowledgements

+ ( ( 5 <

( 8 = 1 <. >

! + ( ( = . ? 2 ? ;

© 2004 CRC Press LLC

? ( 0 ? ? 0

? ( 5 "

© 2004 CRC Press LLC

About the Editors ? 2 < : 8 0 < /

'5= < /

Æ 9 9 /

/ < 2 5

@ '5= " 8 : 2 " 0 2 8

A 9( ( 9 " : 8 ( ? ;

" + " ( 0

( ( < : < 2 = > < : 8 ( ='" <> B >

? ;

0 < =

8 5 = = < : 9 ;( ( ! : > ;

0

: 9 ;( / / ' " , 8

1 : 2 ' " : ' ( ? @ 2 '5= < ! " 8 ( 9

/ 9 ;( ' 0 C 6 <

> ;

! 8 $

© 2004 CRC Press LLC

CONTRIBUTORS 0 A 0

/

5 0

' /

> 0

= 0 /

@

+ 5(

@ /

#

# ! $ # ! % & # ' (

5D 5

D

# ! $ $ #)

# ! # ! - %1$ # ! $ / 5 #

8 /

# ! %$

5 5

0 /$

5

8 9 2

1 5 (

= ( 2

2

? 2

?

? @ 2

# ! # ! ' # ! *$ #) ( # + , # >

# ! -. /01$

#

# ! & 3 # #

2 2

$ # ! . $ #)

0 E >

" + 8

>* +

= 0 8

# ! ($ 32 &4 '2 2 . $ /0

= >

# ! $

? /F

# ! $ /

© 2004 CRC Press LLC

3 / # # ! *$ / !

2

/ '

# ! - %1$

? E

# ! / $

? @

# ! $

#

2

@

< E (

5 =

"

/ ==

E %E"& "

# ! &$ / !

< ( =

# ! &

! (

5 # # ! 9

0 "

!

9

0 " (

A 9(

" + " (

? )

H " $

6 * # $ 7 # ! . / #

2 < (

!

" G$ < (

# ! .8 8$ /

= <

% #

<G$ <G(

# ! .8 8$ /

E 5

! / # /

>

#

> @

!

© 2004 CRC Press LLC

/ # ! * 6 / 5 #

! / # ! % 02$

0 A

# ! '

= ( 1 + (

/ #

0 " +

( #

? +

9 # $ #)

9 +

# ! ($

? ;

< H

( #

CONTENTS Preface 1.

INTRODUCTION to GRAPHS

1

2.

! " # & & " # ' &( & ! )& * " & +&& & "# $

56 $%

126 ' $ %

CONNECTIVITY and TRAVERSABILITY ' ' ' '' ' '$ '

© 2004 CRC Press LLC

DIRECTED GRAPHS

4.

GRAPH REPRESENTATION

3.

& ( , " - & # %& % . ' % ( " " # )/ -0& $ ) ( - " # & & ( # %& %

193 ' ' $

5.

COLORINGS and RELATED TOPICS ' $

6.

$ $ $' $ $$

$ % ' ' '' '

+ -

#( . $ "(, 2& -& $ 0$ 1 , -& ( ) * * , 23

484 '% $ '

TOPOLOGICAL GRAPH THEORY ' $ %

© 2004 CRC Press LLC

'

ALGEBRAIC GRAPH THEORY $

7.

* + & * + ! - 0 & & 1 ( ,, " & ( + & # -, ),. / $

340

- & ,4+ ,4+ 4( * *3 !# #

2

, ( * ( ( ) (( " 5,

610 $ $ $' $$ $%' $$ $

8.

ANALYTIC GRAPH THEORY %

.3 6 6 , 5 ! % 5( , % # % %' " ## & * % +

9.

%% % % %$ %

%$GRAPHICAL MEASUREMENT

& * ' ( ' & % (

' )4 ,

10.

787

872 % %% '

GRAPHS in COMPUTER SCIENCE

952

-& ' 5 & + % , ), ( 7 %(( ## % 7 4 ## ,

' + &( & '$ ! $

11.

NETWORKS and FLOWS

*3 4 / " * 4 % ( * & + ) " ' & 5 4 6 * ( )! ",(8!

© 2004 CRC Press LLC

1074 %

Chapter

1

INTRODUCTION TO GRAPHS 1.1

FUNDAMENTALS of GRAPH THEORY

1.2

FAMILIES of GRAPHS and DIGRAPHS

1.3

HISTORY of GRAPH THEORY !" # GLOSSARY

© 2004 by CRC Press LLC

2

Chapter 1

1.1

INTRODUCTION TO GRAPHS

FUNDAMENTALS OF GRAPH THEORY

Introduction !" " " # " " #" $

1.1.1 Graphs and Digraphs % #& # # " ' " " ( $ # # Basic Terminology DEFINITIONS

)

%

* + ,

+ ,

¯ ¯

¯ - " %

# + + , + ,, ./ /

) ' .

) )

% .

" # " . & #

& #

© 2004 by CRC Press LLC

Section 1.1

) ) ) )

3

Fundamentals of Graph Theory

% & %

% # #

.

) # #

#

)

% &

% 0/1 01 # .

01

' " 0 1 " /

EXAMPLE

)

% * + , 2 ' ./ * / * / " /

Figure 1.1.1

REMARKS

)

% 3 / " " " 4 " $ "

) (" " % / 56 7 +. /" 6 , Simple Graphs

8 " # # # # # DEFINITIONS

) )

% / / % .

© 2004 by CRC Press LLC

4

)

Chapter 1

%

INTRODUCTION TO GRAPHS

./ /

Edge Notation for Simple Adjacencies and for Multi-edges

% & # # &./ (" + , " # / ." " # " 2 # EXAMPLE

)

2 /

Figure 1.1.2

*

General Graphs

8 9 / 8" / " + /, + :,

% 0 1 / / " " " DEFINITIONS

) %

/ +' / ,

) )

.

&

/

EXAMPLES

)

2 #3 4

Figure 1.1.3

© 2004 by CRC Press LLC

Section 1.1

)

2

Figure 1.1.4

)

5

Fundamentals of Graph Theory

/ 2 ;

Figure 1.1.5

! " !#!"

Attributes

% # # # #" # # " $ + " , + " , % # + , $ ; DEFINITIONS

) % ./ # #

) % / # # Digraphs

% # %/ ! & $ < # # / 8" " $ " ! # # # # . DEFINITIONS

) % + , " " +, +,"

% # + / .,

© 2004 by CRC Press LLC

6

Chapter 1

INTRODUCTION TO GRAPHS

' "

) ) ) )

% % + , %

/ /

% + , # ' . " 0 1

) +" / , Ordered-Pair Representation of Arcs

' " . . + , + , = / #" +' $ & , 2

#&/ " / > #& / & ? # ! / ! $ +" , = # " / # # - " # " # " 9 > #& EXAMPLES

)

2 @ /)

Figure 1.1.6

© 2004 by CRC Press LLC

" !$

Section 1.1

)

7

Fundamentals of Graph Theory

% #

Figure 1.1.7

% !$ "

Vertex-Coloring

= ./ " $ ; DEFINITIONS

) % ./ $

) % ./ & >

) )

% ./

+, # /#

" + ,"

REMARK

) " " " ¼ + ," # # 01 / #

0 1

EXAMPLE

)

2 A / ./ /#" # %" # / /# / /#B " ¼ + , *

Figure 1.1.8

%" +

, * ¼ + , *

1.1.2 Degree and Distance . $ # ÜC

© 2004 by CRC Press LLC

8

Chapter 1

INTRODUCTION TO GRAPHS

Degree DEFINITIONS

) + , . " +," # # / +2 " " # #,

% " .

) 9 # . /

) . # B . # - /

)

% . 6

EXAMPLES

)

2 C 9

6"

" " " @" @" . 3

Figure 1.1.9

)

" #!&

2 6

. Figure 1.1.10

!" " '"&

FACTS

2 " 5DCC" Ü 7 # .

(

) +-, #

() ()

' " # #

% / 9

© 2004 by CRC Press LLC

Section 1.1

9

Fundamentals of Graph Theory

(

) ' " # 9 #

(

) 9 " / 9

(

) " / " 9 9 " # Walks, Trails, and Paths DEFINITIONS

)

%

9 "

* ¼ ½ ½ ½ * " ½ '" " ½ " ¯ ' " ! # # 9 ) * ¼ ½ * " ½ & ¯ ¼ ¯ + , ¯ % .

) )

) )

# + , %

! . .B "

%

!

% + ,

) ) )

!

($

% . % % !" " .

EXAMPLE

)

' 2 " . 9 ! " . 9

Figure 1.1.11

© 2004 by CRC Press LLC

10

Chapter 1

INTRODUCTION TO GRAPHS

Distance and Connectivity

) ! #

) . . !

) ) )

% # ! %

% . . !

) .

) )

.

.

EXAMPLE

)

2 B #

Figure 1.1.12

" $ &&" %)$ &&"

1.1.3 Basic Structural Concepts = # 9 " " # Isomorphism

' " "$ " % .//. // " DEFINITIONS

) % . #& ) " . & . +, & +, '" #& +, +,

© 2004 by CRC Press LLC

Section 1.1

11

Fundamentals of Graph Theory

) % #&/ ) ) " & #& & +, +,

)

=

)

%

. ! ! + , *

*

.

& 6

) % + , EXAMPLES

)

2

% &

Figure 1.1.13 ' $ . # 9/ ! " 0! & !1 & )

!

*

6 6 6 6

6

6

! *

6 6 6 6 6

6

= # . ! . !

© 2004 by CRC Press LLC

12

Chapter 1

INTRODUCTION TO GRAPHS

)

2 " ! 9 > ./#

Figure 1.1.14

% & "" ) #!" *"

) 2 ; / 9 +' ,

Figure 1.1.15

% % #!& "

FACTS

(

) # #& ./ 9 "+E,

/. /

()

% ! + ," ! /

( )

# " # " 9 ( " " # -. ;

( )

- + , & . 9 . Automorphisms

3 DEFINITIONS

) )

%

+, *

) + , *

) )

% # % #

© 2004 by CRC Press LLC

Section 1.1

13

Fundamentals of Graph Theory

FACTS

() ()

. # ./ # /

EXAMPLE

)

2 2 @" . # " # ./ /

Figure 1.1.16

Subgraphs DEFINITIONS

) % # +F" # # # ,

) ' " * " + ," ./" "

+ + ,, *

+ + ,, * +

,

)

% # + , * + , +%" # " , ) % # # ' " (

#

EXAMPLE

)

2 2 :" # # # " # # #

Figure 1.1.17

© 2004 by CRC Press LLC

! !& !

14

Chapter 1

INTRODUCTION TO GRAPHS

FACTS

(

) ? ) # " # # # # # +# ,

()

' # #

2 *

# #

"

Graph Operations

" " " # " # $ " DEFINITIONS

)

* + ," " . " + . #,

) * + , . #

)

+ , * + , & . + ," + ,

* + ,

#

) + , + , ) % + , .

) )

%

#

+ / , . "

)

+ ,

+ +

)

#

G , * + , + , G , * + , + , + ,

+ ,

+ ,

+ +

G

, * + , + ,

, * + , + , + , + ,

#

+ , + , + , + , + ," + , + , + , + , + $, + ," $ + ,

© 2004 by CRC Press LLC

Section 1.1

15

Fundamentals of Graph Theory

) ./ / & " " ./ /

) %

&

%

EXAMPLES

)

2 A /

Figure 1.1.18

)

+&""

2 C &

Figure 1.1.19

)

, "

2 6

Figure 1.1.20

-" !&"

) ' 2 C #" . /." . . /

1.1.4 Trees / " ! % 3

© 2004 by CRC Press LLC

16

Chapter 1

INTRODUCTION TO GRAPHS

Acyclic Graphs DEFINITIONS

) ) )

% + ,

)

#

%

+" $$ $,

% .

9

+ Ü ," $ $ " . " + 2 ; # , EXAMPLE

)

2 B

Figure 1.1.21

" "% +"

FACT

(

) # . ,

&½ &¾

+ Ü

Trees as Subgraphs

> #/ " . % 9 . # / DEFINITIONS

2 ' " ' " '

) % ' '

)

%

'

/

#

EXAMPLE

) 2 2

" ' # #!" / ' "

© 2004 by CRC Press LLC

Section 1.1

17

Fundamentals of Graph Theory

" " " " / '

Figure 1.1.22 " %" " . . .

(# 2

' " # " / FACT

() ? ' # " # ' # # ' +2" . ' " " / , Basic Tree-Growing Algorithm

# / . # ! !

% /. // 6 Algorithm 1.1.1: /&

+% %" 0"1 2

)") . !") ' +, ./# '3 ' . = # 6 . '3 # )* = ' +, ' ? # % . ' = # . )* G ' ./# +,

+,

'

REMARK

) 53#! " 4!"!" +% 7

= + #! ," % # 9 + " , 9 & " # + ,

© 2004 by CRC Press LLC

18

Chapter 1

INTRODUCTION TO GRAPHS

#!

FACTS

" # # +, () ' . # / .

() % # / # Prioritizing the Edge Selection

/3 / # # / Algorithm 1.1.2: +5" +%

)") " . " 3 !") ' ./# '3 ' . '3 ' = # 6 . '3 # )* = ' F ' ? # ' ? # # % + . , ' = # . )* G ' ./#

FACT

( ) > 3 >

) "*+ $ + **+ * ," *+ $ ++ **+ * ," + *$ ," ,- +$ ** , + 6 ,

References 5A;7 " " H/4 " CA; 5CA7 #I "

" " CCA 5?C@7 ? ?!" " - " 4" CC@ +2 - " J= J#" C:C,

© 2004 by CRC Press LLC

Section 1.1

Fundamentals of Graph Theory

19

5DCC7 K ? K D"

" J" CCC 54C7 2 4"

" J " CC +2 - " % /=" C@C, 5 C 7 L 8 H "

" K = M " CC 5667 = "

" # " 666 5=6 7 ="

" - " J/4" 66 +2 - " CC@,

© 2004 by CRC Press LLC

20

Chapter 1

INTRODUCTION TO GRAPHS

© 2004 by CRC Press LLC

1.2

FAMILIES OF GRAPHS AND DIGRAPHS

! ' /< ' N

Introduction = " O O 9" # % " 3 5=CA7 > #

1.2.1 Building Blocks # # # # DEFINITIONS

) % & #

) ) ) ) ) )

&

& #

&¼ &½ . . /

% #

(

&

) /. " +N " # # ,

)

/.

(

"

REMARKS

) % 01 " / #

© 2004 by CRC Press LLC

Section 1.2

21

Families of Graphs and Digraphs

© 2004) by= CRC PressLLC 01

01 9 " 0 1 0 1 !

EXAMPLES

)

Figure 1.2.1

&" &"

) Figure 1.2.2

" &$&

1.2.2 Symmetry

!

Local Symmetry: Regularity

DEFINITIONS

)

% .

) ) % )

¯ ' . )

)/ #

FACT

()

% (* + Ü ,

EXAMPLES

) )

2

) * 6 " . )/

; & + 6," + ," & + ,

)

5=CA7 . /

Figure 1.2.3

© 2004 by CRC Press LLC

@

"% + ! %" @ '"&

22

Chapter 1

INTRODUCTION TO GRAPHS

)

& & & / :/ . ./ ' / / :/. ./

) ( /

Figure 1.2.4

A " ./

6' + ! &&" %" A '"&

Global Symmetry: Vertex-Transitivity

( ./ # Ü@ DEFINITIONS

) + * , * * = # # # # * = # " B #

) *

% +B * , "

#

. " & ) > ) * % B

) + " , )/. &

&

) + + , /! / # + , 6

)

)

*

© 2004 by CRC Press LLC

&

G &

*

Section 1.2

)

23

Families of Graphs and Digraphs

6/. / 2 ;

Figure 1.2.5

5"

EXAMPLES

) . G Æ / ' /! &

)

% /! J ) / " #" " "

) J ./" + , ' 6" " " "

)

&

FACTS

()

54# 3 7 # 9 & 9 > . +

()

= / # + " )

+ *

& +

&

* 6

1.2.3 Integer-Valued Invariants # / / # ( # Cycle Rank

$$ 6 " + ,

© 2004 by CRC Press LLC

24

Chapter 1

INTRODUCTION TO GRAPHS

DEFINITION

) * + , # G + @ ! ! , 8 " + , " ! # + , + , G + , EXAMPLE

)

! 6

Figure 1.2.6

" %" ! " 6' '"&

FACTS

()

5 3 7 9 ' + " 5DCC" 7,)

' ' '

(

+ "

,

% ' # .

) 5' +, 7 ? # ) +, +,

&½ ' ' ' # # # ' " ' ¼

. & .

¼

+ ,

() ()

! ! %

Chromatic Number and k-Partite Graphs

' " &" " & >

/ . ; ; DEFINITIONS

) % # + , " & +2 " &½ , ' , $ " # ,$

© 2004 by CRC Press LLC

Section 1.2

Families of Graphs and Digraphs

25

) % # . & ' , $" &

) % ) # ) +

, &

) % ) )/ & > % ' ) ½ " &½ ¾

" ," & EXAMPLES

) )

- #

- # #" # #

)

/ & /

Figure 1.2.7

&" +"" & . *

FACTS

(

) 5 3 7 % # + " 5DCC" ;7,

( ) % )/# )/ ( ) 2 ) " # )/ HJ/ k-Connectivity and k-Edge-Connectivity

# 3 $$ *$$ : . DEFINITIONS

) " - + ," # /

) " - + , # /

© 2004 by CRC Press LLC

26

Chapter 1

INTRODUCTION TO GRAPHS

# 0 1 - " - . - - "

)

% - ) ) -9" )/ ) /

) % / - ) " ) )/ /

) % ) + ) , )

) % ) + ) , ) Minimum Genus

# 3 DEFINITIONS

) + , # # # / + : , /

)

% 6

1.2.4 Criterion Qualification % " " # . DEFINITIONS

) % ! . + . ,

) % + ; . ,

) % )/ ) # + ; ,

) % )/ ) . + ,

/

) % )/ / ) // + ,

) 0+ , B 0+ , & . %" # . 0+ ,

© 2004 by CRC Press LLC

Section 1.2

27

Families of Graphs and Digraphs

) % . # + Ü, EXAMPLE

) Figure 1.2.8

"

FACTS

() ¯

5? 37 9)

¯ 5L7 # # . ¯ 5:67 H 2 C #

Figure 1.2.9

()

!& !

%

EXAMPLE

)

Figure 1.2.10

© 2004 by CRC Press LLC

"!" %" " ! '"&

28

Chapter 1

INTRODUCTION TO GRAPHS

References 5:67 ? = !" 3 "

C + C:6,"

CP ; 5DCC7 K ? K D"

" J" CCC 5L7 K L3" Q R IQ = I." ;6 + C," :;PAC 5=CA7 K =" ! " (. F J" CCA

© 2004 by CRC Press LLC

Section 1.3

1.3

29

History of Graph Theory

HISTORY OF GRAPH THEORY

!" #

# ; %

Introduction % A:A" / # #! :; ? - + :6:/A, LS # # # 3 2 # 5?=CA7 5=CC7

1.3.1 Traversability # #! -T ! LS # # # + :;," #9 " # J L! + A6@/C;, = 4 + A6;/@;, " The Königsberg Bridges Problem

#. " " 2 " ! / ! # LS # . O " ! # Ü . " C

c

d g

A

c

D

a B

Figure 1.3.1

© 2004 by CRC Press LLC

b

' 78

30

Chapter 1

INTRODUCTION TO GRAPHS

FACTS 5?=CA" 7

() ( @ % :; ? - 0 #/ 1 % J# " " ! # () ' :@" - " / # () - 5-) :@7 # " # 0 # / 1 % :@" : " # :; () -T " # #"

LS # #

() - #" # # 9 %" " " + , % % & + # # % # % ," % " " & & 4 # 9 ." # LS # # # () ' #" - # # #

. %" " " # # / 0 ! 1" ./ 9 #

() -T ) ¯ ' & #

# # "

¯ ' # # . " & # ¯ '" " # # " 9 & # /"

"

() - " # "

#" # " # % # 43 54) A:7 A: Diagram-Tracing Puzzles

% * $ "%% " 9 # # ! 33 # #! P ." % .

© 2004 by CRC Press LLC

Section 1.3

31

History of Graph Theory

FACTS 5?=CA" 7

( ) ' A6C ? J 5J) A6C7 #)

" 9 $.# # # " & 9 J # # " # ' " 9

( ) ( / 33 # 5) A7 K ? 5?) A:7 #! $ " 0 1 () ' AC" ( 9 ! # # # " # 8 5) A: /7 #

&

() # LS # # # /

33 3 C ' # = = 5) AC 7

% # # 2 # C g

c d e

A b

D f

a B

Figure 1.3.2

" 78

Hamiltonian Graphs

% # # # & . 4T $" " + Ü;," & # L!" # 4T " # FACTS 5?=CA" 7

() % . # / " " ! T # " @ 9 & # # #!

© 2004 by CRC Press LLC

32

Chapter 1

" 5<) :: 7

INTRODUCTION TO GRAPHS

# - 5-) :;C7" %/ <

() ' A;; L! 5L) A;;7

& 4 / # " . # # 0 #1 + 2 ,

Figure 1.3.3 7)9 :& ; 1

() % ! / #" 4

4 #9 " + 2 ," " . R

Q P

Z

W

B

X H J V

S

C

G

D

N

F

M

K

L T

Figure 1.3.4 <"9 &

() ' AA" J /

2 ;,

Figure 1.3.5

#9 # = 5@7 C@ +

!""9 +'" +" $

() Æ # # # %

5 ; 7" ( ( 5(@67" K % < I C:@"

© 2004 by CRC Press LLC

Section 1.3

33

History of Graph Theory

() 4 # " # % /4 + C@6," 4 8 + C:,"

1.3.2 Trees " " ! L> + A /A:," / ! ?" # % + A /C;," K K + A6@/C:," JI + AA:/ CA;," " # Counting Trees

- 9 # > " # # " - ! Ü@ FACTS 5?=CA" 7 5JA:7

( ) = ! # # ! 0 >

1" 5) A;:7 /

( ) T ! " # # # + 2 @, root

root

Figure 1.3.6 ="" " "

? ! # # #" G !½ G !¾ ¾ G !¿ ¿ G 9 F 9"

+ , ½ + ¾, ½ + ¿, ¾ # # !

() % A:6" K $ 0$ $ 0$ () ' A:" 5) A:7 Æ # " " !

© 2004 by CRC Press LLC

34

Chapter 1

() ' AAC" 5) AAC7

INTRODUCTION TO GRAPHS

# # 4 . * @" # # 4 JS 5J A7) # // # # 9 #

() ' C:" JI 5JI:7 # # # JIT ! # K 4 5 :7" # T # $ #& () ? # ( 5(A7 + 54J:7, #9 # 2 4 54;;7" 5@7" Chemical Trees

A;6 ! # . +, " +, ! " # # # % " # # # " %. 2 : " "

H

H

H

C

C

H

H

H

O

H

H

H

C

C

H

H

O

H

Figure 1.3.7 "" " FACTS 5?=CA" 7

() T .

" # . +, # > 2 A H

H

H

H

H

H

C

C

C

C

H

H

H

H

H H

C

H

C H

H

H

C H

Figure 1.3.8

© 2004 by CRC Press LLC

C H

H

H

% > !" !"

Section 1.3

35

History of Graph Theory

() 5) A:7 / Æ +!,

# " B # # ! * A 2 H#

;

C

A

() = L > # # #

# 9 " # ' A:A" 5) A::/A7 & # " ! ) - # .# # " L!I

" /

( ) ' A:A" 5) A:A7 !

" ( ) ?

C 6 C6 % ? K L 5? C7 3 " JI T #/ #

1.3.3 Topological Graphs -T 5-) :;67 " ' . ' C6" 3 # # # L33 L ! + AC@/ CA6," ! P # # H #" J " P . Euler’s Polyhedron Formula

! " # ! # # " " 1 )

G1 *

' : " I " # -T # 4 " " # !

© 2004 by CRC Press LLC

36

Chapter 1

INTRODUCTION TO GRAPHS

FACTS 5?=CA" ;7 5CC7

() " H# :;6" - # # " +, & + , # " " " ' # # # . # # " G * G

() - # ' :; # / " # # %/8 ? 5?) :C7 :C" () ' A " %/? 5) A 7 # -T # /

&

() % " /%/K ? 5?) A 7

. " # 4 -T P " " / + " RT , 2 / " ?

G1 * 6 . " # "

G 1 * # & &$" 9 1 $ $ $ + : ,

() ' A@ / " ? 5?) A@ / 7 '

( " ./

." > 3 # -T ! # $ #9 ' " 4 JI ! ? T AC;/ C6 #

() ! JIT ! " /

# 8 J 4 5 46:7 + , / ) '

* ! 4 # ( <# 5<

7 9 % 8 C @

Planar Graphs

# / & # & + 2 C, # # " #9 # L !

© 2004 by CRC Press LLC

Section 1.3

37

History of Graph Theory

K3.3

K5 Figure 1.3.9

7!"%) &

&

FACTS 5?=CA" A7

() % A6" % 2 8S# 33 / ) ! ' ! # # U 9 ! # # " # 4 3) ! & ! # # U ' " #

&

() % #" #" " " * * $ $ " " # 4 5 7 " C ) 33 " " " =" " -" " %" " " + 2 6, #

&

W

G

E

A

B

C

Figure 1.3.10

+%"+&"&"$

( ) ' C6 L ! 5L67 # # B # # ( 2! J %

&

&

( ) ' C 4 = 5= 7 # #

© 2004 by CRC Press LLC

38

Chapter 1

INTRODUCTION TO GRAPHS

4 " " # # # 8?

() ' C; = 5=;7 3

. " 5;C7 / C;6 # L !/ + Ü@@, Graphs on Higher Surfaces

% +# &, ! " . " # J 4 + A@ / C;;, ? 4> + A@ / C@ , # " # 4 3 + AA6/ C@, /# " # # # D 4 & C@6 H # J 3 L !T CA6 FACTS 5?=CA" :B :7

() ' AC6" 4 54) AC67 #

&

4 # # " #

() ' AC " ? 4> 54) AC 7 #

# " 4 T #

() ' C 6" 4 3 5 67 . 4>T // # " 8S # # & " / 4 4 # L #" # # J 2! 527 C" . ' C;" ' L 5L;7 /# " " @ () 4 /# C; #

# # Æ" 66 8 / C@6" C@A # D 5D@A7" = T 5@7 # C@ $ " " # K ? 5:7 3 # #&" 3 " + Ü:," + = ! 5:7,

() ' 9 CA6 " #

5A;7 " # " 0# # 1 + Ü::, 4 " " # # # " 2 /# " " C:C 4 4 " K J 4!" = 54=:C7 # 6 # # &

© 2004 by CRC Press LLC

Section 1.3

History of Graph Theory

39

1.3.4 Graph Colorings - ! " " # / # # 2 A; " # +, 01 # % L A:C ' # L % = 4! C:@" # ! L" !>" 4 4" " #9 # H #" " J " # 5 CC7 8 " # " 5 AA67" < <3 C@ % " # # D C@A The Four-Color Problem

8 # #! / # / # FACTS 5?=CA" @7 5=6 7

() ! / # % 8 4" (# A; 8 # ! # & # > 2 ! " # # 2" - 4 # () 8 " # #

Æ ( 6 % A@6" # " #!

" # 8 F # J" / #

( ) ( K A:A" ? 8 " ! # # " # # Æ " Æ T

( ) ' A:C" L 5L) A:C7" # " # / !

" # # L #" # . 4 9" ! #"*$ " / LT " " 9 " # + / , () ' AA6" 5) A:A/A67 0 1 / " ( / #

© 2004 by CRC Press LLC

40

Chapter 1

INTRODUCTION TO GRAPHS

? +? ," 2 + ? ," < K 8 =

() ' AC6" 4 54) AC67 # + ! #

" LT " / " 3 # " + Ü , 4 #9 # . #" C6 L " # #

() 6 "

LT O " F # # J =! 5=) C67 + 2 ," # J 2!" # 4 ?#

digon triangle

quadrilateral

two pentagons

pentagon and hexagon

Figure 1.3.11 ?&)9 !' "

$ $+ O # . / ) /. / !> 5) C 7 2 +! !> , #

Figure 1.3.12

/)*

() ' C " !> 5 7 # ) " )" $ $ "

() ' C

" 2! 52

7 # # / " / #

;

() % C;6 4 # # / ( . " 4 54@C7 #

© 2004 by CRC Press LLC

Section 1.3

41

History of Graph Theory

() ' C:@" % 4! 5%4::" %4L::7" K L"

# # A # " # / 9 # / #

() % CC" #" " " 5C:7 /

F # # # " 3 %/ 4! " # # @ #

Other Graph Coloring Problems

% ! / #" #

#

FACTS 5?=CA" @7 52=::7 5KC;7

( ) ' A:C " L 5L) A:C7 # & > / ! # 4 = C # #9 ! / #

( ) ' AA6" 5) A:A/A67 / 9 # .

() ' C @" LS 5LS @7 # . #

+ Ü

,

() & > C6" ! =" J () ' C " ? ! 5 7 #

. G " + Ü; ,

9

() ' C;6" # ./ # % " $ $ "

() ' C@" < <3 5<@7

. # G ' " <3 /

() # / # # 3 # # O ." 8 3 C@6" J - V

© 2004 by CRC Press LLC

42

Chapter 1

INTRODUCTION TO GRAPHS

Factorization

% )* ) # # " ./ % )*&$ )/ # 2 ! # K J 5 AC/ C 67 = 5 C / 66 7 + Ü;, FACTS 5?=CA" 67

() ' AC " J 5J) AC 7 3 " # ' ) " )/ # / 4 / /" 01B # & # () ' ACA" J 5J) ACA7 "

" + 2 ," # /B " " # / + !, / + ,

Figure 1.3.13

5"

( ) ' C:" 5:7 3

/ 2 . 3 )/" )

1.3.5 Graph Algorithms # #! C " 2 3 + Ü , 6 # #" #" + Ü," # # ' # !" " + W ., # #" ! FACTS 5 A 7 5??A;7 5?JA@7

( ) 2 " !

# "

© 2004 by CRC Press LLC

Section 1.3

History of Graph Theory

43

A ' C6" J" 3 %H

3 " 2!" 8 K 5 2K;7 # C ' CA6 # C # J # 5JA:7 + Ü@,

() $$ " " !

/ " # #! ( ! 5 @7 # K L! 5L;@7 % " < KI! + C ," # J + C;:, + Ü 6 ,

() # 3 2! 5 2;7 . $ # !" # - 4 54@ 7 ( 3

/ ! () 2 " " !

C6 C;6" 1 +J - 9, # F H # # # J8 + $ 4, # J H 3 & + Ü ,

() Æ !"

# ! - = &! 5 ;C7 + Ü 6 ,

() #"

" # 8 +8/L L , 5@67 C@6 + Ü,

() ' # /

# &# 9 ! ! LS # J 4 54;7" ! 0 1 54<;67 #& # # + Ü ,

() C@6 # # # Æ/ " - 5- @;7 # / . ! 5: 7" L 5L: 7" HJ/ " " / #/ " * $

J" 4 # HJ/ ' ! J * HJ 2 # 5K:C7

References 5%4::7 L % = 4!" - /#) J " " + C::," CPC6 5%4L::7" L %" = 4!" K L" - /#) J " #" + C::," CPC6

© 2004 by CRC Press LLC

44

Chapter 1

INTRODUCTION TO GRAPHS

5?=CA7 H ? " - L ? " K ="

,-./0,1./" (. F J" CCA 5 7 !>" % # " ! + C ," P@

5 7 !>" # " ; + C ,"

;P A

5?@7 !> ? " " " @6 + C@," ;;P; 5 @7 ( !" ( &I #I I I" " " & 2 + C @," :P;A 5 7 ? !" ( !" # # " : + C ," CP C: 5) A 7 %/? " Q / I" ) #0 C + @, + A ," @APA@ 5) A;:7 % " ( " # +, + A;:," : P :@ 5) A:7 % " ( " # +, : + A:," P@ 5) A:C7 % " ( " # % " + + A:C," ;CP @

,

5) AAC7 % " % " + # + AAC," :@P :A 5: 7 % !" . / " ; P ;A # . "

! " %8" H D!" C: 5) A7 " 5 7 #" & + A," 6CP @ 5CC7 J " #

" # F J" CCC 5 A 7 3 " # " 3 % + CA ," PA 5 2K;7 3 " 2! 8 K" / / #" 3 % + C;," CP 6 5 46:7 8 J 4 " % " ) '

* ! + C6:," ;P 6 5 8) A@67 % 8 " % " " # = = " "

H @C + A@6," ;6 P;6

© 2004 by CRC Press LLC

Section 1.3

45

History of Graph Theory

5 ;C7 - = &!" % # . " & + C;C," @CP : 5 ; 7 % " # " # " +, + C; ," @CPA 5 7 4 - " J." " @ +K C ," C ,"

6 +%

5- @;7 K - " J" $ " : + C@;," CP@: 5-) :@7 ? -" + :@, # " 0 " # A + :; ," AP 6 5-) :;C7 ? -" T 9 9 Q " " 4 ; + :;C," 6P: 52=::7 2 K =" ) 0 ! " J" C:: 522;@7 ? 2 2!" 8. $ A + C;@," CCP6

!"

52

7 J 2!" #" + C

,"

;P @ 527 J 2!" % . #" # + C," @P@C 5K:C7 8 K"

! � " = 4 2 " C:C 54F=:C7 4 4 " K J 4! = " 6 # & "

+, : + C:C," P:6 54@ 7 - 4" 8/ ! $ " " C + C@ ," ;; P;;@ 5:7" K ? " < " C + C:," CP @ 5:7 K ? = !" N ) 4 #" #5 ;; + C:," C P6 5@67 8 " " " 6 + @/ @@ * C@ ,B + C@6," :P :: 5@7 = " (# # " 4 " @C + C@," : P :; 54;7 J 4" ( #" " 6 + C;," @P6 54<;67 J 4 4 - < " #" : + C;6," P ; 54) A;@7 = 4" 8 # +, + A;@," @

© 2004 by CRC Press LLC

"

46

Chapter 1

INTRODUCTION TO GRAPHS

54;;7 2 4" # " " " " " :A + C;;," ;P@ 54J:7 2 4 - 8 J" 54) AC67 J K 4 PA

) " % J" " 8/ " + # +

S 54) AC 7 ? 4>" F# J# H# #" ::P;A6

C: AC6,"

A + AC ,"

54@C7 4 4" F 3 <##" 4 " A 6WA 6WA 6#" # '" 8/</XS " C@C S 54) A:7 43" F# 8S !" ?3 = F# 3 " @ + A:," 6P 5KC;7 K " # " =/'" CC; 5L;7 ' H L " % 4 "

AP

#

+ C;,"

5L: 7 8 L" # # #" A;/ 6 ( ! + - 8 K = ," J J" C: 5L) A:C7 % L" ( # "

+ A:C," CP 66 5L) A;@7 J L!" ( " @ + A;@," P A

# % "

S 5LS @7 LS " F# % 8 " :: + C @," ;P@; 5L;@7 K L!" ( # #" # " : + C;@," AP;6 5L67 L L !" #Q # " ; + C6," : P A

5??A;7 - ? ? " K L ?" % 4 L" + ,"

2 " # 3 " =" CA; 5?) :C7 % 8 ? "

)6 6 6 +

," 2 " J" :C

5?) A 7 ?" I I T IQ T- Q " . IQ #" 6 " " #6 + A ," : P6 5?) A:7 K ? " < 3 " H %# + A:," A PA:;

© 2004 by CRC Press LLC

' " +%#

, 8

Section 1.3

47

History of Graph Theory

5?) A@ / 7 K ? " S ." 7 * ' 6 + A@ / ," C:P A 5?JA@7 ? ?I3 8 J" 8 " !

0 C" H/4 " CA@ 5?) AA 7 - ?" %6 6 6 8 " < " /<" J + AA , 5? C7 % ? K L " ' " #

+ C C," 6 :P 6:C 58@C7 K 8" ? #Q I #"

@ + C@C," ::P C; 5(@67 ( (" H 4 " @: + C@6," ;; 5(A7 (" # " ! C + CA," ;AP;CC 5JA:7 8 = J # " (3 ; / # # # " 3 % @ + CA:," P: 5J) AC 7 K J" S "

6

; + AC ," CP

5J) ACA7 K J" IQ " 6 ; + ACA,"

;P

: 5J) A6C/ 67 ? J" Q " ) # + A6C/ 6, + 6," @PA 5JI:7 JI" L# %3# S " <# " @A + C:," ;P ; 5JIA:7 JI " ) ! 9

" " CA: 5J A7 4 JS " H 3 S# J" # +, : + C A," P 5@7 " ( # / " 0 " A + C@," CCP 6 5 :7 K 4 " / #" C + C :," P;; 5) A: /7 8 " - # # 9 A I T & # TQ Q &" # + , ; + A: /," @P 6 5:7 "

" " C: 5D@A7 K = D " 4 / #/ " # & " :" @6 + C@A," AP;

© 2004 by CRC Press LLC

48

Chapter 1

INTRODUCTION TO GRAPHS

5A;7 H # J " O " "2 ,1;< + ' % ," ? 8 ? H 6 + CA;," # F J" ;P : 5C:7 H #" " J " " / "

9 " 4 :6 + CC:," P 5) AC 7 = = "

% # ! # # +

% )," 8" ? " AC 5=AA7 4 " 8 #3 K =" % ) -T LS / # "

+ CAA," P C 5C7 - " % !" # A + CC," AP ; 5) A::/A7 K K " #" & : + A::/A," A 5) A:A7 K K " ( # 9" + A:A," @P ; 5) A:A/A67 J " ! " # % " )

6 + A:A/A6," : C 5 67 4 3" - ! S# J# LS# / 2S" 0$ C + C 6," ;;P :C 5@7 = " ( " " + C@," CAP 6 5:7 = " 3 " "

+ C:," 6:P 5;C7 = " 8 " " C6 + C;C," ; :P;; 5:67 = " ( "

C + C:6," ACP C@ 5<) :: 7 %/ < " 9 #Q " 6 " =#> + :: ," ;;@P;: 5<

7 ( <#" "" % 8 9 ? C @" H D!" C

5<@7 < <3 " ( / " + C@," ;P6 5<@;7 < <3 " " ; + C@;," CP : S 5=) C67 J =!" F# ! <#3" ;A + C6," P @

© 2004 by CRC Press LLC

Section 1.3

49

History of Graph Theory

5= 7 4 =" H/# " + C ," ;P :

# & " :"

5=;7 4 =" ( # " ;: + C;," ;6CP; 5=CC7 K =" " : - " CCC 5=6 7 ="

© 2004 by CRC Press LLC

:

! + ' 8 K,"

"Æ " J F J" 66

50

Chapter 1

INTRODUCTION TO GRAPHS

GLOSSARY FOR CHAPTER 1 &$&& ) $$ '"1 * + ," #,

./

) / + .

+ ./

, * + ,) / + ,

@&&$ "1 P

. !

./ ½ ¾ )

& 6 @&" )

! + , *

@&" '"&) &) $ ""!") B ( !" P ) "" ) / B # & +2 " &½ # #,

!#!" ) + , . / &" !&" + " $, P )

+

, * + , + ,

+

, * + , + , + , + ,

+ * , P * ) ./ " *

-$$ ½

-$$ ¾)

+ * , # 0 1

&" ) $ " &" '"1 P ) . 9 &"& ! P ) #

/#

&&!" +B * ,) &) # #B $ +& ) ./ & ) + , 01B 01 # " ) + , & +" ,

&") *$"

© 2004 by CRC Press LLC

51

Chapter 1 Glossary

&" "" ) # . & . ' , $" &

&" & ) # " #

&" & ) & #

&" )+"" &½ ¾

) )/

& > % $" " "

&" P ) . # &&" ) # ! )+&&" ) ) " " #

) )/ ' -+ , - + , &" P ) + , # &&"'"$ ) #

&"&$ )+&"& ) # ) / #

+ Ü; ,

&"&$ )+&&" ) .

+ Ü ,

)

&"&$ )+ +&&" ) /

+ Ü ,

)

/

&! ) " $ " &!) " $ &!"+ ) # &!"") $* ( &!"+'"1 + $",) . #

&$&) &$& ) /. "

&$& ) P

+ , G + ,

* + ,

+ , ) # + ,

+ $, P . " +,) #

# / +2 " 9 # #,

#!& P ) 9 # . /

" '"1

* + ,) . " #

© 2004 by CRC Press LLC

52

Chapter 1

INTRODUCTION TO GRAPHS

* + ,) ./ " # /

"

" P ) . $$ $" ! + $ ",) ) & &" "& . " . P )

"

!

&" + $,)

" ' "

&" ) " &" P ) "& P # ) ! # &&"&"$ '"1 P ) .

) # " * + , +& P ) / $

""!") / # # " $

+&" P ) ./

"

+ ,

)+ +&&" ) #

+&&"'"$ '

-+ ¼

,

) # - + ,

)

)

)/ /

+""' ) "$ & ) ") " ! ) ! .

+ . ,

! " P ) &" P ) # )+&" P ) )/ # ") " P ) B

) # / / . .

© 2004 by CRC Press LLC

53

Chapter 1 Glossary

!)

* + ,) $ " "

"

" ) + Ü; . ,

Ü

) $ $&! + ) /! / # $&! ) / +½ , 6 , &&) # P . ) # !& ! P * ½ ) # ./

'" P " " '"1 P ) . 6 & ) # H/ )

*

'" P ) B " #

) #& ) ) " & #& &

+, +,

) #& ) "

+, +,

@½ + " ,)

+ +

G , * + , + , G , * + , + , + ,

G

+ ,

@¾) G 0&1 @¿) 0&1 " %)) # ½ P ) 0+ , ./ / " & .

¾) % # " .

0+

,

) &* " ) &* " ! ! + !, ) # # # H) 2 + , 2 + ,

© 2004 by CRC Press LLC

/ + :

,

/

54

Chapter 1

INTRODUCTION TO GRAPHS

1 ) "

$ " !"+&) !"+ ) " !" ) / . " # /

P .) & . " P ' ) ' ! ) ./ / B &¼ &" ) / / &¾ " '"1 P )

+, *

P ) + , * !" P . ) #

"

"$ &" ) # 3+"" ) ./ 3 # ½ + " ," #

&

"" ") 3*" ") . " &) $ . " ) " . " () /. " +N " # # ,

(

5" ) 6/. / " ;/ "

/ &

) 6" " #

"& ) ! "& ) / O " #" " "

!&") $ " $ ) & +& ) / &

'"1+& ) ./ &

! P ) $$ $" ! ! ) . ' )* . )

© 2004 by CRC Press LLC

55

Chapter 1 Glossary

+) & +!. + P ) & % *

%

) / / ) / / 1) . / Æ - ' )* " (

)" +

/ * ) G * , )/. & )

&

! P ) # . " P ) # " A'"1B P ) # # # # 6

# #

" $ &&" ) . .

!

! )

"

!) - ") $ ") ! ") " P ' ) ' "' &½ ) . "' %). ". ") ! 3 !$ P . #) # + " / ,

) ./ / & " " ./ /

! " P

'&) '"1 ""!") ./ # #

'"1+& P

) ./

$

'"1) # " * + , '"1+""' ) %) P ) 9 * ¼ ½ ½ ½ "

* "

" &) " )

!

!

½

. .

. > .

%)$ &&" ) " % " P ) + $ , #

© 2004 by CRC Press LLC

Chapter

2

GRAPH REPRESENTATION 2.1

COMPUTER REPRESENTATIONS OF GRAPHS

2.2

THE GRAPH ISOMORPHISM PROBLEM

2.3

2.4

THE RECONSTRUCTION PROBLEM

RECURSIVELY CONSTRUCTED GRAPHS

! ! " " " GLOSSARY

© 2004 by CRC Press LLC

Section 2.1

2.1

57

Computer Representations of Graphs

COMPUTER REPRESENTATIONS OF GRAPHS

Introduction

! "

# # $ % #

2.1.1 The Basic Representations for Graphs # & ' & DEFINITIONS

(

(

) * + ! ,

- * +

) * + ! , -

( . ) * +, ' ' * + & ( . ) * +, ' ' & ( / *½ ¾+, *¾ ¿+,, * ½ + ' ½ ' ( ) * + # ¾ ( ) * + # ¾

© 2004 by CRC Press LLC

58

Chapter 2 GRAPH REPRESENTATION

( ) * + ' , # 0 1 ) ' ' 2 0 1 ) 3 #

(

) * +

, ' 4 ' , % & %

(

' , # ' 0 1 ) ' 3 #

4 , 0 1 )

) * +

3 #

EXAMPLES

( 4 # & ' &

! "# # ! !

Figure 2.1.1

(

' ' # #

)

* +

3 3

* + * + * + * + 3 3 3 3 3 3 3 3

FACTS

$(

& '

$(

) * + # % *

& ) * + %

© 2004 by CRC Press LLC

*

5

¾+

+

Section 2.1

59

Computer Representations of Graphs

REMARKS

%

(

4 , 06781, 0

6791, 0:; <31, 0-8=1, 091

%

(

, &

#

, %

%

(

>

¾

) *

/

+ , & ' &

*

+ 6# , # & ', #

# ' , # # & #

*

+ 4 , & '

#

%

(

? &

# & ( '

,

'

2.1.2 Graph Traversal Algorithms @ % A ! ! /

/

4

/ , & # % #

Depth-First Search ALGORITHM A ! . , % B#C > ' , % BC A ! # % # ' &

, % ,

# B ! C

# * + % #

DEFINITIONS A , !

%

, # (

(

(

*

*

+ #

! '

+

* + '

!

(

*

+ '

!

(

© 2004 by CRC Press LLC

60

Chapter 2 GRAPH REPRESENTATION

&$# '

Algorithm 2.1.1:

# ( ) * +, # ) 01 & ' $ # ( !

!( *+ ! () ! !

01 () 2

! () ! ! 01 ) 2 *+2

!( *+ 01 () 2

! ' 01 ! 01 ) *+2

FACTS

$( $(

+ ) * + . # ! ' # B*C B+C, / ! A ! %

*

5

' #

$(

. ! , % REMARKS

%

( A ! =D3 08 681 Æ !

%

( A ! !

Breadth-First Search

! / ' . ! ' # . !

' ,

© 2004 by CRC Press LLC

Section 2.1

61

Computer Representations of Graphs

ALGORITHM

! ! ! ! ! * "+ ' % / " ! ! *"+ / " !

, % 5 . , ! % % Algorithm 2.1.2:

) &$# '

# ( ) * +, # ) , 01 & ' , ! ' $ # ( !

!( # * + ! () ! !

01 () 2 01 () 2 01 () "2

01 () 2 01 () 32 " / "2

! ! * "+2 *" " ! () ! ! *"+2 ! ' 01 ! 01 ) 01 () 2 01 () 01 5 2 01 () 2

! ! * "+2

DEFINITION

"

( ; #$ # , 01 , * 01 + 01 #$ # * +

"

FACTS

$(

! %

© 2004 by CRC Press LLC

*

5

+ ) * +

62

# * + 01

$(

Chapter 2 GRAPH REPRESENTATION

REMARKS

%( ; % ! , ! =D3 -

! " #

%( ! % A &% E E !

2.1.3 All-Pairs Problems # ( #

4 & '

All-Pairs Shortest-Paths Algorithm

< # # # # # %# %# > # A &% E ALGORITHM

# 4> # 4> & ' # ) * + 4 # ' % 0 1 * + . % 0 1, # % 0 1 ! >

4>

, # 0 1

' ' 4 # %

, FACT

$( 4> ' # ) * + * ¿+ * ¾+ REMARKS

% ( 4 4> 06781 0:; <31

© 2004 by CRC Press LLC

Section 2.1

63

Computer Representations of Graphs

Algorithm 2.1.3: $"!&+ # "" #

(

% 0 1

( : '

$ #

) * +, #

)

'

0 1 # 0 1

!( &'*+ ! () ! ! ! () ! !

0 1 () % 0 12 ! () ! ! 0 1 () 32 ! () ! ! ! () ! ! ! () ! ! 0 1 5 0 1 ( 0 1 0 1 () 0 1 5 0 12

% ( ; 0 1 ' '

' , ' > 0 1 ) * ½ ½0 1 5 ½0 1+ . ' # 4> FE

Transitive Closure

. # & # %# # ' ' ' ) * + > ) * + # & ' , # # ' $ $ 0 1 , 3 # > $ & ' # 4> ' , FACT

$ %! *+ * ¾+

$ (

*

¿+

REMARKS

%( < > 0>G 1

© 2004 by CRC Press LLC

64

Chapter 2 GRAPH REPRESENTATION

Algorithm 2.1.4: #

, #- ."!#(

( ) * +, # )

0 1

& '

( ' $ 0 1 # , 3 #

$ #

$ 0 1

!( $ %! *+ ! () ! ! ! () ! ! $ 0 1 () 0 12 ! () ! ! ! () ! ! ! () ! ! 0 1 ) "# 0 1 () 0 1!0 12

%(

; $ 0 1 ) ' '

' , ' >

% 0 1 ) % ½0 1 ! % ½0 1 % ½0 1 # ! . '

# # FE

2.1.4 Applications to Pattern Matching & ! % " ' ' #

< : F

# DEFINITIONS

(

*?4+

) * + #

'

,

'

&

H ) # H !

#

,

) © 2004 by CRC Press LLC

Section 2.1

65

Computer Representations of Graphs

( ?4 * ! ' # *

( (

'

! . + , + *& * & ( ; £ A! ) ) ) ½ , , !

?4

( ; H ! ! #(

H

, ' ) ' ) 4 H, ' . ' + , * 5 + ' + , ' + , *£ + ' +£

> # ' F £ 5, 5 4 ' , ***-£ ++ 5 + # -£ 5 ' - 3 Kleene’s Algorithm

< : F ' ! , # #, 4> ALGORITHM

; ) * + ?4 # FE * #+ # % / % # % 0 1 ' ' ' # ' *' + REMARKS

%( %(

FE 0FDG1

FE , # % 0 1 ' ' # ' , ' % 0 1 ' ' ' *% 0 1+£ ' ' " # ' % 0 1 ' '

© 2004 by CRC Press LLC

66

Chapter 2 GRAPH REPRESENTATION

Algorithm 2.1.5: /"0# "!

# ( ) * +, # ) , ' 0 1 $ # ( ' % 0 1 # % 0 1 '

!( .

*+ ! () ! ! ! () ! !

% ¼0 1 () 0 12 ! () ! ! % ¼ 0 1 () ) 5 % ¼0 12 ! () ! ! ! () ! ! ! () ! ! % 0 1 () % ½0 1 5 % ½0 1 *% ½0 1+ % ½0 12 ! () ! ! ! () ! ! % 0 1 () % 0 12

' , % ½0 1 *% ½0 1+ % ½0 1+ # ( , " , # '

%( 4> FE # % 0 1 () *% ½0 1 % ½0 1 5 % ½0 1 . 4 > # E #

, 4>

*+ %( FE # % 0 1 () % ½0 1 5 % ½0 1 % ½0 1 # 5 !

%( , 6

, 7 FE 06781 %( @ % ! ?4 ' ! ' % ( 4 ! ' 0=3, <79G1

© 2004 by CRC Press LLC

Section 2.1

67

Computer Representations of Graphs

References 0=31 I , ! , DDJ33 , - K I ; #, . , ==3 06781 I , K - 6

, K A 7, >, =8

06791 I , K - 6

, K A 7, >, =9 0<79G1 I , < , K A 7, >, =9G

0:; <31 6 : , : - ; , ; , : < ,

! . , 33 0-8=1 < -,

" :

< , =8=

04G 1 > 4, =8 *< +,

# $ D*G+ *=G

+, D

0681 K - 6

- &, -Æ , # $ G*G+ *=8+, 8 J89 0K881 A K, -Æ # %, $ *+ *=88+, J

%#

0FDG1 < : F, ! , J3 , - : - < K : , 7 , =9D 08 1 - &, A ! , * + *=8 +, GJG3 091 - &, , =9

& ' < .

0>G 1 < > , ,

© 2004 by CRC Press LLC

$ %# #

%# $ =*+ *=G

+, J

68

2.2

Chapter 2 GRAPH REPRESENTATION

THE GRAPH ISOMORPHISM PROBLEM

I ! / . D : '

Introduction , !, Æ # # . ' '$, Æ ' . ' *% + *' *¾ ++, *' * ++

'$ *' *½¾·´½µ+ '$ # ' , , ! #

2.2.1 Variations of the Problem DEFINITIONS

( # ½ ) *½ ½+ ¾ ) *¾ ¾+ , ½ ¾, , , ( ½ ¾ , # * & ½, *& ½ ,**+,*&+ ¾ < & & ,

( #

(

# L ,

, ! . L , $ , # #

# ½ ) *½ ½+ ¾ ) *¾ ¾+

½, ,*+

, ( ½ ¾,

( * +, # , # ½ ) *½ ½+ ¾ ) *¾ ¾+ , ½ ¾,

© 2004 by CRC Press LLC

Section 2.2

, ½ ¾ ¾

,

(

,

(

(

,

,

½ /

½ ¾ / ½

69

The Graph Isomorphism Problem

*

+ )

/ ¾ *

+

½ ¾

½ , ,

,

* +

,

)

* +

#

/

¾ 2

*+ ) ½ ¾

) *½ + 0 ) !½ ! + & ! ( )

, # *

'

FACTS

$

(

#

½

¾

# &

1 1 ½ 1

'

¾

)

½

¾

,

½

# ! ' # & L

$

(

/

'$

$

(

/

'$

EXAMPLES

(

-' # 2

, %2

, %

< , # , * 0F=D1+

(

2

½

)

,

3 # & ' > !

*

*

*

+ *

+

+ '

+ !

'

2.2.2 Refinement Technique / & #

# #

< #

/ , 0:831, 0>8G1, 0>8=1 ,

#

Æ " '$

! * 06781, 0- <931, 0=81, 0:91,

© 2004 by CRC Press LLC

70

Chapter 2 GRAPH REPRESENTATION

0:91, 0AA;881, 04 931, 04M < < 91, 06;<>981, 091, 06>81,0F=D1,

(

08=1, 0 91+ # % N #

'$

*

+

N /

DEFINITIONS

%

(

3 ( %

' #

(

(

3

1

(

(

4

(

5 6

&

4 %½ 4 % * +

* +

3

1

%½ % %

) 0

1 #

,

#

½

%

3 ) 0%½ %1 '

+

'

* + ) 0

*

* +

+3

(

.

3

,

%½ %

) 0

'

* +

5 6

* +

4

#

3

0

* & %

1,

% 5 * 6

#

# $

* + )

(

3 *

+ ) 0

%½ %¾ %

1

73 *

+ ) 0

1 )

4* % *

+

* %

#

*4 , Æ (

" 3

!

* +

, ! 4 3 , ! "

(

5 & 6

(

½

¾

+

.

#

*

,

, ½ (

*

+

3 ¾

*

+

½ ¾ FACTS

$

(

;

3½

3¾

!

+ 3½ *

(

;

3½

3¾

*

¾ ,

, ,

,

+ 4 ,

*

+

*

½ ¾

#

© 2004 by CRC Press LLC

½

+ 3¾ 5 ½* * ½ 6

+

$

½ ¾

#

½

+ )

5 ¾ ,* 6

,

*

¾ 3½

* ++

, .

*

+ )

3¾ *

+

7 3½ *

+ )

7 3¾ *

+

Section 2.2

71

The Graph Isomorphism Problem

$(

; 2½ 2 , #

2 2

REMARK

%(

" / * 0=31+ 4 ' , # '

8*+ ) 08 1

0 12

0 1

# 8 ' . , ' A E % 0A3 1 , B ' # ! ! 'C Backtracking

! / % % DEFINITIONS

(

% % # / 9 2 9, # 9 , #

9

(

# 92 9, # 9 9 "

( 4 9 2 9, ', , 9 , '

9, %

*&( + # " #

. " ,

( ; % 9, * % , : " 9 % # # ( * % * > !' * % , % ¼ : # % , ( % ¼ ( % ; FACTS

% '$ % % # ! Æ # # ' Æ % % # !

$( *0 891, 0 8=1+ . # &

*

+

E %

© 2004 by CRC Press LLC

72

Chapter 2 GRAPH REPRESENTATION

$( *0 DD1+ ; O ! & P, O P , ¾ , ¾ 5 Æ # !' ' "

$ ( . % % , !' ' , % % * +, # ' / , * + ' # , !' ' ' * 0:91, 091+

2.2.3 Practical Graph Isomorphism FACT

$ ( - # (

* - J 06781+2 * 6

> 06>812 4 048G1+2 * ; % 0; 8=1+2 * 6 06=D1+2 D # * 0 931, 4 04 931, 0 331+2 G # ' * ; % 0; 9 1+2 8 # & * , , 0 9 1+2 9 # * 0991+ REMARKS

%( -Æ > ; * "

! + 0=81

%(

(, A F 7 (, 2 (

%( ( # : 7 'L; ' %( # ( #

%( %

0 91, # ! / (

© 2004 by CRC Press LLC

Section 2.2

73

The Graph Isomorphism Problem

% % 2 % % *' , # ' + , #

9

%(

(

" ½ ) 8GD, ( "&) * !, # " ) G888 > , Æ , ! , %

% #

(

2.2.4 Group-Theoretic Approach * 0>;G91, 0>8=1, 08=1, 04 6; 931, 0; 9 1, 0; 91+ '$ , DEFINITIONS

( * + ! FACTS

'$ 4 Æ 08=1 0; 9 1 #

, * + # , % ; % 0; 9 1 # Q ; %E / , #

!

$

( * 08=1+ ' , ! ' * + , # * + , '$ / ! * +, H

* &

;

$(

!*+

* & !

!'

*; %09 1+ . * +

, # #

!

4 *+, * < 3+ # * < 3, ! * + !' + $( ' < 3 !* + ) !* + ! * + ) ! *+

© 2004 by CRC Press LLC

74

Chapter 2 GRAPH REPRESENTATION

> < , 3 ! * ·½+ ! * + ! * ·½ + ! * + 9

$

( ! % 9 , , ! * ·½ + !'

$(

! ! * + '

9 *! * ·½ + ! * ·½ +

$(

06;<>981

*¿ +

'$

. , # ' / , ' , REMARKS

%( % (

4 4 ; %E /

4 G %#

%

( <@:E9 *0; 91+

*' *½¾·´½µ +

2.2.5 Complexity # %# ? , . # ; E 0;8D1 1 ) 41 , 41 ) 41 1 # ' '$

* 0K8=1, 0FM=1+ DEFINITION

( / '$ FACTS

'$

* 0678, =, 89, ::89, 8=, 6<9 , ; 91+

$(

# ( 2 2 * # , +2 # # 2

© 2004 by CRC Press LLC

Section 2.2

75

The Graph Isomorphism Problem

D # * , !*+, # ,*+ ) +2 G *

' +, !*++2

8 * 9 2

!*++2

= 2 3 / * # # , , , , # / +

$( 41 * % # , ( ½ ¾ + $ ( *F %% 0F3 1+ ' %'$, # % REMARKS

%( @

'$ ' ' %'$, ? , $ %# ) 4 = '$

%( . # # % , ; < % *0<8 1, 0<8G1+,

References 06781 I , K - 6

, K A 7, , >, =8 08=1 ; , B: C, , =8= 0- <931 ; , - M, < <%, , $ %# # = *=93+, G 9JGD 0 9 1 ; , A R , A , . # , # +, # $ # # *=9 +, 3J 0; 91 ; - ; %, : , # +- # $ # # *=9+, 8J9 0=81 ; , . I : , F , A I %, : : .. . > ; , 7=83, K =8, 7 M M

© 2004 by CRC Press LLC

76

Chapter 2 GRAPH REPRESENTATION

0=1 A , / / ,

.

D *==+, GJGG 0991 6 ; , ' , %# *==3+, GJG 0891 F < , . , ! / , $ %# # 8 *=89+, 8J 8= 0 DD1 6 %, A $ ! , # # $ # # 89 *=DD+, GJ9 0::891 K : : K : , , " &' 3 *=89+, DJ = 0:91 K : F < , ; , , , $ %# # 3 *=9+, 3J D 0:831 A : : : , Æ , %# $ 8 *=83+, DJG 0:91 A : , , %# D *=9+, DJG 0AA;881 ? A, K A , - ; , # , ( 8 *=88+, GJ3 0A3 1 A ; < E %, ', " & &' * S;... * 33 +, ?# R % < , 9JDG 04 931 . < 4 K ? , !' , # +/ # $ # # 8 *=93+, GJ 048G1 4, ; , J # 0 # . *< , -+, - 7 , =8G 04M< < 91 4M , > < , - < % , ? , # +- # $ # # *=9+, GJ83 04 6; 931 4 , K 6

, - ; %, ' , < : : , , : *=93+, & *=93+, GJ 04 6; 931 4 , K 6

, - ; %, , # /+ !!! # 1 # # *=93+, GJ

© 2004 by CRC Press LLC

Section 2.2

77

The Graph Isomorphism Problem

06;<>981 T , : 6$, - ; %, : <

, > , *¿ + *¿+ , %# $ *=98+, DJD 0K8=1 A < K,

& , 4 , =8=

"

091 , , $ # G *=9+, =J G 0 331 , . ! , # 0/ $ # # * 333+, =J G 0 R=91 K ; K R, ===

" , :: ,

069 1 4 6 , , <, / % , %# $ # *=9 +, DJ 0681 K - 6

: F >, ; , # 2 # $ # # *=8+, 8 J9 069 1 : 6$, . , . & G, < , =9 06=D1 > 6 , *?+

, $ %# # *==D+, J= 0F=D1 F , : M % , M % , , : : . : *: + 7=D3, A" =D, 7 M M 0F3 1 F A %%, ' " , $ %# * 33 +, D3JD G 0FM=1 K FM , 7 < M , K U, , %M , ==

"

0F =1 A F ; 6 , @ ' , $ # 8 *==+, D8J 8 0F 3 1 . F %, ;, A , 7 , $ # DG * 33 +, 9=J= 0;8D1 - ; , @ , # $ # *=8D+, DDJ8

%# #

0;==1 K ; #, , : 3 * D DJG+ , - , ==

© 2004 by CRC Press LLC

78

Chapter 2 GRAPH REPRESENTATION

0; 91 ; #, < ? , $ %# 3 *=9+, J 0; 8=1 < ; % F < , , %# $ G *=8=+, 9J=D 0; 9 1 - ; %, . , %# # # D *=9 +, JGD 09 1 !, # ' % , 3 # %# $ # 4 5/6 *=9 +, DJ8 08=1 , , .

9 *=8=+, J 0 91 A F, , & 3 *=9+, DJ98 0<8 1 ; < % , / ' # / / ' , # +0 !!! ' *=8 +, DJ = 0 891, , @ / ( , # +7

# $ # *=89+, DJD9 0 8=1, , , %, %# # *=8=+, 9J 0 931, , . , # +/ # $ # 8 *=93+, DJ D 0=31, U , A # , 88JD $ * K , -+, > , ==3 0<8G1 ; < % , , *=8G+, J 0>G91, > ;, , & 5 # /6 = *=G9+, JG 0>8=1, > , *+ @ ! , . & $ DD9, < , =8G 078G1 K 7, , %# $ *=8G+, J

© 2004 by CRC Press LLC

Section 2.3

2.3

79

The Reconstruction Problem

THE RECONSTRUCTION PROBLEM

# :& < : A % E F E D ;U"E 2 ?> E ; G A 8 . A %

Introduction . ! # " *=88+, & " :& B

C D , # # > # %# ,

2.3.1 Two Reconstruction Conjectures < # ( # , ¼ / V . # % # %#, , Æ > %#, , %#

¼

Decks and Edge-Decks DEFINITIONS

( ; 4 ' , 4 , ,

( , *+, ' , *+, % . ' , *+ %

© 2004 by CRC Press LLC

80

Chapter 2 GRAPH REPRESENTATION

, % % , ,

. , . # , # # . A!

, &

(%

%

EXAMPLE

(

4 # ' %

Figure 2.3.1

# 1

Reconstructibility

# 06881, 0?891 %#

, # < , ' 0-991, 0991, 0;981, 0=1 0;< 31 DEFINITIONS

( 0 # % .

( # % , CONJECTURES

< 0 *0 + ) *+ @ # > %

, %!#(! .!( 0FD8, 7G31(

0

* # + %

, &%!#(! .!( 06G1( %

* #

. , % *+ , % # % / / , , # # ! # %

© 2004 by CRC Press LLC

Section 2.3

The Reconstruction Problem

81

EXAMPLES

( .¾ , .½ # ,

:& , .¾ .½

( ) .¿ .½ , .½ ¿ * +.½ *+ , .¾ 0 ) 1¿ .½ *# 1¿ +, *+ ) *0 +, 0

- :& , Relationship between Reconstruction and Edge-Reconstruction

. % %2 %, % ' ., 4 :& # , - :& 4 # FACTS

$

( * #E + 0 81 ; # % , , *+ / *+ , ,

$

( *6 E + 06G=1 .¿ REMARKS

%(

> , # ,

%(

- # , %# ' , / > / Reconstruction and Graph Symmetries

Æ % % < , , ' , , / , # DEFINITION

( ; !' , # # , #

© 2004 by CRC Press LLC

82

Chapter 2 GRAPH REPRESENTATION

. # , $

, #

FACTS

$

(

.

·½

,

,

½

,

$

(

0F8, M 8G, =31 4 !'

,

$

(

,

099, =31 .

¿

,

/

%

REMARK

%

(

! ¾ ! ! ! !

4 D ;

%

!

,

¿

!

! !

# #

!

/

# # #

<

, #

#

,

'

/ , #

#

%

!

!

&

&

#

2.3.2 Reconstructible Parameters and Classes Reconstructible Parameters DEFINITION

(

$ %

,

;

0

#

)

, * +

#

0

FACTS

,-

,

4

$

(

* 0;< 31+

$

(

/

*< 0=,

;< 31+

$

(

%

,

* 0;< 31+

© 2004 by CRC Press LLC

'

Section 2.3

The Reconstruction Problem

83

$

( 4 # ,

, #E

$

( % , # #

$(

*FE ;+ 0FD81 ; 0 # *+ < , 0 *+ 0 0 , Reconstructible Classes

> # B C, % % , . ! ! % , , , # ' ! # DEFINITIONS

( " * "+ , * + -/ , " * "+ *+ * *++ #

(

* * + -/ , #% * #% + , # ' , / % * %+

+

FACTS

# # # * #

, #E +

$( $( $( $(

* 0;< 31+ A * 0;< 31+ 0FD81

0G=1 < * , # + #

$(

0R991 &

$(

0 881 # #

$( $ (

04 89, 4 ;9, ;91 ' 0 8G1 @

© 2004 by CRC Press LLC

84

Chapter 2 GRAPH REPRESENTATION

$ ( 0 91 .

E , . , ,

$( 0R 9 1 . ' E ,

#

$( 04=1 # $( 0T=9, T=91

3 H =*H+ *+ =*H+

$( 04> >31 < # % * , #

. +

$( 0:81 .

$( 00-S 991 :# $( 0-6981 REMARKS

%( .

?> E ;, # # # %( # #

* , + ' # * , , + 6# , , , , ' Æ % 0< 91

%( 06881 ?

%( !

, / , # D ,

2.3.3 Reconstructing from a Partial Deck % % . , # * + 06GG1, * ' + 0G=1

© 2004 by CRC Press LLC

Section 2.3

85

The Reconstruction Problem

Endvertex-Reconstruction DEFINITION

( # ' / ' %

FACTS

$( $ (

06GG1 '

0 81 4 , ' # '

D # ' Reconstruction Numbers

% , 6 069D1 ! DEFINITIONS

( , *+, % / , *+, !

( ; , *+, % , # , / , *+, ! FACTS

4 #

/ 5 / # 5

$ ( $(

099, =31 /

0=31 # . ,

/ 5

$(

$( $(

0;3 1 . 5 .

06981 .

, *+ /

0=31

© 2004 by CRC Press LLC

86

Chapter 2 GRAPH REPRESENTATION

$( 06;981 . ' '

*+

' # / "

$( /

* 0;< 31+

. , , *+ ., , .¿ .½ ¿, *+ .

, 5 $( 0=D1 ;

$( 0=1 -

$ # *$ +

CONJECTURE

6 ; 06;991( .

, *$ +

FURTHER REMARKS

%(

, ' # *+ *+ . ,

% ( > % ,

, , # # ¿ , 0991 # %#

% ( 0991 ,

% # ) / < ! # ) ) / @ , # Æ % *+ Set Reconstruction

6 06G1 # , < : & CONJECTURE

# + % . # # . # , #

' %!#(! .!((

%, %# # %

© 2004 by CRC Press LLC

Section 2.3

The Reconstruction Problem

87

DEFINITION

( FACTS

$ (

08G1

$ (

08G1 4 # ' , /

$

( 08G1 / #

$( $( $(

08G1 08G1 ?

< * , # + #

$( $( $(

0831 0 8G1 @

0 :81 7 * , + Set Edge-Reconstructibility

, , % >

# > # FACTS

$( $ (

08G1 /

0A43 1 / # / / # #

' , / /

$

( 0A I=G1 . # #

, Reconstruction from the Characteristic Polynomial Deck

< #% 0< 8=1 %, # # 6 #% # / % , # #

© 2004 by CRC Press LLC

88

Chapter 2 GRAPH REPRESENTATION

FACTS

$

(

0< 8=1

%

$

(

. %

#

,

% * 0;< 31+

$

(

0:;=91

%

$

(

0< 1 .

;

%

Reconstructing from -Vertex-Deleted Subgraphs

'

>

< D # #

6 #

FACT

$

(

09=1 ;

/ Æ

' * + , #

. ,

* +

2.3.4 Tutte’s and Kocay’s Results . FE ;

, #

0 8=1 # # FE ; '

, # / . 0F91, F E #

F E

' 0=1 0;< 31

Kocay’s Parameter DEFINITION

(

;

+

* +

)

) *

½ ¾

+ / * $

* + (

© 2004 by CRC Press LLC

½ ¾

/

* +

) *

+

)

*

+

Section 2.3

89

The Reconstruction Problem

FACTS 0 8=1, 0F91 * 0=1, 0;< 3 1+

$(

; ) *½ + / * ( *+ ; >* + ! * +

#

# % * + ) * + * +

>

# !

$( $( $ ( $ (

6 # !

The Characteristic and the Chromatic Polynomials DEFINITION

( #

4

, * +

* +

FACTS

$(

0<G1 * 0 =1, =+ ;

8

5

8

8

5

5

Æ

)

*+

5

# '

$(

0> 1 * 0 =1, 88+ ;

- * 5 - * 5 5 - * -

Æ

- )

*+

# '

© 2004 by CRC Press LLC

90

Chapter 2 GRAPH REPRESENTATION

$(

0 8=, F91 *< 0=, ;< 3 1+

$(

0 8=, F91 *< 0=, ;< 3 1+

2.3.5

Lovász’s Method; Nash-Williams’s Lemma

. /

;U " 0;8 1 # , , % ' 7 ;U", M 0M881 . / * ;U "+, ?> 0?891 # ;U"E M E # ' ' 0-991 0=1 0;< 31 The Nash-Williams Lemma DEFINITIONS

0

0

( 4 ,

! & ' ,

0

0 00 1

(

0

, #

* + , &

*+, & '

0 0

0 1

0

0 #

REMARK

%(

0

0 0

? 0 1 0 1 , 0 1 , 6# , 0 1 ,

0

0

0

0

FACTS

$(

0;8 1 ;

0 *+ 0

0 1 )

© 2004 by CRC Press LLC

* + 0 1

0

Section 2.3

$(

91

The Reconstruction Problem

0?891 ; 0 0 % 00 1 ) *+ 5 * + *00 1 01 +

)%/ -

*+,

*< 0=1, 0;< 31+

$(

0 ,

*: ?> E ;+ *< 0=1, 0;< 31+ < ?> E ;, ,

* + * +

$(

0

0 < 32 , 01 < 32 1 ; *+ < ¾ ; , 0 1

0;8

$ (

0M881 ;

,

< W

$ (

0=31 6 # Æ % Structures Other Than Graphs

, # ! * X + # ! , X , , # , Y E X / #

/ , X

*

. , # , ' , X

# ,

DEFINITION

( . ,

;

FACTS

> # # ! , # ? > ; , % # / ' ' 0=, ;< 31

$(

0

0;< 31 ; # , # ' * +

0

< 0; $( 0:F 9=1 ; * X + © 2004 by CRC Press LLC

<

0

<

X

92

Chapter 2 GRAPH REPRESENTATION

$ < 0< =91 ? , +, ",

$(

0< =91 <

$(

0< =91 4 ,

$(

#

$(

?

?

0< =91 4 , ? =@*+ , * +

@

0 ==1 ! + * , # ! ! " +

$(

03 1 : # +¾ =3 ! +¾ / D , 2 ,

, D 0:F 9=1 ' The Reconstruction Index of Groups

;% * X + X DEFINITION

( :*X + X # , * X + FACTS

$(

- :& ( A ) , , A , A , :* + )

$

( 0=91 ' ' D 4 ' , 0:=G, =G, 98, = , =D1

2.3.6 Digraphs & # , #

© 2004 by CRC Press LLC

Section 2.3

93

The Reconstruction Problem

, '

DEFINITION

( 0=81 4 / * * + * ++, FACTS

$ (

0<88, F9D1 ' !

$

( 06G81 ! *< 06881+ < 0A =3, =G, I ==1 CONJECTURE

, 4 &%!#(! .!( ! # * +(

4

* #

%

2.3.7 Illegitimate Decks DEFINITIONS

(

% ( #

% FACTS

$(

09 1 A # %

$

( 06<9 1 / % # ' % 0FM < =, F 6=1

References 0:F 9=1 ? , R : , . F %, R , : , %# # 5 # (6 8 *=9=+, DJG

© 2004 by CRC Press LLC

94

Chapter 2 GRAPH REPRESENTATION

0A I=G1 ; A , < A , A I , @ & , %# # $ # # # 3 *==G+, J= 0 :81 - & A : , 7 6 E :& , # $ # (# 8 *=8+, D=JD=G 0;3 1 F K % K ; , @ # , # G * 33 +, 8J9 06981 A > , - % %, ; 6 6, : , %# " *=98+, J 3 0 =1 ? ; , " , : 7 , == 0=31 U , , %# " *==3+, J 0G=1 K , @ FE # # # # GD *=G=+, 98J=8 0G=1 K , @ 7E & , 8 %# $ # *=G=+, 9J 99 0=1 K , E , J D A F# *-+, 9 , : 7 , == 06881 K ; 6 , N , %# " *=88+, 8J G9 0 81 , @ & , %# # *=8+, =J 0:=G1 K : , < # &# *==G+, J 0:81 T : , #

, 8J8 : , K 4 F % *-+, : " , < I , =8 0:;=91 A :% U ; U , <% ' & , (# # # $ # & # # $ # *==9+, =J33 0A43 1 : A , @ 4 , A , @ / , $ # D= * 33 +, =J33 0A =31 A : A : , @ , %# # # # D *==3+, 3J 0-991 ? - , , # & # G *=99+, J 3

© 2004 by CRC Press LLC

Section 2.3

The Reconstruction Problem

95

0-S 991 ? - , ; R S ' , :#

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

= *=9D+, DJD 06<9 1 4 6 , , <, / % , %# $ # *+ *=9 +, DJ 06G=1 ; 6 , @ , # # $ # # *=G=+, 9DJ98

© 2004 by CRC Press LLC

3

96

Chapter 2 GRAPH REPRESENTATION

0FD81 K F, , 8 %# $ # 8 *=D8+, =GJ=G9 0FM< =1 K FM , 7 < M , K U, "

, %M , == 0F91 > ; F , @ , *=9+, 3J F9D1 > ; F , @ < % E , %# " = *=9D+, 8J8G 0F81 A F , ? , $ # & # ): = *=8+, DDJG3 0F ;3 1 . F %, ;, A , 7 , $ # DG * 33 +, 9=J= 0F 6=1 A F ; 6 , @ ' , $ # 8*+ *==+, D8J 8 0;91 K ; , ' ..( , %# # 5 # (6 3 *=9+, =GJ 0;981 K ; , N # / # , 5 (6 *=98+, DJG 0;< 31 K ; < , " :

, : 7 , 33 0;8 1 ; ;U", %# # 5 # (6 *=8 +, 3=J3 09 1 !, # ' % , 3 # %# $ # 4 5/6 *=9 +, DJ8 0831 , , # %# $ #

*=83+, DDJG3

08G1 , @ , %# # 5 # (6 *=8G+, DGJGD 0991 , ( , # & # G *=99+, 88J98 0=G1 , 4 4 : , A , 7 - , ? # , ==G 0 881 A F, : %# " *=88+, 9J 9 0981 I % , , $ # & *=98+, =8DJ=93

© 2004 by CRC Press LLC

Section 2.3

The Reconstruction Problem

97

0= 1 I % , , # # $ # = *== +, 9J8 0=D1 I % , % , %# # # ; # 3 *==D+, GJ 8 0=91 I % , , # # $ # D *==9+, =JG 0=1 , , < ' # %# "

* + *==+, 8J3 0=D1 , , %# "

=*+ *==D+, 8DJ9 0M8G1 I M , , # $ # )9# 8 *=8G+, 83=J8= 0M881 I M , # ¾ 6 , %# # 5 # (6 *=88+, 9J 9 0991 > K , 9 : , A , 7 > , =99 09=1 > K , , 9 *=9=+, J 8 0=31 > K , # ! , %# " *==3+, =JGG 0-6981 > K , ? - A 6$, , %# " *+ *=98+, 9J3 0?891 : < K ?> , , : 9 ; > % K > *-+, " , , ;, =89 0=31 ; , , %# " *==3+, 8J8= 0< =91 K $ A < , ? , %# # 5 # 6 9* + *==9+, G=J98 0< ==1 K $ A < , , 3 ! %# # *===+, 8 0=81 < , 4 " & &#*# *==8+, J = 03 1 A , @ 6 , %# # 5 # 6 == * 33 +, J=

© 2004 by CRC Press LLC

98

Chapter 2 GRAPH REPRESENTATION

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

© 2004 by CRC Press LLC

Section 2.4

2.4

99

Recursively Constructed Graphs

RECURSIVELY CONSTRUCTED GRAPHS

! ! " " " < " 4 : -/ : "

Introduction A! , &

, # ** &+

* &

DEFINITIONS

( ! * ! + , - ! # # !

( -

# #

REMARK

%(

, , & < 3

2.4.1 Some Parameterized Families of Graph Classes Trees DEFINITION

( # ' * + # * + ; * + # *½ ½+ ! *¾ ¾+ % & ½ ¾ *½ ¾+ # ) ½

© 2004 by CRC Press LLC

100

Chapter 2 GRAPH REPRESENTATION

, * + A!

6# , ! ½ * +

EXAMPLE

(

4

Figure 2.4.1

%(#- !#(! !

Series-Parallel Graphs

4 , . 0A GD1 4 2 $ ! # , , #

Figure 2.4.2

2!&##& """ ##& """ #

4# , # ! DEFINITION

( # * + ! #( " * ) )

+ * + #

" * + # * + # #

" * + # * + # # #

" * + # * + # 2 #

( & %% ! # # , , !

EXAMPLE

(

! 4 ! 2 #

© 2004 by CRC Press LLC

Section 2.4

101

Recursively Constructed Graphs

Figure 2.4.3

.!!#! ! !# ! ##& """ #

-Trees and Partial -Trees DEFINITIONS

( ' , . , # 5 * + ' & . ,

(

. , .

(

*

+

( / " ' / FACTS

$( $( $( $(

, < .

, , * + EXAMPLES

(

# 4 , #

© 2004 by CRC Press LLC

102

Chapter 2 GRAPH REPRESENTATION

Figure 2.4.4

.!#(! ! & " &

A 4 B

C # .¿ ! ' 3 , # * + ' @ 4 ,

(

4 D # 2 * + # ! & ! -'

Figure 2.4.5

##& """ &

Halin Graphs DEFINITION

( & ) $ % , # $ # ' % $ FACTS

$( $(

6

6 % , , 6 6 EXAMPLES

( 6 4 G #, # # 2

© 2004 by CRC Press LLC

Section 2.4

103

Recursively Constructed Graphs

Figure 2.4.6

3 "

(

4 8 # 2 ' 6 2 !

Figure 2.4.7

! 3 "

Bandwidth- Graphs DEFINITION

(

* + ' ' ! $ *!+ *+ *# =+

(

EXAMPLE

(

# # 4 92 #

Figure 2.4.8

© 2004 by CRC Press LLC

) *& *& #

104

Chapter 2 GRAPH REPRESENTATION

Treewidth- Graphs

# % < *, 0<9G1, 0<9G1, 0<=1+ !

# , # % E # % , , > E

& , DEFINITIONS

(

"

" "

( (

) * + *

) *

$ +, # $ # '

* &+ # * & $ ,

(

'

# %

#

REMARK

%(

, , , ! ' * + @ , # , , , # * , # # +

EXAMPLE

(

# 4 = 4 , , ' ( ½ ) ½ ) ) ) ) ) # ' 4 =, * + , # 2 ,

$

Figure 2.4.9

© 2004 by CRC Press LLC

$

# " &!!#!

Section 2.4

105

Recursively Constructed Graphs

Pathwidth- Graphs DEFINITIONS

(

#

/ '

( (

,

½

¾

' ½

(

, !

# %

#

EXAMPLE

(

# 4 3 ' !

#

$

Figure 2.4.10

&!!#!

Branchwidth- Graphs DEFINITIONS

$ +, # $ &

( ) * + * # ' '

(

$

. '

$

$

, *

$ +

( ; * + ) * + ' ½ ¾

$ , # * ½ + * ¾ +

$

( $

$

*$ + '

© 2004 by CRC Press LLC

$

106

Chapter 2 GRAPH REPRESENTATION

(

(

# %

#

FACTS

$

# 3

$

( 0<=1

( 0<=1 # ' # /

$

( 0<=1

#

#

EXAMPLE

(

# # 4 * +2

Figure 2.4.11

) *& # &!!#!

-Terminal Graphs DEFINITIONS

,

% *# ++ ! # , #

( ) * $ + ' $ ) ½ ¾ , # $ ( ! + ) ½ ¾ # #

EXAMPLE

(

# 4 I # 2

© 2004 by CRC Press LLC

Section 2.4

107

Recursively Constructed Graphs

%(#- !#(! ! & "

Figure 2.4.12 REMARKS

%

(

,

$

)

#

.¾

*,

%

(

½

%+ *

+ #

% # + *

+, #

4

½

(

(

#

*

+ )

¾ *

+

$

$ ½

,

) *

$

) *

+,

+ #

¾

)

* + (

* +

$ $ $

(

½ ½ ¾ ¾ ½ ¾ ½

"

½ ¾ ½

"

$

'

"

+

& ' ;

"

*

+

,

#

,

)

*

+ )

¾ *

+

$ ½ $½ ¾ $¾

(

% +*½ ¾ + & *½ + *¾ +

*

.

' ,

*

+ #

½ *

+ )

Cographs DEFINITION

(

!

#(

"

# '

"

.

"

.

½ ½

¾ ¾

, &

,

%

½

¾

¾

½

¾

, #

*

½ ¾

+ #

½

:

© 2004 by CRC Press LLC

½ ¾ ½ ¾

108

Chapter 2 GRAPH REPRESENTATION

EXAMPLE

(

4 ! *+ #

Figure 2.4.13

.! !#(!

FACTS

$ ( $(

0:; 91 0:; 91

Cliquewidth- Graphs

. + # 0:-=1

% DEFINITION

( ; 0%1 ! #(

#

# *+ ) *+ 01 / # % . ½ ¾ / # 01, *+ & ½ ¾ / # * + *½+ / # , # *½+ ½ *½ ¾+ *½ + ) *¾ + ) *+ *½+ / # , # *½+ ½ # #

" "

REMARK

%(

A! 9 ! / # / # / # / #

,

, ' , , ,

. , . . %, / ! ( # &

© 2004 by CRC Press LLC

Section 2.4

109

Recursively Constructed Graphs

- # , # #

< , / / # ,

/ # # / . + * 0:@331+ EXAMPLE

(

/ # 4 -' ,

! *+ #

Figure 2.4.14

"4(*& !#(!

-NLC Graphs DEFINITION

( ; 0%1 # 0%1 0%1 '( $ % ! #(

# *+ ) *+ 01 %?;: . ½ ¾ ?;: 01, & ½ ¾ ?;: , # ½ ¾ ½ ¾ *½ ¾+ # ½ ½ , *½ + ) 2 ¾ ¾ , *¾ + ) * + *½ + ?;: , # ½ # #

" "

"

EXAMPLE

(

4 ?;: . 4 D,

, , , , , , # # % ) # ! # , * &+ # * + !

© 2004 by CRC Press LLC

110

Chapter 2 GRAPH REPRESENTATION

Figure 2.4.15

&25. !#(!

-HB Graphs A B ,C DEFINITION

( & $ % # *¾ + #

/ # * 5 + REMARKS

%(

# , ,

%(

/ # ?;: , ' '

%

( ' * +

%

( / , # / # $ 6 4 , 0K< 3 1

% (

6# , 6 ! @ ,

/ # , / # 6

2.4.2 Equivalences and Characterizations Relationships between Recursive Classes

/ ! < # 7 !

© 2004 by CRC Press LLC

Section 2.4

111

Recursively Constructed Graphs

, 4 = * + 0 ;< ==1 FACTS

$( $( $(

# - # # #

! * 5 +

* 0> 69G1, 0> 981+

$( $( $(

# # & %%

< #

$( $ (

< #

6 2 !

$ ( $( $( $( $( $( $( $( $(

0:-=1 : / # 0:@331 - # / # * ½ 5 + 0<=1 - # # ; 0<=1 - # #

5

0>=1 : ' ?;: 0>=1 - # * ·½

+?;:

0>=1 - / # ?;:

0>=1 - ?;: / # 0K< 3 1 - /

# 6

Characterizations

< " DEFINITIONS

( ,

) * +

, # * + )

* + )

( # # ' *B C+ ! ,

# ' # & , #

( 0 ! / '

© 2004 by CRC Press LLC

112

Chapter 2 GRAPH REPRESENTATION

REMARKS

%( ! & * + F > # ( < # 0 , 0 , ' ! 0½ 0 0 , 0 %( < *0<991+ ! > E & # , " 7 , , , ' 2 # %# " %( , < " !

%( %# * 4 #+,

' %#

FACTS

$ ( 0:; 91 : 1

$ ( .

$(

,

.

$( . . $( ( # 4 G

.

Figure 2.4.16 $! !# ! " &#

2.4.3 Recognition . , Æ , # "

© 2004 by CRC Press LLC

Section 2.4

113

Recursively Constructed Graphs

REMARKS

%( <

4 ' , 6 " !

* , +, ! , * 0:? 91+

%( ", ,

# * 0A GD1+( ' * + * + # * +2 2 ' , , 2 # , # # . # .

%( <

* 0 9G1+ # *0<=G1+

EXAMPLE

(

/ # 4 8

Figure 2.4.17 %(! ! !# ! " & Recognition of Recursive Classes FACTS

$( " $( < " $( # , # , # , # "

( 4 !'

*

+

4 G

$( 4 G " % !' (

4 8

$( > , 4 G * 0=1

$ ( >

) 2 0<=G1 #

) +

, # 4 G

$ ( # 0<=1

© 2004 by CRC Press LLC

114

Chapter 2 GRAPH REPRESENTATION

$( < " ! , " * 0<991+

( 4 # < 6# , ' '

$( 0> 981 - # ¼ ¼ 4 ' , ' , ¼

B C 4 ' , $( 0:<9D1 : "

$( ' " / # $( ' " ?;: $( 0K< 3 1 . 6 , ! # / # ,

# #

/ # *+ 6

/ # #

References 0 : 981 < , A : , % #% , : ' ! % , $ %# $ *=98+, 88J 9 0 : <=1 < , : , % #% , A <,

, %# $ *==+, JG 0 6 =1 < , < 6 , % #% , *+, -Æ % , $ # *==+ 0 9D1 < % #% , : " % , & *=9D+, G=J8D 0 9G1 < % #% , : " , $ %# $ *=9G+ 3DJ 0 :=31 < , % #% , A : , 4 " , $ # *==3+, J= 0@=91 ; < @ , @ # # 1 , $ # *==9+, J 0 81 ; > % - , " % , $ *=8+, JD

© 2004 by CRC Press LLC

Section 2.4

Recursively Constructed Graphs

115

0 <=1 A %, ? , A < , , Z % ' , %# # ( *==+, 8J 9 0991 , ( , A A , < . : < , . , =99 0=31 6 ; , : # # , # :# :)) =2//, A : < , 7 7 , ? , ==3 0=1 6 ; , # , *==+, J 0=G1 6 ; , ! # , $ %# *==G+, 3DJ8 0 6F=D1 6 ; , K , 6 6 , F%, ' # , # , , %# *==D+, 9J DD 0K< 3 1 , K ; K, I , K < , / # % ,

* 33 + 0F=G1 6 ; F%, -Æ # # , %# *==G+, D9J3 0M=1 6 ; 6 M , # #

, $ %# $ # *==+, 9J9G 0=1 , % , , :, A , $ %# $ # *==+, 9JD3 0 ;< ==1 M , I ;, K < , " 9, <. A

, <., *===+ 0:=31 : , .( " ! , # # *==3+, J8D 0:= 1 : , ...( , , ' , # > # *== +, D8J 9G 0:=D1 : , I...( @ , . *==D+, 3J 0:=G1 : , S( ; , *==G+, 98J 0:-=1 : , K - , " , 6 # , %# *==+, 9J 83 0:F 91 A : A F % %, 4 ! , & *=9+, 8J G

© 2004 by CRC Press LLC

116

Chapter 2 GRAPH REPRESENTATION

0:; 91 A : , 6 ; , ; < , : , $ # *=9+, GJ8 0:=1 : , , *==+, =J9 0:? 91 : &, A ?, > %, 6 , $ *=9+, 98J = 0:@331 : < @ , 7 / # , $ # * 333+, 88J 0:<91 A : , R , ; F <# , : ( , , & *=9+, =J D9 0:<9D1 A : , R , ; <# ,

, $ %# *=9D+, = GJ= 0A GD1 K A Æ, , %# $ # # # *=GD+, 3J9 0==1 : 7 , @ / # , . & GGD *===+, DJ8 0 <%=1 A A <% F , ?: " , $ %# $ *==+, JD 06R981 S 6 R R, # , *=98+, DJ9 0F.79D1 R F& , . " %, < 7, : " % , 8=J9 # ?=-, =9D 0F=1 F%, # ' , . & , < I , == 0FF =D1 F% A F , # , %# *==D+, GGJ 9 0=1 K [% , ! , %# *==+, J 0 =1 % #% , / ! , D=JG33 # $ @ " $ , < ==, : < , < , == 0= 1 , 4 ' # / %, J 9 # A , == 0=1 , # ( ?# : <

, 98JG 9 +AAB, , : 7 , ==8

© 2004 by CRC Press LLC

Section 2.4

Recursively Constructed Graphs

117

081 A K , @ " % , $ # *=8+, 8J 0<91 ? A < , . -' , %# # ( *=9+, =JG 0<91 ? A < , ... # , %# # ( *=9+, =JG 0<9G1 ? A < , .. # , %# *=9G+, 3=J 0<9G1 ? A < , I -' , %# # ( *=9G+, = J 0<9G 1 ? A < , I. A &

, %# # ( *=9G+, DJ9 0<991 ? A < , I.. A & , %# # ( *=99+, J D 0<991 ? A < , SS > E & ,

, =99 0<=3J1 ? A < , .I # # / , %# # ( *==3+, 8J D 0<=31 ? A < , .S A &

, %# # ( *==3+, 3J88 0<=3 1 ? A < , I... F #%

, %# # ( *==3+, DDJ 99 0<=1 ? A < , S @ , %# # ( *==+, DJ=3 0<=1 ? A < , SI. -' ,

, == 0<= 1 ? A < , SS.. .

% ,

, == 0<=1 ? A < , S. A , %# # ( *==+, 8 J3G 0<=D1 ? A < , S... & , %# # ( *==D+, GDJ3 0<=1 ? , A < , , Z % ' , %# # ( *==+, J9 0<=G1 A < , @ # , $ %# $ # *==G+, 3J8

© 2004 by CRC Press LLC

118

Chapter 2 GRAPH REPRESENTATION

0< =31 < ; , " , & ' *==3+, ==J 0< 991 < \ , > # V, # 1 $ # , % F/ *=99+ 0<=1 A < , : , *==+, 8J 0>=1 - >%, %?;: , $ # *==+, DJ GG 0> 981 I > , ; % , A A , A : < , : 7 , =98

© 2004 by CRC Press LLC

119

Chapter 2 Glossary

GLOSSARY FOR CHAPTER 2

) * +(

, ' 2 ' , %

&

"# J

J ) * +(

' , # 0 1 ) ' ' , 0 1 ) 3 #

-# !#(! ( J

%

( /

""& # #!#& # !"( #

"" !#(! ( J ( 1 J ( & '

*& ( # ' '

!

$

*!+ *+

(

( # # &!!#! J ) * +( *$ +, # $ # ' ' & $

, "( # ' $

*& ( # # * J ( # %

&6# # ( ! ', ' '

&6# ( ' !

! " ( "! (

/ *+ ) ½ ¾ , # ) *½ + 0 ) *!½ ! + $ & ! ( )

! " ( J ( '

.7( # # 6 ! #!!#(

½ ¾

, /*½+ ) /*¾+ $

½ ¾

/

#

! " ( " ## &!#(! ( J ( % # , # , /

© 2004 by CRC Press LLC

120

Chapter 2 GRAPH REPRESENTATION

( % # , #

, /

" ## !#(! ( J

" *& (

.½ ¿

"4(* J ( Æ

, # , & ,

( ! # *0%1 +( # *+ ) *+ 01 / # % . ½ ¾ / # 01, *+ & ½ ¾ / # * + *½+ / # , # *½+ ½ *½ ¾+ *½ + ) *¾ + ) *+ *½+ / # , # *½+ ½ # #

"4(*&

" "

! ( !

" # '

½ ¾ , & ½ ¾ . ½ ¾ ,

½ ¾ , # % ½ ¾ *½ ¾ + # ½ ½ ¾ ¾

" . "

!"! (!!# !" .7( !

, #

!"! " ## J (

( 3 ( % ' % * +2 , 3 ) 0%½ %1 '

!"! J

, - " J ( ' !"!&#- ( # %

%

J / ) *½ ¾ + * # $ +( / ) *½ ¾ + * + * + ) * + ) 2 * +

!- !

!## J ( & #

1 J

( *+ '

! - ( & #4( J

© 2004 by CRC Press LLC

( /

, #

121

Chapter 2 Glossary

-!

3

J

5 6

# &6# !# &6# # ) *

* + ) 0

) 0

%½ %

1(

4 %½ 4 % * +

* +

1

+( #

¾

( ! ! (

', %

& ',

'

&6#

(

!

&! !

J

#

) *

+(

# ' *B C+

,

# ' #

& , #

&1 &" #(

J

( *

J

+

(

2 % #

&&" #(

& !

J

J

&!8 " " ##

) *

(

( ,

&!#("

(

+(

,

#

&!#(" #

( ,

,

&%!#(! .!(

( &

&!#(! ( %

J

#

(

/

&!#(! ! 0 &!#(! !" ! #(( (

*

# % *

,

X(

&&!#(! !"

/ ,

X

" - -&1

+ J #

/ #

X

(

,

/ ,

( #

J

( ' #

J

© 2004 by CRC Press LLC

(

122

Chapter 2 GRAPH REPRESENTATION

-&!#(" (

/

' %

$"!&+ # "" "!( *

+ # '

' ,

!* J ( &

'

#!!# !" 9':( Æ

# #

&!( J (

# #

3 " ( # ,

# ,

&3)

( # 2 A! 3

½ , %

"" 1(

"" 1 !"( #

%

) * + ' , # 0 1 ) 3 # J ) * + ' , # ' 0 1 ) '

J

9':(

3

# #

# #

0 , 0 #!!# ! " " # 0 ( , ( 0 , , ,*+ #!! #( #

#!!# ! #" #( ' & &

#!!#&!" !"( / /" "!#( ! # ! ##

(

)

½

'$

/"0# "!( '

#

" " ( # L , #

, !

" & "!(

+

*

! J %(

(

* 5 +

! /

)

%

!("( # , '

!!!# * !

( &

© 2004 by CRC Press LLC

0

J , # * +

!

123

Chapter 2 Glossary

!

0

!

!

* + * + , * + * + # 0 1

0 0

!!!# J

0

0 ! 0

( ( , * + * + 0 1 (( * /

B , C+

!

0

!!! J '

4 *+

(

0

& .

&25. !&" "&!!"" ( ! # *01 , # 01 01+( " # *+ ) *+ 01 ?;: " . ½ ?;: 01, & ?;: , # * + # , * + ) 2 , *+ ) * + " * + ?;: , # # # !# 6 (! !( * # +

# , , ! ,

2&!#(" (

$

* * + * ++, , Æ / ! J * + ) * +( ' $ , # * + * +

$ $ $ " &( ( / * + * + * &!!#!( #

+

*& ( # * J

2

( # %

#

( # /

" ' /

" - ! ( ' '

!"! " 1( * +

%

!"! "& "!(

** 5 + +

) * + ! J ( # # , # !8 " " ## ! #( , ,

%

© 2004 by CRC Press LLC

124

Chapter 2 GRAPH REPRESENTATION

!#(" ( , # ,

!#(" (

#

%!#(! .!(( & #

!#(! J X( #

,

* X +

( % /

( 0 # %

!#(! ( J

#

!#(! !

(#-" !#( " ##( ! * ! + #, 2 ! # # !

6 ! !"! J #

( # # #

(" ##!(

, , F

##& """ # , * + J !

(

"

"

"

"

*½ ¾ + * + # ) ½ ) ¾ *½ ½ ½+ # *¾ ¾ ¾+ ½ # ¾ # ½ ¾

*½ ½ ½+ # *¾ ¾ ¾+ ½ # ¾ ½ # ¾ # ½ ½

*½ ½ ½+ # *¾ ¾ ¾+ ½ # ¾ 2 # ½ ½

# &!#(" J ( %

# !#(" J ( %

# # ) * +( # # "8 ! ! !"! 3( ! . 3

# " !"!( !

#((( * X + # ! , X

,

& " (#- ( ,

*< A! D +

© 2004 by CRC Press LLC

125

Chapter 2 Glossary

#- "!#( !

( ,

(

*

+

# , #

(#-" 6(

# '

& #

&

* !+(

.

2 ,

2 ,

' &

.

,

"( &!!#! J ,

"

)

" * "

* J

* &

+

) *

$

+(

,

*

$

+,

# '

* & $ #

,

( # %

2 #

*& ( # # -&" #( J ( '

2 (% #

&-&" #(

J

(

* 1" &!#(" J (

*½ *¾

#

J * J *

© 2004 by CRC Press LLC

#

* 1" !#(" J (

$ $

+( '

+( '

$

Chapter

3

DIRECTED GRAPHS 3.1

BASIC DIGRAPH MODELS AND PROPERTIES

3.2

DIRECTED ACYCLIC GRAPHS

3.3

TOURNAMENTS GLOSSARY

© 2004 by CRC Press LLC

Section 3.1

3.1

127

Basic Digraph Models and Properties

BASIC DIGRAPH MODELS AND PROPERTIES

Introduction ! "

"#

$ %&'(! $)*&&(! $+,( " - $ ),(

3.1.1 Terminology and Basic Facts

-

.! # " - !

" / 0 1 "

"

Reachability and Connectivity DEFINITIONS

2

3 ! ¼ 4

5 ¼ ½ ½ ¾

½

" ! / 0 5 ½ / 0 5 5

# 6 " "

2 2

- # 6 4

# 7 #!

2

" - # 6 - # 6 7" " / " # 60 %

© 2004 by CRC Press LLC

128

2 2

Chapter 3

DIRECTED GRAPHS

" # "

74" !

"

2 % # " 5 !

" " " " EXAMPLE

2 # ! ! ! 8 "- "- -

! # -

Figure 3.1.1

FACT

2

% -

4"

! 4"

Measures of Digraph Connectivity

+ # " " 9 9:

; # # 6 /0

/'90 DEFINITIONS

2 # "! / 0

© 2004 by CRC Press LLC

Section 3.1

129

Basic Digraph Models and Properties

2 5 / 0 " "- !

/ 0 -

2 -" - 5 / 0! /0! < " " / " -" 0

2 -" ! /0 <

+ ! "- - "

! " ! "

= >

"- - / 0 Directed Trees DEFINITIONS

2 2

#

" " !

! " " ! -

.

!

!

REMARKS

2 ! - 4

2

6

Tree-Growing in a Digraph

! # #! - # / 0! 3 ! ! # " " " /! -# 0! "

DEFINITION

2 # #

© 2004 by CRC Press LLC

130

Chapter 3

Algorithm 3.1.1:

DIRECTED GRAPHS

! " #$ %

2 " !2 # "-

3 < " +

, " 3 <

25 + % /0 /# 0 " +

" 25 ? "-

+

" ! # " 6 #"

#

EXAMPLE

2

#

@ " # +

" ! #

" / ,0 8 # # "- "

Figure 3.1.2

&& '&

FACTS

2 % # "

" " " "-

2 3 ! ! " REMARK

2

7 "

#

© 2004 by CRC Press LLC

Section 3.1

Basic Digraph Models and Properties

131

2 # - # " ! . @ # 7 - #! A

" /,0! A - A ! ! ! $ )&&(! $)*&&! 9(! $& ( Oriented Graphs DEFINITIONS

2 ! " # "! #

2 ! - ! # " "! " "

2 - EXAMPLE

2

. # ! ¾

Figure 3.1.3

(&) ¾ &) '&

8 ¾ " - 3 ! - Æ < # " B7 & & FACT

''* " $ &(

2 -

Adjacency Matrix of a Digraph DEFINITION

2

5 / 0! ! "

$ ( 5

© 2004 by CRC Press LLC

5 - 5

132

Chapter 3

DIRECTED GRAPHS

FACTS

2 #- 1 4 "! - 4

2 % # 1 " $ ( # 4 - # 6 EXAMPLE

2

1 9 "

'! " - # 6 4 ! # / 0-

5

, ,

Figure 3.1.4

,

,

, , ,

, , ,

+) ,

REMARK

2 #! #

! #

" / 0 # '9

3.1.2 A Sampler of Digraph Models 3 ! # #

Markov Chains and Markov Digraphs

6 " 6 # ! # -

1 /! $ :C(! $+&9(0 DEFINITIONS

2

4 " ! 5 , ! 5

5 , ! " 3

! -

/ 5 5

© 2004 by CRC Press LLC

5 5 0 5 / 5 5 0

Section 3.1

133

Basic Digraph Models and Properties

2

A

! / 5 5 0 5

2 5 / 0 # - # "- 5 ! - 5 ,!

2

6 " #

EXAMPLE

2

2 # D

# # 3 ! # D E #! D B

DC %

! # 5 5 , 9 C! 4 - 6 " 6 " 6 " # C

,

, :C , , ,

9

C

,

, , :C , , ,

Figure 3.1.5

, , , :C , ,

, , , , :C ,

9 C , , C , , C , C , C ,

$'& * , - ./

Equipment-Replacement Policy

+

< # -A EXAMPLE

2

F # D'!,,,! #

DC,, 1 " 6 # # ! A" .

D',, / 0 D&,, / 0 D,, / 0 D',, / 9 0 D,, / C 0

G D !,,, / - - 0 D!,,, / - - 0 D&!,,, / - - 0 DH!,,, / 9- - 0 D'!,,, / C- - 0

- &2 "!

'!

$ ' ' A

© 2004 by CRC Press LLC

134

Chapter 3

DIRECTED GRAPHS

# ! ! # " " # " ! # " # 6 ! " 5 #

? ? " ' # # " C # -#

D,,

Figure 3.1.6

0 & # & '&

A / - 0 " " ' 6 ! "

6 16 F , The Digraph of a Relation and the Transitive Closure

.

" /

0 DEFINITIONS

2 2

# A

# A # " ! #

E ! # " " / 0 # " !

# # ! ! / 0 # " "

2 #

" ! " " $ ! $

2

#

#

# A / 0 # 4 5 5 % / 0 #! 5 , % 74" ! " #

#

"

#

2

%

# " # #

© 2004 by CRC Press LLC

Section 3.1

Basic Digraph Models and Properties

135

! / 0! 5 "

-

! - & "

EXAMPLES

2

# 5 " " / 0 / 0 / "0 /" 0 /" 0

# " # :

Figure 3.1.7 " / &

2 -"

-# 6 # A - # 6 ! " "

! " 6 - " A

" # -

Constructing the Transitive Closure of a Digraph: Warshall’s Algorithm

% -" # " ½ ¾

Æ

! +

$+ '(! 4 ! &¼ &½ & ! &¼ 5 ! & ½ & ! 5 ! & " & & & ½ & ½

/ 0 / & ½ 0 #" & ½ ! " "

Figure 3.1.8 " / 0

& ½

© 2004 by CRC Press LLC

136

Chapter 3

Algorithm 3.1.2:

DIRECTED GRAPHS

1 &&* " / 2& 314

2 -" & # " ½ ¾ !2 " & 3 < &¼ 5 5 3 / 0 & ½ % 5 3 / 0 & ½ / 0 & ½ / 0 & Activity-Scheduling Networks

3

1! 6 & # 6 G 6 " " 6 6 #! " # 6 @ # 6

Figure 3.1.9

/) '&

Scheduling the Matches in a Round-Robin Tournament

! # 6

/

" 0 3 @ ! 1" " 4 # + "# ÜC'C DEFINITIONS

2 - #

2 " # " # #

© 2004 by CRC Press LLC

Section 3.1

2

137

Basic Digraph Models and Properties

5 / 0

# " @ ) <

ÜC

ÜC!

ÜC9

REMARK

2

- - -

! # "! < 6 " / C'0

ÜC'

- '

Flows in Networks # 6 6 # 6 7 !

1 ; # # 6 1

6 ! # 6

DEFINITIONS

2

!

5 / " " 0 # "-

! - ! " " 2

" 2

2

) ! " 2

! 5

( !

) A

¾

/0 5 ,

/ " 0 /

#

0 # " " 2 " !

" !

2

!

!

(!

# < !

# <

!

; #

- # 6 6 ; #

% 0 ; #

4 ; # /

/

2

0

! A %

; # # 6 / " 0 4

/

0

Software Testing and the Chinese Postman Problem ! # F ; # " # "

!

3 # ! #

6

DEFINITIONS

2

#

© 2004 by CRC Press LLC

# 6

138

Chapter 3

DIRECTED GRAPHS

2 / 0 # 6

2

)" -#

!

"

" A -#

- &

2 # F ; # ! # "! !

A 4 # " 6

< 4" I I ! #

-# 4 REMARKS

2 6 ! # < / 0 3 ! # 4

2 J ! ; # F ! #

2 7 ! # !

Ü9! " " I I

Ü9 Lexical Scanners

# ! ! < # & # + # - # 6 < # " "

" '

! , . " ! @ " ! # " A " ( ! " A 7

# 6 3 A ! " A

Figure 3.1.10

© 2004 by CRC Press LLC

# 5 6

Section 3.1

139

Basic Digraph Models and Properties

3.1.3 Binary Trees A

! 3 ! 3

!

B # #

Rooted Tree Terminology DEFINITIONS

2

3 !

2

! /! ,0

"

! 4

/#

4 0

2

3 "

2

0!

"

"

2

/

"

4

/

3! !

0

5

!

!

# "

A

2

# " ! - - " #

2

# " # !

2

' )

Figure 3.1.11

/0-

"

FACT

2

7"

© 2004 by CRC Press LLC

"

140

Chapter 3

DIRECTED GRAPHS

Binary Search

$ # A . A ! & - 6 !

-

# DEFINITIONS

2 /0 ! # "

6 ! 6 " 6 " ! 6 "

2 " "! "

@ EXAMPLE

2

- # 6 2 H & 9 H C 9, 9'

Figure 3.1.12

'& ' )# '&

Algorithm 3.1.3:

! )#7 #" 7

2 - 6 !2 " %/0 5 ! 8J%% 25 / 0 + / 5 8J%%0 / 5 %/00 3 %/0 25 * " /0 7 25 + " /0

- " # " ! 3

! # " # B! # -

- ,/ * 0 # "

© 2004 by CRC Press LLC

Section 3.1

141

Basic Digraph Models and Properties

6 ! ,/ 0

References $ )&&(

G ) ! ! 7 ! -+ ! &&& $ ),( K -K ) )! ! " ! -G

! % ! ,, $ %&'( ) % % 6! # 7 !

B

! % L8# * 6! &&' $ :C( 7

! $ ! I-B

! &:C

$)*&&( K % ) K *

! ! &&&

I!

$ &( B 7 ! # Æ ! ! %! % 9' /& &0! HMH $ :'( ! % % ! I-B

! &:' $ H9( ! ! I-B

! &HC $& ( B K ! ! 6L ! && $#&( N

8 # ! ! K + O ! && $+ '( +

! ! & ' % & /&'0! M $+,( +! 3 ) ! 7 ! ,,! I-B

! / 7 &&'0 $+&9( + % + ! . 2 ! 7 ! I! &&9

© 2004 by CRC Press LLC

142

3.2

Chapter 3

DIRECTED GRAPHS

DIRECTED ACYCLIC GRAPHS 7 ) I 9 .<

Introduction + !

! ) + ! 4

" " ! " " 6 )

"

3.2.1 Examples and Basic Facts DEFINITIONS

2 2 2 2 2

#$ " < " <

" " " " EXAMPLES

2 ( )

1

6 # P 6 . 1 " 6 Ù Ú 6 Ù Ú ! ! ! " 1 #

# )! # ! 1 2 " 6 # " "

© 2004 by CRC Press LLC

Section 3.2

Directed Acyclic Graphs

143

Figure 3.2.1 & /

2 $ $ / ! ! 0 / ! <! 0

3 4! # ! ! ! / Ü 0

Figure 3.2.2 )

2 $'

! # " 6 % & !

" 6 &

# ; # # & )Q

8 ! " # A ! # I

) ! /Ü 90

Figure 3.2.3 " &&

2 * ' # # 9 #

© 2004 by CRC Press LLC

144

Chapter 3

DIRECTED GRAPHS

# 3 ! # !

# = # #> = > " 6 !

Figure 3.2.4 && %'

2 = > ! #

" # #

!

! ! 6 # 6 # B #"! # )! # ! " # "

&

2 $ % " !

! ! # A

" # 6 = > L

# 3 " " A /! - - 0! )

&

FACTS

2 7" ) 6 2 7" ) 4 ! !

2 7" ) ) 2 " ) ) 2 ) " # 6 2 ) " !

1 !

< " /

Ü 9 A 0

2 ) C #

Figure 3.2.5

© 2004 by CRC Press LLC

Section 3.2

2 2

145

Directed Acyclic Graphs

)

/

! # 6 0 /

! # 6 0

2 2 2

) " ! " )

7" # - " ! !

8 ! " ½ ¾ # REMARKS

2 2

)! $B &9!

'( $ :'! ÜM (

!

3 !

# E Ü 3 !

! ! 4 #

3.2.2 Rooted Trees 3 & ! & )! F " " B #"! ) " ! $)*&&! Ü ( DEFINITIONS

2 2

#

# " !

! " " ! 4

3 # # 6! #

# # 3 ! # # - -! '! "

Figure 3.2.6

© 2004 by CRC Press LLC

"% %)

%

146

Chapter 3

DIRECTED GRAPHS

2

"

# # ! ! #

2 - #

"!

#

EXAMPLES

I" 7 B

2 ! #

!

# ! ! # : # A # " - - !

7 # 6 1 $ 3 # 6 ! A A # "

3 : # " / "0

@ + 6 # ! 4 @ ! # # " # 7 '! # # ! #

Figure 3.2.7

2

" 6 % / ## ' &/&

1 A A # H #

© 2004 by CRC Press LLC

Section 3.2

147

Directed Acyclic Graphs

Figure 3.2.8

FACTS

2 2

7" )

! " / 0 ,!

" DEFINITIONS FOR ROOTED TREES

2 " ! ! 4

2 2 2 2

" 3 / 0 !

G " 3 "

2 2 2 2

" !

" # , / 0 " - # " " - #

- -- # " " -

" " &

Figure 3.2.9

2& & ) 8# )9

2 # " 6 @

2 - #! " # " ! 6 @ #

© 2004 by CRC Press LLC

148

Chapter 3

DIRECTED GRAPHS

REMARKS

2 ! ! ! 6 A A ! # ! # ! /9'C0 "

/9'C 5 C'90!

2 , #

/ ! 0 B #"! ! 5 5 ! # !

! @

!

@ 3 ! " " E

Figure 3.2.10 : FACTS

2 -- - " " % 2 % -" --

?

- -

#

-- Spanning Directed Trees

" ! " 3 !

"! # B #"! ! #

# # # 4 # 4 #

" + E , $)*&&! ( 3 #

E '9 B # # 6 FACTS

2 3 & ! A

#

A 2 " " & "

&

© 2004 by CRC Press LLC

Section 3.2

Directed Acyclic Graphs

149

Functional Graphs

! " @ ! DEFINITION

2

# "

EXAMPLES

2

+ A . ! A & # " . # / 0 + / 0 5 A ! " . B! & /# 0

2 A

! " ! @ @ , & 3 #

Figure 3.2.11

" & '& 8 9

FACT

2 % & ! 3 & ! " -

3.2.3 DAGs and Posets " # ) 7" ) ! " ) " # ! $ &,! :M( DEFINITIONS

2

, - 2

! E

2

! ! 5 E

2

$ ! $ ! $

© 2004 by CRC Press LLC

150

Chapter 3

DIRECTED GRAPHS

2 ! / 5 / 0

!

2 2 2

$

7 7

/

! #

2

!

!

5

/ 5 / 0 # "

2 $

/ 5 / 0 # "

1 "

2 % / - # " 1 # # EXAMPLE

2

% 5 9 C H , , ' B / 5 / 0 #

Figure 3.2.12

2 '&) ;

FACTS

2 2

3 ! )

7" B )

/

#0

2

7" ) &

# / " &!

3 ) & / !

!

" ! # 6 " & & / 0 &

3.2.4 Topological Sort and Optimization 3 )! " # "

# J ! < "

! 6

© 2004 by CRC Press LLC

Section 3.2

151

Directed Acyclic Graphs

" ! < Æ DEFINITIONS

2 " " ½ ¾

#- - "

2 ! ! # / !

0

# $ H9! Ü'( FACTS

2 2

)

) A Algorithm 3.2.1:

"& &

2 & ! 2 & )E # 0 25 &E % 5

%& 5 1

" 0 - 25 " 0 < 3 " ! 2 & ) 0 25 0 8# 0 ) 0 # % 25 %?

,

REMARK

2 # ) ! # 1 #

! " ! #

$)*&&! : M :'( Optimization

! # -# ! # " " L )! "

-

$B%&C! ,( "

# "

# A

" A

3 ! A! " " 3 ! # " "

© 2004 by CRC Press LLC

152

Chapter 3

Algorithm 3.2.2:

DIRECTED GRAPHS

! ) 0 <

2 ) & # " ½ ¾ E # /0 " / 0 ! ! 2 " 3 < /½0 % 5 / 0 # / 0 ! % Algorithm 3.2.3:

! ) 0 7 <

2 ) & # " # /0 " / 0 !

! 2 "

3 < /0

0 25 & % 5 25 0 0 ) J / 0

# / 0 0 0 25 0

EXAMPLES

!

# # ) "

Figure 3.2.13

2

& / 20-

. $ ! # #

# 2 "! ! 6 # " ! 6 "! 6 ,

/ 0 6 6 ! 6

" B # 46 " Q 6

© 2004 by CRC Press LLC

Section 3.2

Directed Acyclic Graphs

153

! # /0 # " # 4

# %

/ 0 5 / " #0 3 < 2 /½0 5 /½ 0 5 ,! /8 2 ½ 5 0 J 2 / 0 5 / 0 ? /0 / 0 3 3 < 2

! /0 5 /0! J 2

/ 0 ! / 0 5 / 0 / 0?/ 0 ! # /0! ! / 0 A " A ! % $B%&C! &( ! 7 !

2 . $ " 3 6! 6

! " # # 1 ! # # # %

/ 0 5 / #0 3 < 2 /½ 0 5 ,! J 2 / 0 5 / 0 ? / 0 / 0 3 3 < 2

! /0 5 ,! J 2

/ 0 ! / 0 5 / 0 / 0?/ 0 ! # /0

2 $ + # # "

! # # Q 3 # 6! # / " !

0! / ! #

0 3 )! # 6 " / " 0! A

# % ¼

/ 0 5 / #0 3 < 2 /½ 0 5 ,! J 2 / 0 5 / 0 ? / 0 / 0

© 2004 by CRC Press LLC

154

Chapter 3

DIRECTED GRAPHS

3 3 < 2 / 0 5 ,! /0 5 5 ! J 2

/ 0 ! /0 5 /0 / 0?/ 0 ! # " / ¼0

2 + # # "! #

# Q " )! #

7 9 /

# # " #0 2 B # # " "Q 3 )! " ! / 0 5

3 < 2 /½0 5 ! /½ 5 0 J 2 / 0 5 /0 / 0 3 3 < 2 / 0 5 ! /0 5 , 5 ! J 2

/ 0 ! /0 5 /0 ? / 0 ! # " /0

2 %- + # # " # #

# # "Q

- "

- 3 9 ½ " 9 3 ! # #

! # # " # "

Figure 3.2.14 " , /&

, /&

© 2004 by CRC Press LLC

Section 3.2

155

Directed Acyclic Graphs

3 )! " !

/ 0 5 " 3 < 2 /½ 0 5 ,! /½ 5 0 J 2 / 0 5 /0 / 0 / 0 3 3 < 2 / 0 5 ,! /0 5 5 ! J 2

/ 0 ! /0 5 /0 / 0 / 0 ! # " /0

2 % - + # # " # # Q - 4 " # 6! # " ! # 6 ;

# ; # 3 9 ½ " ' 7 : ! !

/0 < , FACTS

2 M " < ) / "0 2 3 1 )!

2 ) " 1 6 /1

# 1

! #

1 '

60

References $ &,( N ! ! B K " "! &&, $)*&&( K % ) K *

! ! &&&

I!

$B &9( B ! ! I! &&9 / ! + ! &'&0 $B%&C( B

) % ! ( ) ! 7 ! ) #-B

! &&C $ :'( ! % % ! I-B

! &:' $ H9( ! ! I-B

! &HC

© 2004 by CRC Press LLC

156

3.3

Chapter 3

DIRECTED GRAPHS

TOURNAMENTS A 7 I ! ! " 4 9 " ! 6 ! C N! . ! ' : H G

Introduction

# - / 0! - ! ! 1 " !

! # 6 # " E !

K + F

$ 'H( &'H 3

; # 6! ; C 4 " " "

B #"! "

/ $B 8 'C(! $B ''(! $+:C(! $:&(! $H(! $R &(! $)&C(! $ )&'(! $&'(0 # 6 <

/ $ )&H(0 ! # " " " < ! 6 -K ) $ ),(

3.3.1 Basic Definitions and Examples

" " #

/ 0

DEFINITIONS

2 ! ! # " " / 0

2 " #

2 " / 0 " / 0 + / 0

© 2004 by CRC Press LLC

#"

Section 3.3

157

Tournaments

2 " " "

" " "

2 / 0 " " 3 /0 8 ! #

#

/0 / 0 " " 3 / 0 / / 00

2

& / 0 - 0! # " ! !

- /

2 "- # " " " " '

2 " &! ,/ 0!

" ! ! 2 / 0!

"

3 &! - "

! ( / 0 / ( / 0 & 0

FACTS

2

/ ¾ 0 @

-

!

! "

"

2

$ C9( / 0 - /

0 - "

" " " "!

/ 0

/ ¾ 0 / 0

/ ¾ 0 S 5 S

A # " / 0 "

/ 0

9

C

'

:

H

&

9

C'

9C'

'HH,

&C '

, &:

EXAMPLES

2

9

© 2004 by CRC Press LLC

,C'

158

Chapter 3

Figure 3.3.1

DIRECTED GRAPHS

"

2

) < - !

3 "! /

' / 0'

0 3 ! " " / $ 'H(0 -

- CC

! ' ! C Regular Tournaments DEFINITIONS

2 #

/! /0 5

" / 00 / 0 # /0 /0 5 ¾

2 #

" 1 " /! % ,/ 0 ,/0 5 %!

" 0

2 % 5 - ? # , %

-- , " ! ? 5 , ! - - ! # "

, & # "- /&0 5 - /&0 A 2 / 0 /&0 &

# #/ 0! #/ 0

2

% 5 / 0 A A # ! # ! / 90! % " !

" 4 /

* 0

#/ 0

& FACTS

2 2

#/ 0! # 5 !

-

$:&( 3 -

- !

/ 90 "! -

/96 ? 0 M /96 ? 90 /96 ? 90 B ?F F BB 5 /96 ? 903 B ? B 5 3! # 3 / B

6#-B 0 $:(

© 2004 by CRC Press LLC

Section 3.3

159

Tournaments

REMARK

2 4 ! 5 - ?

#

/ 0 6 ) !

EXAMPLES

2 &- #

" " 9! "! 5 ) 5 9 ' H

#

/ 0! #

0 1 8 2 7

3

6

5

4

Figure 3.3.2 " & &

#

/ 9 ' H9

2

:- # 4 -

# / 0! # 5 /:0 5 9 ." ! -

,/ 0 ,/0 5

"

4 :-

Figure 3.3.3

/ 99

#

Arc Reversals

- - 4 " FACTS

2 $ ' ( 3

# - # 4! 4 " -

© 2004 by CRC Press LLC

160

Chapter 3

DIRECTED GRAPHS

2

$: ( 3 # - % A # % ! 4 " %-

REMARK

2 $HH( # "- B A

- # " B #"! / $H9(0 CONJECTURE

T = =* 2+ $'9( : 7" - "

# "

3.3.2 Paths, Cycles, and Connectivity I - " #

" "

-K ) " $ )&'( 6 $ ),(

A # DEFINITIONS

2 / 0 &

" & / 0 &

" & /B 9C0

2 & / 0 " " &! EXAMPLE

2

# ! #/ 9 ' H0 " , '

C 9 H : ,

" "

!

REMARK

2 : 4 A " " FACTS

2 $T 9( 7" "! "

© 2004 by CRC Press LLC

Section 3.3

161

Tournaments

2

# 4" - 2 / 0 /0 /0 $ C&( /0 " " " %! % ! % $ 'H( / $B ''(0

2 $) :( 2 - 4 4 / '0

2

$ '(

-

2

,/ 0 A !

,/ 0 A / $ ),( $ &(00

2

$G ,( 7" - / ? 0'

Condensation and Transitive Tournaments DEFINITIONS

2 3 # " ! # - ! ! ! %- # "-

# #"

"

"

2

" ! ! $ ! ! $ ! $ EXAMPLE

2

&- "-1 - ! ! ! # " " " " " " !

" " " " " A C / 0 / 0 / 0! " - # "- ! ! ! # ! FACTS

2 2

$B 8 'C( "

# A" 4" - $ 'H( / 0 " /0 /0 4 /0 4 /, 0 /0 "

! /! $ ( 0

© 2004 by CRC Press LLC

162

Chapter 3

DIRECTED GRAPHS

2 7" / ½0- " - Cycles and Paths in Tournaments FACTS

2 $ ':( 7"

- ! !

-!

- / $H,( 0

2 $K :( 7"

- !

-! 9 - / $H,( 0

H!

2 $ :9( " / 0

- ! # " -! $H,( 0

-

!

-

:! /

2 $H,( " / 0

- ! #

" / $)G &:(0

-

-

!

-

,!

Hamiltonian Cycles and Kelly’s Conjecture CONJECTURE

>&&)* + / $ 'H(0 -

- / 0' ! # EXAMPLE

2 - 4 :- #/ 90

2

,

9 C ' ,E

, 9 '

C ,E

, 9 C '

,

REMARK

2 N

F 1

# 6 7" 1 2 & / ! $H(0E " - ! C! # -1 $RH,(E

- ',,, -1 $H( "

#

FACTS

½ !

-1

2 $BU & ( " "! -

"

"

2 $HC( 7

-

© 2004 by CRC Press LLC

Section 3.3

163

Tournaments

Higher Connectivity DEFINITION

&

2 % / % 0 " " &! &

% #

FACTS

2 $H,( 7" - "! A - " " 9- "! A -

2

$H:( 3 %- 3 % # ! - 3

2

$H9( " % ! ½ ¾ ! / 0- ! ½ ¾

%

2

$ & ( -! - ! " - - ! '! # "-1 - -! 4 # / 90 / - 5 ! # # $HC( $ ),,(0

2

$ ) %,( 3 %- - # "-1

"

H

%

!

%

Anti-Directed Paths

, ! " 3

! /

0!

3 ! %

% DEFINITION

2 / 0 & 4 & & EXAMPLE

2

# - -

9

© 2004 by CRC Press LLC

164

Chapter 3

DIRECTED GRAPHS

Figure 3.3.4 "% # # )& FACTS

2 $B ,, ( % @ - !

C ! 4 #/ 90 " / # A "

- &: $):(0 2 $IH9( 3 " '! " - - / 1 # A

" C, $: (! # "

" H $ :9(0 2 $B ,, ( 7" - ! 'H! " /0

/0 # / # A " ½ $H'(0

3.3.3 Scores and Score Sequences ,! B ) % $% C (! 8 <

/ " $&'( 4 $)&&(0 FACTS

2 $% C ( 4

/ 0! # ! 4 -

! # ! 4 -

%

%

5

2 $B 8 'C( 4

%

%

5

5

5

/ $B ''(0

2 %

5 / 0 4 " # ! - 5 4 - # 4

© 2004 by CRC Press LLC

Section 3.3

165

Tournaments

½

# - ! 4 / 0- / $:&(0

2 $"H,( 4

5 / 0 4 - !

4 /,0! /! ! 0! /! ! ! 0! /! ! ! ! 0 / $&H(0

2 $VH9( % 5 / 0 4 7" - # 4 4

5 /

0

2 $* HH( $* H&( 7" - " EXAMPLE

2 3 " 4 /! ! ! 9! 9! 90 A

,! ! 4 '- 3 ! '- # "-1 - ! 3 ! # " " " " 3 4 /! ! ! 9! 9! 90 A 4 ! '- # 4 /! ! ! 9! 9! 90 REMARKS

2 " - - #

! # 4

- - #

!

- # 4 ! 4 - )" - " ! # 4 " # $ :H( 2 ) & )" - ! #

6 6 6 # J ! 6 " 6 /# : " 0 # 6 4! # " $ 9( 6 "

! # ! # # " #! " 6 ! # F - . A - " " 6 # !

" $ 'H(

© 2004 by CRC Press LLC

166

Chapter 3

DIRECTED GRAPHS

The Second Neighborhood of a Vertex DEFINITION

2

% " & ! / 0!

" & - - / 0 5 $ ! ( ( / 0 ( ( / 0 ¾

(

FACTS

2 2

$&'( 7"

"

/ 0 ( / 0

(

$B ,,( 3 ! # " ' CONJECTURE

7) * '

+ / $&'(02 7" "

$

#

(

/$ 0

/$ 0

(

&

-

3.3.4 Transitivity, Feedback Sets, Consistent Arcs 3 - / - 1 " # 0! 1 # / # F

"0

6 !

" " -

DEFINITION

2

EXAMPLE

2

% # "- # % #" %! ? ! 5

/ ? 0 0 6 !

/ ? 0 6

Smallest Feedback Sets

6 - 4" A " - / 0 / 0 5 / 0 " #

© 2004 by CRC Press LLC

Section 3.3

167

Tournaments

6 # $ " " " $ B+ &'( + 6 -# 6 -# $ )B+ &:( $*,( FACTS

2

# " "

6 / $ B3 &C(0

2

6 4

" / $ B3 &C(0

2

$ B3 &C( 3 #

6 ! "

# -

2

$ B3 &C( &

6

Acyclic Subdigraphs and Transitive Sub-Tournaments DEFINITION

2 & & FACTS

2 $:L:! H ( 3 */ 0

" - */ 0 ! " "½ "¾ ½¾ ¾ ? "½ ¿¾ */ 0 ¾½ ¾ ? "¾ ¿¾ "! "

# /$'&! :(02 (

*/ 0

9

C

'

:

H

&

,

9

'

&

,

9

,

C

99-9'

2

$I :,! 8&9! T &9( 3 / 0

" - " - # / 0 "!

9 : 9 H / 0 5 C 9 : ' H ¾ /' ':0 / 0 ¾ ? C9 / 0 ¾ / 'CC0 ? : CC

2 $ B &H( $)) +&H( 4 5 /½ ¾ ½ 0! # ½ ¾ ¿ ½ ! " " - " " "! ! #" / 0 / $,( ! $ B &( -

0

© 2004 by CRC Press LLC

168

Chapter 3

DIRECTED GRAPHS

Arc-Disjoint Cycles

" ! / 0 -1

! "/ 0

6 !

/ 0 5 / 0 "/ 0 5 "/ 0! # 6 "

-

REMARK

2 8 4 / 0 4 -1 /0 ! # # $ )B:( 4 / ' 0/ 0' FACTS

2 2

! / 0 "/ 0

$N :'( ,! / 0 ! "/ 0 ! ,! -

6

-1 / $:C( $ ),( $3&C(0

3.3.5 Kings, Oriented Trees, and Reachability N = > "

# # =6>

/ # 6 $% C (0 $ H,(

7 # " - " " = -> 6 = " "> " / H #0 "! 6 < !

- /! $&'(0 DEFINITIONS

2 " - -

2 "

- -

" " !

" " !

2 % ! % ! " " 6 % " 6 FACTS

2

$G C( 7" 6 3 ! " " 6 $% C (

2 %

$ H,( " % ! - # 6 % ! % 5 ! /% 0 5 /9 90 / $H(0

© 2004 by CRC Press LLC

Section 3.3

Tournaments

169

2 $H,( , % ! - #

6! ! " 6

# A2 /0 % ? ! /0 5 % 5 ! / 0 5 % 5 5 9 % ! ! ! /90 / % 0 / 9 0! /C 9 ,0! /: ' 0 / $ & ( 0 %

2 $H9( %- " 9% ? " "!

# 4" 2 /0 %- /9% ? 0 /0 /9% ? 90 /9% ? 90 6#- B / 0

/9% ? 0- 2 $ H,( 3

" " 6 / $ 'H(0 3 ! " %!

%- $H9( REMARKS

2 " 1 -

# 6 " # $H( $+ H9( ) #

" ) 6 - # $H,( # 6 6 $&'(

2 A ! 6 ! Tournaments Containing Oriented Trees

/ 0 ! / 0 - #

" CONJECTURE

7 * + / $+ H (02 7" / 0- " " EXAMPLE

2

H 9 # C '-

Figure 3.3.5 " REMARK

2 8

#

Æ 1! 4 = - > 4½ ½E

/ 0- " ."

© 2004 by CRC Press LLC

170

Chapter 3

DIRECTED GRAPHS

, "

/ $B ,,(0!

# 1 FACTS

2 $B ,,( 3 + / 0 - " -- -

" " ! + / 0 /: C0' /7 @ + / 0 $BU

&( + / 0 ' $B ,,(0

2 $%+ I ,,( 7" - ! H,,! - ! # !

" " -

2 $I,( 7 / ? 0-

? # ! # 2 $ &( - -

-1 ! A " !

/ 0 ! / 0 -

- 3 - - ! " " ! - 1 - Arc-Colorings and Monochromatic Paths CONJECTURE

0 ? + / $ + H(02 " %! " /%0 " - " " %

/%0 " # " " ! " REMARK

2 1 " " ! /0 5 C' # /0 5 &- - # - # / 0 / $ + H(0 3

! 7U

6 / 0 5 3 " 6 # /%0 A % 1 # / $% &'( $,,(

" 4 0 FACTS

2 $ + H( 3 # # ! " " " 5 ! / $H9( 0 2 $HH( 3 #

- " - #

! " " " 5 !

© 2004 by CRC Press LLC

Section 3.3

171

Tournaments

3.3.6 Domination 3 "

" B #"! !

# 5 / 0 5 % " % 6 #! # "! " 6 #

" % /0 Ü& DEFINITIONS

2 " "

" "

2 "- ! 1 #"

2 /0 # #

" "!

2 ! 5 / 0 EXAMPLE

2

% " - # "- # #" " / 0 - # " " " - ! " / 0 " - / " 0 ! 6 !

" " ! 5 / 0 5 FACTS

2

$):( " %!

%¾¾ ¾! # / 90 4 - % / $BB,(0

2

$%&H( 6 # #

"!

3

! - ! 6 # #

"

2

$%&&(

6 ! !

# "

" 1 " # < $K%&H(

2

$<<'C( 3 - # A 5 / 0 *¾ *¾ *¾ ?

© 2004 by CRC Press LLC

!

172

Chapter 3

DIRECTED GRAPHS

3.3.7 Tournament Matrices # 6 )

K + 8 K I

/ Ü E $):(0E # 6 B K $ '9( # " # < =>! ! F " E # 6 6 $'C( # # < < V % $%H ( # 6 6

6 < " " <

# 6

DEFINITION

2 4 6 5 /- 0 ,F F! # ,F - ? - 5 !

" "! ½ ¾ ! ! 6 5 $- ( ,- " - 5

, #

! 1 " " EXAMPLE

2

' # '

Figure 3.3.6

,

REMARK

2 " 6 6 ? 6 ? 2 5 7 ! # 2 ! 6 6 ! 7

F "! 1 6 1 ( " ( ! ! / 6 5 / ½( / ! "

© 2004 by CRC Press LLC

Section 3.3

Tournaments

173

FACTS

2 $)'H( % ½ ¾ " ! # ½ ¾ , ! ½ / 0'! / 0' ½¾ ! 5 "! 6 2 " 6 ! ! 2 6

2 $ )NI &(

! " E " B /! 6 6 5 2 0 # " 6

2 $&C( 3 ! 6 4 / 0' 5 , 74 A " / 0' /#

0 7 " 6 / 0' / 90 A ! # " / 0'9!

/ 0 / 0 # ,F / ! / 0

F " , / 0 0

# # / 0 " F

, / 0 / ! ,F0 2 $&(

-

" -

< " 4

=Æ >

4 /'0/ 0! 4 - " -

3.3.8 Voting + 6 ! " - ! " " " A " !

+! ! ,$ ! ) - , Deciding Who Won

" 6 => " / "0 " " " => " "

=#> "

" 7 - ! " ! 4 # 2 "

/

0! " " 1 ! " # 6 " ! # " = > " /

0! " "

" = ">

! "

© 2004 by CRC Press LLC

174

Chapter 3

DIRECTED GRAPHS

" - / 6 0! "

/ 4 0 K- % F $% &:( DEFINITIONS

2 / "0 " / "0 ! " 1 "

2 & - !

# " ! "- ! " " & 1 -

2 & " & 1

" &! " /

" " " & @ " " 0

2 " /! 6 0!

" F 1 /! 1 0 REMARK

2 3 A 1 ! " A = ">! - # " =" > ! 1 " F 1 " 3 " ! 1 EXAMPLE

2 :

1 " C- ½ ! ¾ ! ¿ 3 # ½ ! ¾ ! ¿ ! ## " " # ! ! " " "

! " " ! ! " 1

! #

1 " ! 1 "! ! 1 "

Figure 3.3.7

© 2004 by CRC Press LLC

" + )

Section 3.3

Tournaments

175

Tournaments That Are Majority Digraphs FACTS

2

$C&( 7" - /! " 0 1

? ! ! ? ! "

% -/ 0

-" 1

-/ 0 # " ! */ 0

- 1

*/ 0 # "

2

! -/ 0 /CC ' * 0 $C&(! " -/ 0 ! /" ' * 0 $7 '9(

2

$ 'H( */ 0 # ! */ 0 5 */90 5 */C0 5 ! */ ? 0 */ 0 ? ! -/ 0 */ 0

2

$ &&( 3 1 ! ! ' ! !

- ! "

8 "

! / 0 8 # 3 ! (

" -

Agendas DEFINITIONS

2 " /! " 1 0

2 " 4 " #! " /½ 0 "! " A " ! # " ! # " !

2 )" 1 - / 0 " " " ! " ""

" /! / 0 " 0 " 1 "

3 ! ! ! " 6 / 0 EXAMPLE

2

)" / " 0 1 # H! " " #

© 2004 by CRC Press LLC

176

Chapter 3

Figure 3.3.8

DIRECTED GRAPHS

-+ )

FACT

2 $::( ! " "

/ ! $& (0 Division Trees and Sophisticated Decisions DEFINITIONS

2 )" /½ ¾ 0! /½ ¾ 0

!

! ! "

- 4 /½ ¾ 0E

/½ ¾ 0E ! , ! " " #

4 /½ ¾ 0! /½ ¾ ¿ 0! # " " ? !

/½ ¿ 0!

/¾ ¿ 0

2 % 1 - /½ ¾ 0 " " " "

" /½ ¾ 0 ! #

" " " 1 # # " 6

" " E " ! , ! !

" " " 1 #

# " " ? FACTS

2

$ HC( " "

" 4 " " -

2

8 " " /." $::( $H,( " $&(0

2

$&:( # "

" - !

!

EXAMPLES

2

" / $ 0 # & )" 1 #! " , "

© 2004 by CRC Press LLC

Section 3.3

177

Tournaments

"

!

" 8 $ ! #

:

Figure 3.3.9 / + )

2 1 9- # ,

"

: ! / 0!

Figure 3.3.10 + ) # Inductively Determining the Sophisticated Decision

# # A /

2 /$ 0 - " $ 0 FACT

2 $+H9( % 1 - /½ 0

" 6 " 3" A 4 $ $ $

#2 $ 5 ! ! !

$ 5 $

#

2 /$ 0

$

References T T ! I ! MH ' " ! ! $'9( T $ ! < I ! &'9

© 2004 by CRC Press LLC

178

Chapter 3

DIRECTED GRAPHS

$ ':( !

! ! %! /! , /&':0! H MHC $ :H( N ! 4 ! &! /&:H0! 9M9& $ :9( ! N I

! ! &! ! / : /&:90! MH $ H( ! 9- ! %! /&H0! M& $ B &H( B ! " # ! ! 9& /&&H0! HCM& $"H,( I " ! C:M'9

4 ! &!

$ HC( K 6! " ! 0 ' /&HC0! &CM ,'

9 /&H,0!

$

$ &( K -K! 7-1 - -

! &! ! / C /&&0! M $ B &( N ! % + 6 B ! # ! G" 3 &! 1 /&&0! H M&9 $ ),,( K -K! * ) ! * ! " ! %! 9 /,,,0! ::MH: $ ),( K -K ) )! ! ! ,,

" "

$ )&'( K -K ) )! I ! ! ! ! C /&&'0! M:,

$ )&H( K -K ) )! ) < 2 " ! H /&&H0! :M,

&!

$ B3 &C( K-I T T ! . B ! ) 3

6! ! ! " ! ! %! /&&C0! &M:' $:( K ! . W F ! %! ' ! $! : /&:0! CMC $:C( K- ! ! 'CMH, + 2 $ ! 8 B

! &:C

$H( % + 6! 2 " " ! 9MCC E # ! &H! 3 %! $! 3 C! J" I! ! &H

© 2004 by CRC Press LLC

Section 3.3

179

Tournaments

$N :'( K- * N @! J F " F ! )! ! ! )! (! ' 4 4 , /&:'0! H M& $H( K- C /&H0 M9

!

2 " !

&!

$+:C( % + 6 K + ! " ! M9H ) ! ! $ ! 1567! ! &:C $)'H( 3 ) ! . ! /! ! %! $! :9 /&'H0! M C $):( 3 ) ! 6 ! ! C /&:0! M H $%H ( V %! J ! / /&H 0! 'M::

3

&! !

$VH9( V %! # " ! &MC ! I! &H9 $,( K ! % F 4 " - ! &! H /,,0! 99MC9 $ C&( I ! ! ! )! ! $! I 9& /&C&0! CMC $ )NI &( ! ) ! K N6

! 8 K I

! K ! " ! 3 ! '& /&&0! :&M& $ )B:( ) ! )

B! ) # ! &! ! / , /&:0! M9 $ ) %,( ) ! K ) B %! I " ! &! ! / H /,,0! M, $ )B+ &:( 3 - ! )T ! . B + ! 8# ! %! 'CL'' /&&:0! &M C $ B+ &'( 3 - ! . B + ! .

! %! ' ! $! /&&'0! MC' $ C9( % "!

! M9,

/! %! / ! ' /&C90!

$7 '9( I 7U % ! . ! ! %! ! ! ! $! & /&'90! CM

© 2004 by CRC Press LLC

180

Chapter 3

DIRECTED GRAPHS

$H ( + <

G ! . ! &! ! / /&H 0! HM $&'( ! 4 2 F 1! &! /&&'0! 9 M9H $%&H( ! K %! < N !

! &! & /&&H0! , M, $%&&( ! K %!
2 " ! &! & /&&C0! 9HMC,C $)) +&H( ) ! ) T T

! T I + ! - - " ! &! ! / : /&&H0! HM&' $)G &:( * ) % G 6 ! ! ! %! :& /&&:0! :M C $B ,, ( B "! . B ! &! ! / /,,,0! M $BU & ( BU 6"! B ! ! ! ! /&& 0! CM $B ''( B % ! ! ! %! % : /&''0! M9' $B 8 'C( B ! R 8 ! #! $ % ' ! K + O ! &'C

© 2004 by CRC Press LLC

Section 3.3

181

Tournaments

$B ,, ( B " T! . B 2 F 1! &! ! / :H /,,,0! 9 M: $B ,,( B " T! 2 F 1! &! C /,,,0! 99MC' $BU &( BU 6" ! ! /&&0! M , $BU &:( BU 6" ! . B ! &M C " " 8 " 1559:!

J" I! &&: $3&C( ) 3

6! 6 ! 7 &! ! /&&C0! & $K :( . K 6 ! &:

! ! J" !

$K%&H( ) K< K %! # ! ! /&&H0! M $% C ( B ) % ! .

333 ! /! %! / ! C /&C 0! 9 M9H $% &:( K- % ! $ %. ;! ! &&: $% &'( G %6 ! - ! ! 99 /&&'0! CMH $%+ I ,,( X %! -+ + K I ! ! ' /,,,0! 9M9: $ H,( ! 6 6 ! %! %! C /&H,0! ':MH, $ &&( K

! . - 1 " ! %! $! $! : /&&&0! &M99 $ &( * 6! - A B ! ! %! ' /&&0! &&M, $::( 8

! ) - " ! ! &! ! $! /&::0! :'&MH, $H,( 8

! # 1 " 2 - " ! ! &! ! $! 9 /&H,0! 'HM&' $&C( ! 6 ! ! %! % , /&&C0! ' :M' & $ 'H( K + ! ! B ! ! + ! &'H

© 2004 by CRC Press LLC

182

Chapter 3

DIRECTED GRAPHS

$ '( K + % !

! %! /! C /&'0! 'M'C

!

$8&9( G 8 % ! I 6 ! ! , /&&90! ' M '' $I :,( 7 I 6 N ! 1 7U ! &! ! & /&:,0! CM H $I,( G I "! C&'

# ! ! /,,0! C&M

$IH9( G I "! B ! C&M'& " $" 15<9! J" 8 " ! 8 " ! &H9 $,,( N ! - !

! 9' /,,,0! M9 $T 9( % T! 7 6

3 $ ! <

:

$'&( N ! . ! ! %! /! /&'&0! 'M': $: ( N ! 74" - " 6- " ! ' /&: 0! ' MH,

%!

$H,( N ! # 6 ! ! & /&H,0! H,&MH' $H( N ! 7" " 6!

%!

H /&H0! & M&H

$H9( N ! ! " ! $ ; 3 % ,: /&H90! M $HC( N ! # # - ! :M ! " C %! $!! 8 -B

! &HC $& ( N ! 1 2 " ! %! $! $! /&&0! M& $&( N !

# # " 1 " # ! ! %! /&&0! MH $&'( N ! 2 ! 6! < ! ! C /&&'0! :M

$&:( N ! 74 2 " ! $ 0 ' 9 /&&:0! ' M :H $:&( N % + 6! ! '&M,9 ! I! % ! &:&

© 2004 by CRC Press LLC

$

Section 3.3

183

Tournaments

$:( N 7 #!

4" 6# B ! &! ! /&:0! M H $H9( N

! 6 ! :MH ! I! &H9 $BB,( N ! ! B B! ! ! ,, $+ H ( N 8 + ! 7 - ! $ $! %! H /&H 0! ::M H: $ :9( ! - B ! / ' /&:90! 9M9

&! !

$ '9( B K ! < ! , M9 ) % - ! J" + I! &'9 $T &9( T <- ! .

" ! ! , /&&90! ':M :' $ + H( ! 8 + #! . ! &! ! / 9C /&H0! ,HM $HH(! ! . - ! / /&HH0! ,HM $&( % ! . ! CM 'H

3 !

&! !

'L'9 /&&0!

$+H9( N + ! J " " # ! ! &! ! $! H /&H90! 9&M:9 $*,( 3 /3 - 0 . * /. B 0! " # / 0! ! ! ! ( ! H /,,0! : M& $ & ( R !

! / /&& 0! HMC

&! !

$:L:( K ! . 6 ! /&:L:0! CM H $C&( ! " ! !

%! %

'' /&C&0! :'M:'

$<<'C( 7 <6 ) <6! . U 7U ! 9& /&'C0! &,M& $&H( I ! < 4 ! /&&H0! C:MC& $: (

%! =!

&! ! / :

! B ! , /&: 0! M H

%! !

© 2004 by CRC Press LLC

184

Chapter 3

DIRECTED GRAPHS

$H,( ! B - ! &! ! / H /&H,0! 9M' $H( ! 7-1 B ! ! 3 %! $! 9C /&H0! CM'H $H9( ! " ! ,CM ! ! &H ! I! &H9

$HC( ! B

! %! : /&HC0! C&M' $H'( ! I ! ! ! %! $! &' /&H'0! ':MH, $HH( ! " ! %! : /&HH0! : MH' $G C( B 7 G ! . #

- ! + &! %! /&C0! 9,:M9 $G ,( % G 6 ! 2 ! %! 9C /,,0! &MC $+ H9( N +

! ) 6 ! ! 9C /&H90! M H $* HH( X * ! F 1 / /&HH0! 9HM9H9

0! > -

$* H&( X * ! . 1 ! $! /! 9 /&H&0! H,9MH,H $RH,( V R ! 7"

# -1 B ! &! ?' " $ ( ! ) ! /&H,0! :,MH $R &( N- R R- ! $! : /&&0 HHMC

© 2004 by CRC Press LLC

M " ! &! . @!"

185

Chapter 3 Glossary

GLOSSARY FOR CHAPTER 3

/)

2 # 6

1

/) @

2 # 6

1

+) ,

M 2

#

M " 2 " /! "

1 0

& &

/

02

M " 2 4 " #! "

½

/

0 "! "

" ! #

A

" ! #

" !

/ ,

M 2 " 4

E

# )&

M

#

&

M

&

&2

4

&

&2

4

&

) & # @ # '

2 #!

!

#

# !

5

2 "

2 " 8

2

2 - 2

" " "

"

' ' )

2

2 # " # !

!

'&

2 " "! "

@

' )#

/02 ! # " 6 !

6 "

!

& / ,

!

6 "

6 " M 2

" #

&

E

M 2 # ! #

'&)

© 2004 by CRC Press LLC

M /

02

# "

186

Chapter 3

'& &

02

# " "

-

M /

& -# )

2

--

DIRECTED GRAPHS

" "

&

2 # "!

# ½ ¾ 2 ½ ¾ / 0 / 0 " " 2 # "- ½ ¾ " ½ ¾ / 0! # - ! # " #"

"

"

M

# "-

5

2 ,

2 " -

/! 6

0!

" F 1 /! 1 0

2 %

M " 2

/ "0

/)

/ "0

!

"

"

1 "

# 0 M - 2 < "

/ " -" 0 /0 "- /

"

/0

&

M

A% % . /

2

&2

&

02

M /

# "

1 "

/

M /

$

02

$

202 & 0 -

2

I 2 # 6 " A

6 # # 1 8 ! - " A A

$

2

M " 1

/

½ ¾

!

"

/

6

0

M "

- !

0 "2 !

2

# ½ ¾ 0

/

/! 0

4

/ ,

© 2004 by CRC Press LLC

M 2 "

E

"

4

Chapter 3 Glossary

187

2 ! )&2 # ! ! ! .-/ ! 2 # &

!

& # A 2

# " ! #

%.2 & ! %.&) 2 # E !

)& 2 # 2 0 0 '&2 A -#

" #

)&2

/ 02 ! # !

# 3 #! # #

2 #

2 #

%&. M ¼ 2 4 ¼ ½ ½ ¾

½ " ! / 0 5 ½ / 0 5 ! 5

¼ - # 6

/ M " 2 ! A :

M 2 " " " "

/ $ 0M " 2 " !

M 2 "-

!

' M 2 E

! 1 #" 5 / 0

'&)# & 2 #/) M -" 2 <

/0 /0 / 0 M 2 # ! !

#

& 2 # 6 # '. M 2

A% % .2 & © 2004 by CRC Press LLC

188

Chapter 3

M

" #

DIRECTED GRAPHS

2 #

& 2 # " & )& / 0 M &2 "

&

& / 0 M &2

"

&

; M 2

- # "

1 # #

2 M 2 #' / 0 M 2 #

"

# M " /!

E

/ 0

/

# M " &2

2

"

/ 0 #

"

2 / 0

& / , M 2 - # 2 $ %# 2 . M 2 " " - -

0

"

E

!

& M 2 " # , & ' M " 2 -

& %&.2 - # 6 4 &/& / , M 2 & , M 2 " " ½ ¾

#- - "

& 2 " -# ) 2 + ) & M - !

# # "-

- ./ 2

"

1

"-

"

-

&

2

# - " #

- " E 6 "

,#A% '&2 ; # "

- # 6

6 ; # % 0 ; #

4 ; # /

##A% '&2

A

%

; # # 6 / " 0 4

© 2004 by CRC Press LLC

189

Chapter 3 Glossary

&&) '& / M

2 " " # 6

" " 7" "

# /" " # 60

' 2 % .2 5 / 0

" # 6 ; # E

" " ! " ; # !

!

# A%2 # 6 5 / " 0 # " " 2 ( ! " !

! # < !

" !

&! # <

A% 5 / " " 02 # "- ! "2

!

!

)

-

" 2 2 )

" ! "

A%2

# 6 5 / " " 0 " 2 ( ! " 2 2 ) A /0 5 ,

)

(

¾

/

0 5 ,

# " ! "

¾

2 "

!

A

2

# " A

2 M 2 ! 6

2

#' / 0 M 2 # /

$ 0 M " &2

" E

,/ 0

( / 0

/ # => 0

# 2 !

# # / , M 2 "

& 2

E

!

"

&&) 2

/

;"! !

0

2 ! % #2 % 2

/ &02

# 6

#& M 2

# " @

© 2004 by CRC Press LLC

190

Chapter 3

DIRECTED GRAPHS

/

M 2 " " "

A,/ & #2 #!

! #

& 2 #

! & / 02 # /0 /0 5

¾ ! '&)#2 #

" 1 " /! % ,/ 0 ,/0 5 %!

" 0

& M 2

# ! #

' M " 2

-

2

2 " " !

! " " ! - .

! $ ! $

! -# )2 # " " - # E

-

& 2 # / 0! #/ 0 E #

! A

A% % .2 & " 2

" /! /0 # 0 0 / 0 / B / 0 M - 2 - / 0! # " ! !

/ 2 * ' M "

&2

" & - - E ( / 0

M 2 " " " ! - -

'& M 2 & 2 # - - & 2 # - - M " 1 - /

0 " " " 2 " ""

" /! / 0 " 0 " 1 "

. M 2 " < M " 2 ! A H M 2 " < ' M 2

"

© 2004 by CRC Press LLC

191

Chapter 3 Glossary

. )&2

/0 # #

" "!

& 2

# " ! - - " #

M 2 2

M 2 2 ! %#2 " % # "

&) 2

# " # "

! ! # "

&) '& #

)'& M 2 ! A &2 & & 2

#

2 ,2 4 6 5 /- 0 ,F F! # ,F - ? - 5 !

/! 1 0

2 # "

'&2 $ ! B

2 E ! A ! '&2 # "- # - !

½ ¾ " " ½ " " ¾

! %#'&2 # " " 6 % " 6! # % ! #2 ! ! -"

/ & M "

&2

&

/ 2 #! / 0 / 0 ! / 0 / M 2 " / & #2

#!

$ ! # #$ ! #$ / 2 " "

! ! $ ! ! $ ! $

M 2

" " " -

& & 2 #!

" ! #

© 2004 by CRC Press LLC

192

Chapter 3

DIRECTED GRAPHS

/ ,# M 2 " #

% M 2

" !

< <

© 2004 by CRC Press LLC

Chapter

4

CONNECTIVITY and TRAVERSABILITY 4.1

CONNECTIVITY: PROPERTIES AND STRUCTURE

4.2

EULERIAN GRAPHS

4.3

CHINESE POSTMAN PROBLEMS ! " #

4.4

DEBRUIJN GRAPHS and SEQUENCES $ %&' ( '

4.5

HAMILTONIAN GRAPHS ' ' )*

4.6

TRAVELING SALESMAN PROBLEMS

4.7

& +' $

FURTHER TOPICS in CONNECTIVITY

GLOSSARY

© 2004 by CRC Press LLC

4.1

CONNECTIVITY: PROPERTIES AND STRUCTURE

Introduction ! " # *,-* $% " ! Æ $ & ' ( (&% )! * % + !& ! ! & ! , ( , -. / 0 -* 110 -2 340 -+ /0 -* 2 110 ( ! #&

5' -6770 -8970 -$ 9/0 -8* 3/0 -(/10 -(/ 0 -(/40 -:/0 -;/70 -2<//0

4.1.1 Connectivity Parameters " , !

2 # = >

= > ; ! ! Preliminaries DEFINITIONS

' # ! & !

© 2004 by CRC Press LLC

Section 4.1

195

Connectivity: Properties and Structure

' 6 ! # !

' : ? = > 6 6 # " ? ! !

' : ? = > 6 6 ? = > # ? !

' = >

# ! @

'

#

' = > ! @

' = >

FACTS

' A !

'

Vertex- and Edge-Connectivity

6 ! B ,- ' - DEFINITIONS

' 6 = > !

' 6 = > !

+ # ! C D " = > = >

= > = >

EXAMPLE

'

( ! # !

© 2004 by CRC Press LLC

? ?

196

Chapter 4

Figure 4.1.1

CONNECTIVITY and TRAVERSABILITY

?

?

FACTS

'

?

? 1

½ ½

½

? 1

½

$

#

?

?

'

"

?

!

'

"

?

"

"

8 !

½ =

> ? 1

! !

'

6

Relationships Among the Parameters

6

! !

Æ

Æ

=

> + #

="

Æ

=

>

>

FACTS

'

-+0 (

'

Æ

-* 730 (

!

?

?

Æ ?

1

#

DEFINITIONS

'

! ? Æ

'

!

!

'

?

!

?

Æ

!

Some Simple Observations 6 ! ,

FACTS

'

© 2004 by CRC Press LLC

AB

Section 4.1

'

"

'

197

Connectivity: Properties and Structure

·½

?

!

E

!

'

A & !

Internally-Disjoint Paths and Whitney’s Theorem DEFINITIONS

'

# ,

#

'

6

=

½ ¾

5

>

!

# 6

=

>

=

> ?

?

FACTS

'

-+ 0

'

!

+ % &

'

!

5 ! 5

! & !

Strong Connectivity in Digraphs ( # #& -:/70 -* F 730 -8 210

DEFINITIONS

'

"

'

! &

! &

!

'

(

?

=

>

, !

?

=

> =

>

!

'

:

6

=

> =

> C D

© 2004 by CRC Press LLC

=

> !

198

Chapter 4

CONNECTIVITY and TRAVERSABILITY

REMARKS

'

" # ! !

" = > ? = >

! = > ? = >

Æ Æ 6 , Æ ? Æ Æ

'

F Æ 6 ! 2 * = > ( 9 FACT

'

-2* 910 (

Æ

!

! ? Æ

? ? Æ

An Application to Interconnection Networks

6 !& = > ! ! & = > - & !& 8 *

-8* 3/0 " # !& 6 B = >

!& Æ

B = > # !&

4.1.2 Characterizations + ! 5 5 " 6 ! 5 5 + ! & Menger’s Theorems DEFINITION

' : ! 5 => => !

@

( 5 =>

© 2004 by CRC Press LLC

?

=>

Section 4.1

199

Connectivity: Properties and Structure

( ! 5

=>

#

FACTS

' '

? => ' 5 -$% $90 ( 5 => ? => ' => $ , ! B (

'

'

-$ 90 A 5 ! => ? => =>

( => !

= > @

=> # ' -'.

' '

=A $% > -A(47((470 => ? => (

= > ? =>

REMARKS

'

. $% #

'

6 $% ! ( (& -((470 & "-/& F!& )! Other Versions and Generalizations of Menger’s Theorem

" ! # $% # -. /90 -(/40 -$3 0

$% -;10 DEFINITIONS

'

2 # #

'

= = > > #

' '

! ?

# = B

# >

'

# ! 5

6 # = > 5 = > = > = >

© 2004 by CRC Press LLC

200

Chapter 4

CONNECTIVITY and TRAVERSABILITY

FACTS

'

6

5

'

"

6

=

> ?

=

=

=

>

>

'

6 $ B

" # # !

'

=

> ?

=

>

2 -2 70 5 ( 4 C# D

!

=

=

# 5

B #

>

>

=

> : G

-:970 5

=

>

> 8 5 ! $

'

-$ 93$ 930

=

>

=

>

=

>

=

>

REMARK

'

6 $% =( 3> (

&

5

Another Menger-Type Theorem

(

=

> #

5

B

5

=

(

>

FACTS

'

6 # ! ! B

*!

?

=

>

=

> !

=

>

=

>

=

> ?

=

>

6 $ B

( /

'

-AH 99:F 930 6 # 5

B = @

>

Whitney’s Theorem " # ! # ! 5 ! ! =( >

$% !

&

! !

+ "

FACTS

'

-+ % +0

=

© 2004 by CRC Press LLC

5

>

Section 4.1

201

Connectivity: Properties and Structure

# 5

' =A + % >

= > : 6 # ½ ¾ ? ½ ¾ 5 = ! # > = > ? ?

' =6 ( : > :

Other Characterizations

! 5 ( & $ 6 5 ! : G

2I =! !& > ( =( > FACTS

! E ½ ¾ ½ ¾ ½ E ¾ E E ? ½ ¾ = > ? = >

' -:992930

!

? = > ½ ¾ 5 = > ½ ¾

' -/90

4.1.3 Structural Connectivity

* ,

Cycles Containing Prescribed Vertices

6 , . ! ( 4 FACTS

' -. 710 :

6

! 6 ! E ! ? !

' -+ $790 :

Cycles Containing Prescribed Edges — The Lovász-Woodall Conjecture

: G -:9 0 + -+990 5 '' ' = ! # >

© 2004 by CRC Press LLC

202

Chapter 4

CONNECTIVITY and TRAVERSABILITY

REMARK

' : G -:9 :990 , ! +01-(' . ? 6 5 ! ! ? -A234:/10 ? 4 - /70 B *I & 6 -* 630

= E > =! > $ 5 J ! = ! > FACT

'

! -J 1 J 1 J 1J 10 :

6

$

Paths with Prescribed Initial and Final Vertices

2 ! # 5 = > $% $% ! 5 ,# = > F! ! # ! DEFINITIONS

' B ½ ¾ ½ ¾ @ # ? # 5 ' = > #

5

' ,

5 !

' 6 ! FACTS

' '

& ! = > -: $ 910 -H910 = > ( # => => &

'

6 -631 0

& 6 , -340

( => => ! & &

© 2004 by CRC Press LLC

Section 4.1

203

Connectivity: Properties and Structure

CONJECTURE

= E > ? => ? E

-631 0 ( FACTS

' -;&3 ;&34;&390 " ½ ¾ ½ ¾ = > ! 7 = > = > # 5 ' -*/0 ( = E > E => E ' -;&33;&/1 0 (

= E > = E > E => => = E > E ' -610 A => =, ( /> ! # & ' -/90 : 6 =½ ½ ½> =¾ ¾ ¾> = > ! ½ ¾ ½ ¾

½ ¾ @ ? = > ? # 5 = > ? = >

Subgraphs

* =( 9> =( 3> *! # FACT

' -$ 9 0 A

REMARK

' ! 6 -6330 " $

4.1.4 Analysis and Synthesis B ! = > C D 6% ! ! ! & +

!

( !

© 2004 by CRC Press LLC

204

Chapter 4

CONNECTIVITY and TRAVERSABILITY

Contractions and Splittings DEFINITIONS

' 6 ,

=& 5 ? > :

' 6 ' # ! Æ ! 5 ! 5 5 $ ! ? Æ E ! &

' ( # 5 # ! ½ FACTS

'

" #

! = > " ! ! ? Æ

' '

-6310 A

-630 A - = >

'

-670 A ! , B # EXAMPLE

'

" ( " ! #

Figure 4.1.2

!" "

REMARKS

'

" #

© 2004 by CRC Press LLC

Section 4.1

205

Connectivity: Properties and Structure

'

6 ( 3 J !& % ( 3 6% =( 41>

' 6% # ( 5 ! -J10 ! = !>

'

( 41 ! -670' ! !

'

-9 0 B ( 4 *! : G -:9 0 $ -$ 93 0 = !> Subgraph Contraction

6 DEFINITION

' # FACTS

'

/

'

-$;/ 0 A !

-6630 A !

'

-J110 A

CONJECTURE

-$;/ 0 ( Æ Edge Deletion DEFINITION

' # = #> FACTS

'

-$ 9 0 A !

© 2004 by CRC Press LLC

! E =! >

206

Chapter 4

CONNECTIVITY and TRAVERSABILITY

'

-;&330 : ! : ½ ? = ? > 6 = > 6 # => 6 # ' -;&/10 : ! "

! ? ? = ? > ! E # ' -*;&/0 ( # ! # ! #

REMARK

' ( ;& -;&/40 B

Vertex Deletion FACTS

'

-J : 90 A

#

' -630 A = E > => ! ' -A390 A = E > ! REMARK

'

( 4/ ! 5 : G 6 ( /

Minimality and Criticality

B # = > !

# ! DEFINITIONS

'

= > = >

= > = >

'

# => ? => ?

FACTS

'

-$ 9$ 90 A = E

© 2004 by CRC Press LLC

>

Section 4.1

207

Connectivity: Properties and Structure

'

-$ 90 A #

'

'

E

A # !

! -* 30 A

REMARKS

'

* -* 7/* 110 # # ! : & -: 90 8 ! $ =( 7>

' ( 7 * = > $ % =( 7> # 6 # # J -J 9 0 Vertex-Minimal Connectivity – Criticality

$ -$ 990 '

' - ' ! ! DEFINITION

'

= >

# ! ! = > ? + ? !

FACTS

' '

-$ 990 6 = >

½

6 C& D = ¾·¾ - 0> = > = E >

' -330 6 E B !

'

-$ 990 " = > 7 ¾ 6 , = >

REMARKS

'

= > ! 5 -$ 3 0

'

( 77 5 ·½ = > $ ! ! , =( 79> ' ( 73 ! $ , !

© 2004 by CRC Press LLC

208

Chapter 4

CONNECTIVITY and TRAVERSABILITY

Connectivity Augmentation

+ -(/ 0

* " ' - * * !

? = > , ! ¼ ? = > . !&

! ! + F & -+ F 390 ,

" Æ 6 B ! -(/0

References -. / 0 . $% =// > 4K47 -8 210 H 8 H 2 2 L : 11 -8970 8 F! <& /97

A F *

-8* 3/0 H 8 F * : !& 4 =/3/> 19K -* 730 2 ( * 2 ! 7K 7 !""# F! <& /73 -J : 90 2 J . : & $ % =/9> 7K73 -:/70 2 : : &

* :MF! <& //7

& 6 A

-. 110 . A 2 6# $ L 9 L F! <& 111 -. 710 2 . " & 2 I

N $ ' =/71> 7K34

2

-A(470 A ( A # )! !& ()* ( "6K =/47> 9K/ -A390 < A ! ! = + % , =/39> 9K99

© 2004 by CRC Press LLC

>

Section 4.1

209

Connectivity: Properties and Structure

-AH 990 A . H & 2 (*** % =/99> 71K 7 -A2340 : AI A 2I

$ 7 =/34> K -((470 : ( . (& $ # )! !& $ 3 =/47> //K 1

+

-(/10 ( & & K 9K11 8 J : : G * H I 5 =A> -. /0%(0 8 //1 -(/0 ( & B $ 4 =//> K4

%($ +

-(/ 0 ( & !& K7 H 8 J2 $ =A> $ 1 % !!2 6 N $ // -(/40 ( & !& )! K99 2 $ 2I : : G =A> A 8L //4 -2 70 6 2 $ # $ I

( & 2 $ % =/7> K9 -2* 910 . 2 ( * 8 =/91> 14K

0 ' $

37

-2 340 2 N //7 -2<//0 H : 2 H < 8 ///

(

-2930 A 2I ; 34K / F * /93 -* 110 * $ , + 4 =111> 3K4 -* 7/0 * 4

+ 9 =/7/> 41K

-* 30 < ; * OB P # G G + % , 1 =/3> K1 -* F 730 ( * Q F . ! 5 4 . /73

© 2004 by CRC Press LLC

( 3 4

210

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-* 630 *I

& 6 , $ =/3> /K

-* 2 110 $ * * 2 ! # ' ! B + =111> K41 -*/0 *& Æ ! & & 9 =//> K4

-*;&/0 *& * ;& # 5 $ , 3 =//> 4K43 -H910 * H A L Q I 2 $ 39 =/91> /4K1 -J 1 0 J J ! ; ! 5 ' : G

+ 5 + % , 3 =11> K -J 1 0 J J ! 6! ' : G

+ 5 -J 10 J J ! A# ! ' : G + 5 -J 10 J J ! : G

+ 5 -J 9 0 6 J F * % +

% , 9 =/9 > K -J110 $ J % , 31 =111> K 3

+

-J10 $ J # 3 =11> K1 -: $ 910 . 2 : $ ; # , ! & 0 $ % 1 =/91> K71 -: 90 . : & $ =/9> 93K3 -:9 0 : : G 4

$

+ ) . $

=/9 > 3

-:970 : : G ; $

% 3 =/97> /K3 -:990 : : G

% 1 =/99> K4

$

-:/10 $ L : / =//1> 4K

© 2004 by CRC Press LLC

4

Section 4.1

211

Connectivity: Properties and Structure

-:/0 : : G

//

*6 A F *

-:F 930 : : G L F : $ . $ $ / =/93> 7/K97

$

-$ 90 + $ $ & I 2 ,# / =/9> K3

-$ 9 0 + $ A# I 6 2 I J $ % 7 8 9 =/9> 37K/9 -$ 90 + $ A& 2 I 2 $ ,# =/9> /K -$ 90 + $ 2 & Q 2 14 =/9> /K -$ 9 0 + $ J + 2 7 8 =/9 > 39K1 -$ 990 + $ A & I

I K4

& 2

$ %

$

-$ 93 0 + $ $ =/93> 4K7 I -$ 930 + $ N $ # & 5& ,# 1 =/93> 4K7 I -$ 930 + $ N $ # & ,# =/93M9/> 39K 1

+

+

$

/ =/99>

$

$

-$ 9/0 + $ , 77K/4 * ! 2 3 * /9/ : $ : F 3 /9/ -$ 3 0 + $ ; 3/K/3 ! 4( 5 6789: 6 ;F /3 -$ 990 8 $ H ; $ 1 =/99> 44K7

-$3 0 + . $ $% 6 =/3 > 9K /

+ 3

-$;/ 0 + . $ J ; % , 71 =// > 13K -$90 J $ Q J

+

- $ 1 =/9> /7K4

-;10 ; ; L $% 6 : + 8 & H + =A> %

© 2004 by CRC Press LLC

212

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-;/70 ; ; ' ' 7 =//7> K4 -;&3 0 * ;& + , 9 =/3 > 4K9 -;&340 * ;& "" 9K4 H & =A> ' + , /34 -;&390 * ;& """' # =/39> 4/K3/ -;&330 * ;& + % , 4 =/33> 4K44 -;&/1 0 * ;& A ! & & 7 =//1> 9/K34 -;&/10 * ;& 4 /K44 8 & * =A> L * //1 -;&/40 * ;& = E > =//4> K91 -340 F . 2 R 4K9 % 8 !9: =2 ! /34> : $ : F 1 N /34 - /70 . ; , $ 4/ =//7> //K 4 -9 0 H , 3K/3

+ 9 =/9 >

-330 H H % 5 ;6 * # =/33> 794K793 -/90 S < $% $ 94 =//9> /K/7 -610 6 6 AI G =11> K -631 0 6 & * + =/31> 9K93 -6310 6 , , +

% , / =/31> K9 -630 6 F + 4 =/3> 4K4

© 2004 by CRC Press LLC

Section 4.1

Connectivity: Properties and Structure

213

-6330 6 /9K : + 8 & H + =A> % ((( : /33 -6630 6 8 6 F + , =/3> //K -670 + 6 6 ' < (

$ =/7> K 44 -6770 + 6 6 8 N 6 : /77 -+ $790 $ A + & . $ $

+ $ / =/79> /K3 -+ F 390 6 + F & A + % % 4 =/39> /7K -+0 * + + $

4 =/> 41K73 -+ /0 + 6 # K " $ *

L J& =A> 0 ' % 7/ $ ( // 8 // -+990 . + , + % , =/99> 9 K93

© 2004 by CRC Press LLC

214

4.2

Chapter 4

CONNECTIVITY and TRAVERSABILITY

EULERIAN GRAPHS 8 ., A 6 A 6 A ; 2 2 4 L 6 A 6 . 7 6 A 6

Introduction A JI 8 ' ( ( = > A

C

D

B

(a)

(b)

Figure 4.2.1

! &

! & T : A ! 97 -A970U 6 ! C , D ( # -(/1 (/0

4.2.1 Basic Definitions and Characterizations

6 #

? = > ! #

= >

5 C D C D CD

© 2004 by CRC Press LLC

Section 4.2

215

Eulerian Graphs

DEFINITIONS

' = > ! & = > , # #

' ' '

# #

" # B

" ** !

B

' = # > = # >

' = > = > = > Some Basic Characterizations FACTS

( ! -69 $3 +/1 (3/ (/10

'

1 =-A970 -* 390 -L L0>

:

6 ! B

'

= >

=>

=>

' ' ' ' '

? = > = > =-9/ (3/ (/10>

' = > => =>

(

% ! B

% % %

© 2004 by CRC Press LLC

216

Chapter 4

CONNECTIVITY and TRAVERSABILITY

REMARKS

'

( A =-A970> ! ( = > ( => ! * -* 390 6 B

( => ( => L =-L L0>

'

8 ( ( 7 !

' '

F

6 # # ( Characterizations Based on Partition Cuts DEFINITIONS

' : = > 6 ! = > ! ? = > # ,

'

# ! = >

' 6 = # > = > ! = > 6 = > ! = >

' : # # 6 Ú = > ! ?

"

Ú Ú FACTS

' '

' = > => =>

= > :

= > = > = > ? = >

# 6 ! B

'

= > ¬ = > = >¬ = > ¬

¬

REMARKS

' + ( => ( 9=> = > = > # ( 1=> =! ( 3 / >

© 2004 by CRC Press LLC

Section 4.2

217

Eulerian Graphs

' ( 1

! # & "-/& ; , % U % -((70

4.2.2 Algorithms to Construct Eulerian Tours + ! !

= -(/10> Algorithm 4.2.1: #"$%"& $!"' (#)*

#' ! 5'

# + # # % A % #

, '-< = 1> ! & EXAMPLE

' 6 & !

R CD 6 ? & & & & % ? ! ( 6 ! % # ¼ ? & & & & # ¼ ½

Figure 4.2.2 REMARKS

' 6 * % -

© 2004 by CRC Press LLC

218

Chapter 4

CONNECTIVITY and TRAVERSABILITY

'

6 (% ! !

(%

-:3/ 0 Algorithm 4.2.2:

$"& $!"' ( $)*

! ' = ? ¼ : ? ( ? ' : ? = > " = > ? : ? = >

#' A 5' A

A

>

? = > A# ?

The Splitting and Detachment Operations

6 ' * DEFINITIONS

' : ! # => 6 !

# ! ¼ ¼ 5 6 = ( >

Figure 4.2.3

$! " "! $

' : # ! => => => => ! ! ? => =! > 6 6 = ( >

© 2004 by CRC Press LLC

Section 4.2

219

Eulerian Graphs e3 e3 e4

e2 e1

e4

v

e5

G

Figure 4.2.4

e2 e1

v1 v3

v2 e5

H

+" '

' B # = > ( -F 9/ F 34 F 340 FACTS

'

+** : # ! => ½ ¾ ¿ Ú = > " # ½¾ ½¿ => " # ½ ¿ @ & ½¿

' : # = > ! => " ½¾ ½¿ ½¾ ½¿ # ( " ½¾ ½¿

' '

REMARKS

' 6 : =( >

= -(/10> " !

= ! > ( : -(110

' ., ( ! ! " = > & = 5 > ! -' @ =>

' 6 ! ! # ! ! # !

© 2004 by CRC Press LLC

220

Chapter 4

Algorithm 4.2.3:

CONNECTIVITY and TRAVERSABILITY

,$! $!"'

#' A ! ' = > 5' A " ? ? ¼ : ? ( ? ' " = > ? : ? = > = > A " # ? = > = > A ? @ & '? = E > A# ?

REMARKS

'

K '

' 6 @ ! (% # 1 ' B (% !

" ! : ! = -(/10> ;

4.2.3 Eulerian-Tour Enumeration and Other Counting Problems 6 8A6 6 # " , * =( 4>

= > 6 ! ! # ! +

) 8 5 = >

.F B

B .8 5 DEFINITIONS

' 1

!

' : % =%> 5 # ! =%> ? 6 =%> ! ,

?

© 2004 by CRC Press LLC

? ? = > U

Section 4.2

221

Eulerian Graphs

'

: ? ½ ! ! B ! # B

' : 6 % = > ! U ½ ( ! ½ % 5

# ½ ½ # ¾

( # = > #

!

FACTS

'

, * 2 % =%> ? ½ ? =%> J @ # 6 %

' " % B

' '

% ? 3) - * -A84 6 0 : % =%> 6 ! =)= > >V (

' ( ! * ' = > = > ,# ,

'

6 8 5 %

=% >> !

= >

==> ? )=> ?

' 6 ! 8 5 B 8 5 % B

8A6 6 8 5 B

½ = V>

EXAMPLE

' 6 8 5 10>

© 2004 by CRC Press LLC

% % ! (

4 = -;/

222

Chapter 4

CONNECTIVITY and TRAVERSABILITY

0000

000

1000 D 2,4 :

00

D 2,3 :

001

0100

010

1100

10

01

1101 011

11

0010

010

1010

101 110

0001

1001

100

001

100

000

0101 1011

101

0110 110

111

0011

011

1110

0111 111

1111

Figure 4.2.5

-". !" % %

REMARK

'

.8 5 ? ! ! B 6 % ! .F B 6

! ! ! B ' % J ! ! = -S10>

4.2.4 Applications to General Graphs " ! U

# !* ! * = > " ! B ! B Covering Walks and Double Tracings DEFINITIONS

' = >

! &

' ! & # ! !

© 2004 by CRC Press LLC

Section 4.2

223

Eulerian Graphs

'

! & ½ ? = ? >

' '

6 !

= >

= >

FACTS

'

: ! $ 1 6 ! B U ?

' A "

' - 990 = >

' -6390 " !

B

' -6770 -6390 ! = > 1

'

-L940 : = > = > ! = > ; ( = & ? = > REMARKS

'

6 =., /> ! ! ! ! ! ! !

'

6 B ( 4 , ! 5 Maze Searching

" # *1 ! ! # 6 % 5 *1- * = -(/0 # >

" => ! # , =>

© 2004 by CRC Press LLC

224

Chapter 4

CONNECTIVITY and TRAVERSABILITY

""& $!"' ( ) *

Algorithm 4.2.4:

#' 5'

Ú

=

"

>

? 1

?

=

> ?

+ =

+ =-

>

=

>0

=

?

'?

'?

:

>

?

>

-

=

A#

>0

=

>

E

?

?

=

>

'?

'?

E

Covers, Double Covers, and Packings DEFINITIONS

'

+

'

+

# !

+

'

.

+

=.>

5

!

'

+

+

!

+

CONJECTURES

! "

!

=.>' A .

' A

.

#

!

'

A .

Three Optimization Problems DEFINITIONS

6

'

:

! ! !

!

! &

'

'

+

6

$%

6

+ =

+

© 2004 by CRC Press LLC

=

! !

>

=

> ?

¾

=

=

> ?

'

,

=

¾

>

= >

6

>

& =$+ >

,

>

$ %

&

! + +

+ =

& &

> #

=$+ > ,

Section 4.2

'

225

Eulerian Graphs

6 & & , ! ! & ! => = >

FACTS

'

-(370 : 6 . & + . + 6 . 5

'

-(2340 6 N $ # + &

$ +

'

-(2340 : ! ! ! " N + $ # + & =+ > ? = > = > ! = > ! = > '? => ¾

'

-(2340 ( + $ + N =+ > ? = >

'

( ! ! N + $ + =+ > = > 6 = > ! B ==+ > ? = > ? 1 >

Nowhere-Zero Flows DEFINITIONS

' : ' =%> , = > ? = > =%> ¾ ¾

%

6

'

' : ' = > - : % !

=%> = > , = > '? => 6 ' )! % ' )! ( => ? 1 = > ' ' )! => = > ¼

¼

CONJECTURE

)* +, 4 )! -64 0

=FQ4(> A !

FACTS

' '

-30 A ! 7 )!

-64 0 " => ! )!

' ! )!

! )!

© 2004 by CRC Press LLC

226

Chapter 4

CONNECTIVITY and TRAVERSABILITY

' 6 FQ4( . !

'

-9/0 : ' = > ! ! B

' = > 6 # + = >

.

6

# => + => ( = >

¾ ¼

= >

¾

= >

# =>'

¼

REMARKS

'

. K A ! ! # -2Q/ 0 -Q/90 )!

' F! )! ! = = >> => !

4.2.5 Various Types of Eulerian Tours and Cycle Decompositions DEFINITION

' : % " = >

%

% FACTS

'

-J470 : ! # = > ? 6 5 ! ½ = > ? ¾ = > U ! !

'

-3 ((/10

&

'

: # # !

'

6

-A840 : %¼ !

% ? =%¼ > 6 # %

© 2004 by CRC Press LLC

Section 4.2

227

Eulerian Graphs

'

- (/40 : ½ = > ! ? ½ #

= > / U = > / ! /

REMARKS

' '

( 3 : =( >

( / # = -Q/90>

' % -(+3/ (/10 *!

%¼ ! = =%¼ >> + (

= 8A6 6 -( 30> Incidence-Partition and Transition Systems DEFINITIONS

' ( # => ?

6 = > ?

Ú => ==>>

¾

' 0 = > 0 = > ? => = > => ?

=> A => ' + 0 0 ! A 0 0 ! + 0 0 + ' : = > = > = = >> => = > = > , ' + 0 0 ?

¾

6 C D C D

' = > ,

# = > , = > => = > => = >

© 2004 by CRC Press LLC

228

Chapter 4

CONNECTIVITY and TRAVERSABILITY

FACTS

' -J730 = > Ú = >

=

> ,

+ > # +

' -(310 2

=

' -(310 :

=

! =>

$

" => 1 > #

' -((/10 :

=

> 6

! > = > ,

=

' -(310 : " => $ = > EXAMPLE

' 6 ( 7 ! 0 = > ? E ¼ = E >¼ ' 4 7 ? 0 = > ! ! ( 7 1

5

3’ 5’

1’ 2’

2

4’

4

3

Figure 4.2.6

/! 0 =>"! $ $ '

REMARKS

' 6 + ! => 1 = > H - 3/0' W%

W% ! ' ( 4 ! ( 3 =( > ' (

( + = # > &! '= * . "

. 5 F! Q 4 (! 5 -(:3 (33 (1 (10

© 2004 by CRC Press LLC

Section 4.2

229

Eulerian Graphs

' ( 7 ! # # ' ( ! ( 7 &

' + ( 7 9 # ! = -Q/90>

' = > -(/10 F ! ., # = > Orderings of the Incidence Set, Non-Intersecting Tours, and A-Trails DEFINITIONS

' 2 # ,# B ½ Ú Ú Ú 1 => " 1 => &!

' : # ! => ! Ú 1 => ? Ú 0 = > 1 => ! 2 0 = > 6

0 = > 0 =

>

'

: ! 1 => 0 = > 0 = > ! 1 => ! => + 0 0

'

: ! 1 => 0 ? E ? = =>>

' ! # #

'

= >

EXAMPLE

'

B ! ( 9 ! 7 1 3 / 9 4

© 2004 by CRC Press LLC

230

Chapter 4

CONNECTIVITY and TRAVERSABILITY

2

1 11

10 9 8 7

12

6 4

5 3

Figure 4.2.7

" $ "

FACTS

' 2 #

1 => =

' " ! => B

=

>

>

' -(/40 6 ! - ' -(/30 U ' :

# Æ = > 3 ! 3

# ! # 5

# 6 REMARKS

' ( 3 : =( > B : , = > ' 6 ! 0 = >

= > ' ! U : &! , 0 = > = 0 = >> 5 0 = >

© 2004 by CRC Press LLC

Section 4.2

231

Eulerian Graphs

4.2.6 Transforming Eulerian Tours The Kappa Transformations 6 &

6

( -(/10

DEFINITIONS

'

6

? ?

+ + + '

:

?

?

B

'

=

=

?

=

+

'

:

?

=

> ?

'

¼ =

>

+

=C ! D>

?

=

>

>

!

=

>

=

> #

:

=

? =

> 6 ! !

+

>

2

6

+

6

>

> 6

6

?

'

=

> ?

> ?

=

A ¼¼= >

¼ : £ ¼ £ ¼ ? = > # = > ? ¼ ¼¼ ¼ £ ¼ ¼¼ + + ? + + 6 ? = > ? = = >> '

'

:

!

'

6!

0½ 0¾

?

@

© 2004 by CRC Press LLC

=

>

6

?

=

>

£ ?

=

>

@ ?

232

Chapter 4

CONNECTIVITY and TRAVERSABILITY

REMARK

' 6

, !

# *

FACTS

' : ½ ! @ > = -J310 -&30 -(/10>

B

= #

' : ! = >

@ = > 6 B ! + ! + ? B = > ! 1 => > @ 6 B ! + ! + ? B

' :

=

' : ! @ B

6

=! > ! = >

# 6

' "

EXAMPLES

' 6 ! ? 3 =!

B> ! ( 3= > 6 & ! 6 ¼ ? 3 9 7 4 = > = ( 3=>>

Figure 4.2.8

' 6 A# 4 8

+ ? ! ?

? 4 7 9 3 =! B> =( /= >> © 2004 by CRC Press LLC

Section 4.2

/=>>

233

Eulerian Graphs

¼¼

? 9 3 4 7

=(

Figure 4.2.9 Splicing the Trails in a Trail Decomposition

+ ! 6& ! = > Algorithm 4.2.5: 0"& $!"' (*

#' 5'

: ? ½ + ! ? = > = > ? : ¼¼ : ? = > '?

'?

References -J310 H J 6 A $ =/31> 74K7/ -2Q/ 0 8 : 2 O Q 2 ! $ % ) =// > K4 -(/40 : . * ( 6 F , ,

$

=//4> 1K -(/30 : . * (

$ =//3> //K

© 2004 by CRC Press LLC

234

Chapter 4

CONNECTIVITY and TRAVERSABILITY

- (/40 $ * ( , + =//4> 9K -;/0 2 ; ;

" $ $2 ! * // -A840 6

A F 2 8 5 % %8 =/4> 1K9 -A970 : A =97> 9 3K 1 ? ; " L 9 K1 -(330 ( .# + 33 49K7

-(310 * ( A : J I & . I J + - ) =/31> 4K79 -(3 0 * (

. H 8 N $ =A> /3 K 7 -(370 * ( 5 +

-) =/37> /K1 -(330 * ( ! ' =/33> 3K 3

-(3/0 * ( A = > - $ % /39

$ =/3/> F K 4K/ -(/10 * ( * ) $ F * //1

/

-(/0 * ( * ) $ F * //

/ = .

.

8 1 * ) )57('1 %

$ . =A> J!

-(110 * (

111 /K39

-(10 * ( = > 6 = > $ =11> K -(10 * ( 8 % 5 $ =11> 99K3 -((/10 * ( ( & ; + - =//1> 4K4

© 2004 by CRC Press LLC

Section 4.2

Eulerian Graphs

235

-(2340 * ( $ 2 ; ! =/34> 7K79 -(+3/0 * ( A + $ , # =/3/> 44K71

%

-((70 : ( . (& -. '. N FH /7 I -* 390 * N $I & : + N $ 12 =39> 1K -J470 J A ? ! B =

&> $ -> %8 =/47> F K7 ? -J730 J $ ! $ -> =/73> F 97K31 G : )4 4 $4 (/ 2 KL , -:3/ 0 $ A 3/ -$3 0 6 $J ' " ! $ =/3 > 9K -F 9/0 H F + H + =A> ; N U $ J /93 ) ' $ ( =/9/> 39K/9 -F 34 0 H F + . A % 8 " =A> /34 $ % 0 ' % N : =/34> 9K4 -F 340 H F + A + 0 $ % =/34> F 9K/ - 3/0 H . 6 I 2 $ =3/> /K1 - 990 2 6 ! & & $ 5 ) @@/ * =A> ( ; N * K /97 -9/0 . ) H 8 N $ =A> F! <& /9/ K47 -3 0 . A + - =/3> 9K3 -30 . F! 7 )! + - =/3> 1K4 -9/0 * & =/9/> 19K13

© 2004 by CRC Press LLC

236

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-&30 . J & A

- % J $ J * < =A> $ /3 : F $ 8 F! <& /3 3K4 -6 3/40 2 6 : P '8 $ =3/4> 39K/1 -6340 6

=/34> 7K73 -690 6 A + - ( =/9> K 4 -6770 . H 6 ; $ $ =/77> /9K // -6970 6& ! A $

$ =/97> 73K7 1 -64 0 + 6 6 +

$ =/4 > 31K/ -6 0 + 6 6 8 ; !& $ $ =/ > K9 -L0 ; L B $ 34 =/M> 37K/ -L0 ; L $ % "" =/> K/ -L940 . L

. A $ =/94> K -+/10 . + $J% $ =//1> 9K1 -S10 H S % ( '. F!& 6 J! . 11 -Q/90 O Q ( -. 8 $ .&& " F! <& //9

© 2004 by CRC Press LLC

Section 4.3

4.3

237

Chinese Postman Problems

CHINESE POSTMAN PROBLEMS

! "

#

6 8 " L N . $ #

Introduction 6 !* ! * = > " #

!

6 , 2 = J! $ J> /7 -270U

C D H & A =-A74 0>

4.3.1 The Basic Problem and Its Variations DEFINITIONS

'

! &

'

2 , ? = > ! ! ' & & & !

.

6

' 6

' 6

&& - && && ! &&

' 6 && $ && *,' FACTS

' '

N .

= > $ F = >

© 2004 by CRC Press LLC

238

Chapter 4

CONNECTIVITY and TRAVERSABILITY

The Eulerian Case DEFINITIONS

'

= > ! & , # ! &

= >

#

'

#

'

! &

'

"

? =

B

" #

!

> #

( 3 3 (

& =

*! # C D

>

=

>

( 3

FACTS

'

#

'

'

"

= #>

REMARK

'

# $ #

Variations of CPP DEFINITIONS

'

'

6 B

= >

'

. '

,

; '

'

. '

, B

= >

'

'

" N

6 !

# CD = !

>

" ! ! @ !

'

'

&' * *

6

! B

© 2004 by CRC Press LLC

Section 4.3

239

Chinese Postman Problems

6 ! ! ! => # !

'

'

6 #

FACTS

'

6

"

½

¾

, = > B , #

!

4

!

4

½

¾

! = , > !

Æ

, ! !

"

U

* >

'

½

! ,

¾

&"

=

(

! , B

=

'

-AH90>

-+ /0 6 ! F

'

6 F

M ! = -2 H9/0>

REMARK

'

" # #

!

# -A 2: /4 0 -A 2: /4 0 A#

-(/0

4.3.2 Undirected Postman Problems 6 2 ! # = >

6

) ! ! A =-A74 0> !

DEFINITIONS

'

$

# =$

'

© 2004 by CRC Press LLC

! !

>

#

240

Chapter 4

CONNECTIVITY and TRAVERSABILITY

Algorithm 4.3.1: ,$ 5677 #' 5' $ !

! !

: ! ( ( 3 ( ! ( 3 ( # ! ! ( ! 4 ( 4 . :

6

! ! @ ! !& A = -A74 0-A74 0> 6 , 6 ! ( = -J 790> ' 2 ! & #

! =(%

> EXAMPLE

' ( U ! , 6 ! !' L#

: 4

Figure 4.3.1 $ $!"'

! ( ! 4 6

© 2004 by CRC Press LLC

Section 4.3

241

Chinese Postman Problems

½ ½

6 ! ( 6

= ! 1> ! & !'

REMARKS

' 6 !

# ! ! ! & = & > ! = -2 H9/0>

' " ( B

B # ! A H =-AH90>

' 6 # ! & N ! * , ! ! , ' !

! ! &

" !

8 ! ! ! = > ! ! !

4.3.3 Directed Postman Problems 6 . " !

6

FACTS

' ! &

' 6

! U

6 & " /& ; = >

© 2004 by CRC Press LLC

242

Chapter 4

Algorithm 4.3.2:

CONNECTIVITY and TRAVERSABILITY

,$ 677

#' ! !

5' $ ! " A

( ? 5= > )&5= > ! '

¾

(

(

¾

( 1

¾

( ?

( ( = >

Producing an Eulerian Tour in a Symmetric (Multi)Digraph

B = > 6 ! = -A840> DEFINITION

' ! # Algorithm 4.3.3:

7"! " $" !"

#' A 5' A

# £ ( £ ( # ? £ : ! 5 = > : £ # £ ! !

!

? £ ' # = > = > # U © 2004 by CRC Press LLC

!

Section 4.3

Chinese Postman Problems

243

EXAMPLE

' ( , #

#

# , ! ' ( ? ( ? U ( ? U ( ? 1 ! ! , = > # ! ( 6 ! Æ# ! # , = > ! # B'

Figure 4.3.2 $ $!"' REMARKS

' 6 !& )! A H =-AH90>

' 6 B ' " B ! # # ( # (

!

4.3.4 Mixed Postman Problems FACTS

' 6 # $ - U 6"("8":"6< = - 970>

© 2004 by CRC Press LLC

244

Chapter 4

CONNECTIVITY and TRAVERSABILITY

' $ - ! # = > # ! ! = -2 H9/0

Deciding if a Mixed Graph Is Eulerian DEFINITIONS

' 6 #

' '

#

# # # B

#

' # , + = > @ ! + = > + = > + + 5 +

= > + = -((70> FACTS

'

=> # ,

'

$ #

EXAMPLE

'

# ! # 6 ( , # B ½

Figure 4.3.3

$" 89 +"

6 = > B #

, # ! 6 ! & ! , 6 !& )! !

© 2004 by CRC Press LLC

Section 4.3

245

Chinese Postman Problems

! 89 +" 2 $"

Algorithm 4.3.4:

#' # 5'

( ? 5= > )&5= > : ! ! !& )! '

¾

(

(

1 (

( ?

= >

" = > ( " ( ? ; A ( ? ; A : A = >

EXAMPLE

'

6 = #> ( ( " ,

! # # 6

( ( , = > = > ! , 6 # !

Figure 4.3.4

$ $!"' !" !"

© 2004 by CRC Press LLC

246

Chapter 4

CONNECTIVITY and TRAVERSABILITY

REMARKS

'

6 ., & & = @ ! > 5

'

" !

= > ! =! @> # ! = -AH90>

The Postman Problem for Mixed Graphs

$ F ! + & ! !

,* * ! ! , ; # *! ! B @ # ! = -A 2: /4 0 -A 2: /4 0> DEFINITION

' , REMARK

' " $ !

# " = ( 4> ! ;! ! M FACT

'

#

6 # = > 6 , # ! ( & =-(9/0>

© 2004 by CRC Press LLC

Section 4.3

247

Chinese Postman Problems

EXAMPLE

' 6 B # # ( 4 U ! " # ! F! ! ! ,U ! B ; ! ' U N !

Figure 4.3.5 2" : ''" /!"

" ! ! !

!& )! U

-AH90 " # @ ! ! M

Approximation Algorithm ES

6 ! # ! 5 M 6 ! = -AH90 -(9/0> 4 C M D Algorithm 4.3.5: "9' $!"' ,

#' #

5'

! M !

N # , : ; M :

6 # 4 /#

© 2004 by CRC Press LLC

248

Chapter 4

CONNECTIVITY and TRAVERSABILITY

REMARK

' = > & A " ! ! ( & FACT

' -(9/0 6

A

# "

A# 7 ! EXAMPLE

' # ( 7 ! ! , 6 A = > ; ! ! E 6 6 ! * ! ! 6 , ! " ! E 16

Figure 4.3.6 $ "9' $!"' ,

Approximate Algorithm SE

# A ! =#/> = -(9/0>

6

7 = -(9/0> M 4

© 2004 by CRC Press LLC

Section 4.3

249

Chinese Postman Problems

Algorithm 4.3.6: "9' $!"' ,

#' #

5'

! M !

# # : N : = >

EXAMPLE

' ( 9 A *!

Figure 4.3.7 $ "9' $!"' , Some Performance Bounds FACT

' -(9/0 6 A

! # A# 9

! EXAMPLE

' ( 3

A ! A ! ! A

Figure 4.3.8 $!"' , , "& :"

© 2004 by CRC Press LLC

250

Chapter 4

CONNECTIVITY and TRAVERSABILITY

REMARKS

' " ! # 4 =A>

7 =A> *! A# 3 ' # ! A = > ! A ! ! ! A

' 6 A# 3

B !

!' + A A ! @ T "

C D ! + !& @ B @

6 , ( & =-(9/0> ! ! ! ! ! !

# ' " =-(9/0> ( & 6 ! !

' +

-(9/0 ! U ! ! ( / ! 8 /// ! , ( & # !

L = L//0> ! 6 ( /

Figure 4.3.9 :" " ' $!"' , ,

' (! $ &

*! !

F #

! B &

6 $

" , '

$ B @ = -8 6/0 -8 6/0 1 > 6 - - &'-

© 2004 by CRC Press LLC

Section 4.3

Chinese Postman Problems

251

References -889 0 A : 8 : . 8 F!& L $ + '. =/9 > 74K/ -8 6/0 8 8 & 6 . . 2 %($ + $ =//> 3K41 -8 6/0 8 8 & 6 2 : 6 . 2 ( 9 =//> 444K43 -8310 8& 6 $ # F!& ( < % 0 ' % AA L F! <& =/31> 4 K77 -8 $3 0 , A 8 L A $ ; $ $ # % $ 5 > 0 ' ( % :! F! <& =/3 > -A74 0 H A 6 5 ) =/74> 9 -A74 0 H A$ # $ ! 1 L + ) ' , % 7/8 =/74> 4K1 -A74 0 H A 6 (! + $ 9 =/74> /K 79 -AH90 H A A H $ A 6 $ 4 =/9> 33K -A840 6

A F 2 8 6 ; : 2 % %8 3 =/4> 1K9 -A 2: /4 0 A $ 2 2 : "' 6 5 ) =//4> K -A 2: /4 0 A $ 2 2 : ""' 6 5 ) =//4> //K -(/0 * ( A 2 6 L $ 41 F * =//> -((70 : ( . (& -. '. N FH =/7>

© 2004 by CRC Press LLC

252

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-(9/0 2 ( & # +

$ 7 =/9/> 43K44 -2 H9/0 $ 2 . H ( 1 ' +* ( F! <& =/9/> -270 $ 2 2 N A ; $ =/7> 9K99 -23 0 $ 2 + $ ) *6 =/3 > K/ = > -23 0 $ 2 ; +

$ / =/3 > K 7 -J J9/0 J 2 J 6 $ #

$ =/9/> 3/K1 -J 790 J - F! <& =/79> -: Q330 < : < Q F! . 5 ) 4 =/33> 499K43 -$ 9/0 A $ & 6 $ # F!& $ % 4 =/9/> 7 K7 3 -F /70 < F H ; $ # '. 9 =//7> /4K13 -;9 0 ;@ ( L '. 4K7

=/9 >

- 970 * ; # A 6 + $ =/97> 4 K44 - / 0 + :

& $ # F!& 5 ) 0 7 =// > K - :/40 + : $ : $ # F!& 5 ) =//4> 9/K 3/ - L//0 8

H L ¾¿ # $ # %($ + $ =///> 4K - /0 6 J ; $ # 5 ) 0 =//> K9 -+ 3/0 Q + ; + A 2 $ =/3/> /9K

© 2004 by CRC Press LLC

Section 4.4

4.4

253

DeBruijn Graphs and Sequences

DEBRUIJN GRAPHS AND SEQUENCES $ %&' ( ' .8 5 2 8 2 8 5 B F 2

Introduction F 8 5 , = > ( ! 8 5 " ! 8 5 ! 8 5 B

4.4.1 DeBruijn Graph Basics DeBruijn Sequences DEFINITIONS

'

5 , & #

? !

6! 8 5 B C BD

' " 7 / $ & &¼ & 6 ! ¼ 7 & , 7

' " 7 / $ B ! ! 7

/

' ? ½ ¾ ¿ ? ·½ ? 6

' '

½ ¾ ½ ¾

© 2004 by CRC Press LLC

½ ¾

½ ¾

? ½

?

½

254

Chapter 4

CONNECTIVITY and TRAVERSABILITY

FACTS

'

! 8 5 B @ B

' 6 8 5 &

& 8 5 B

EXAMPLES

'

1111 8 5 B " 111 11 11 1 1 1 11

'

11111111 8 5 B

DeBruijn Graphs

8 5 B

! B DEFINITIONS

'

=> !

! B L# 5 #

8 5 =>

A ,

# ! ! # !

' 6 8 5

' 6 8 5 8 5 EXAMPLE

'

( ! 8 5

FACTS

' 6

8 5

' A # 8 5 6 , ! 1 ,

' '

A # 8 5 A 8 5

© 2004 by CRC Press LLC

Section 4.4

255

DeBruijn Graphs and Sequences 0000

000 0001

1000 1001

001 0010

100 0100

010 0011

0101

1100

1010 101

1011 011

1101

0110

0111

110 1110

111

1111

Figure 4.4.1 -". !" ""

' A 8 5

' 6 = > 8 5 => ! 8 5 B 6 B , # ' -".& "' -8 90 ( 8 5 B

½

¾

4 7 7 1 3 791337

¾

½

REMARKS

' 8 5

8 5 *! 8 5% ! 8 5 B

' 8 5 8 5 => ! 8 5 ! #

4.4.2 Generating deBruijn Sequences Æ 8 5 B , 8 5 & A 8 5 #

© 2004 by CRC Press LLC

256

Chapter 4

CONNECTIVITY and TRAVERSABILITY

FACTS

' -2 70' ! # A ' " 8 5 => B #

# =6 B , , 8 5 > ' 6 B A 8 5

8 5 B E

EXAMPLE

' ( 8 5 B 8 5

0000

000 0001

1000 1001

001 0010

100 0100

010 0011

0101 1011

011

1100

1010 101

0110

0111

1101 110 1110

111

1111

Figure 4.4.2 $" "

=>

REMARKS

' 6 ( ( Æ A =>

# ! ! ( ( ! B 8 ( ( ! W1%

W% ALGORITHM

' 6 8 5 B (% =B > A 8 5 = > 6 B A =(% >

© 2004 by CRC Press LLC

Section 4.4

257

DeBruijn Graphs and Sequences

Necklaces and Lyndon Words

( & J -(J990 & 8 5 B DEFINITIONS

' = B

>

' B

' 0 & ! : & + & #

B

& FACTS

'

& : ! ! !

' 8 8 & 8= >

! 8= >

- 0

'

-(J990' " =# > : ! ! # ! 1 8 5 B # REMARK

' 6 - = > & ! # ! + - =1> ? 13 - =4> ? /

-

=4> ? 3 !

EXAMPLES

'

( , B

4 01101 11010

0

1

1

10101 01011

0

1

10110

Figure 4.4.3

'

0$ ""

6 : ! 1 + 11 8 5 B

© 2004 by CRC Press LLC

258

Chapter 4

CONNECTIVITY and TRAVERSABILITY

'

6 : ! 11 1 + 1111 8 5 B

'

+ ! ( (4 ? : !

" #

111 11 1 1 " ! ! ! ! # 8 5 B 11111111

4.4.3 Pseudorandom Numbers ( $ = >

@ 8 $ B .8 5 B ! ! DEFINITIONS

' B B

' -2790 6 1 B

!'

6 %

@ 1%

B 6 ! 5 = E > ! ! = E 11 11>

' B ! ' 1 2 8 5 B # 1% # %

'

-390 6 .

+

,

= >

7 / /

! 7=/> 6 , / B +

© 2004 by CRC Press LLC

Section 4.4

259

DeBruijn Graphs and Sequences

FACTS

'

; 8 5 B

B

'

8 5 B , 2% , ! (

% # B 1% !

B #

'

" = # > 8 5 B

&

;

B 2 5 8 5 B ! ! B

'

-H/0' 6 !

B

8 5 B

4.4.4 A Genetics Application 6 .F # Æ 8 # # !& 6

.F ! 6 #

, B # ! .F B B #

6 Æ 6

+ - 6 + 10 , 8 5 !

B 7

B

DEFINITIONS

!)

'

'

( + ?

, B 2 6

½

7 7¾ 7

.F B ! , +

+

6!

5 .F B 7 ! ,

REMARK

'

A B & +

8 5 Æ !

6 = > B

! # ! @ + 8 5

© 2004 by CRC Press LLC

260

Chapter 4

CONNECTIVITY and TRAVERSABILITY

References -;0 " & & : ! 8 5 B

!!! M M M&MF& M -390 2 H ( N /39 -. 70 F 2 8 5 ' <

=/ 7> 943K97

/

-(J990 * ( & " J :# 8 5 B + % =/99> 9K1 -H10 H H ; B /7K1 8 * B!A = : 8 . > L 11 -2 H9/0 $ 2 . H ( 1 ' + * ( X /9/ -2790 + 2 % ) % * . /79 -2 70 " H 2 F + 0 $ % =/ 7> 79K 9 -22$ ://0 H ( 2 Æ + $ 2 H * $ :! $ + * ( /// -2<//0 H : 2 H < (

/// -* 790 $ * H 8 /79 -$ 10 Q $& A ' ( # C % , 9 =11> //K1 - 6 + 10 * 6 $ + A .F ' % /3 =11>

© 2004 by CRC Press LLC

Section 4.5

4.5

261

Hamiltonian Graphs

HAMILTONIAN GRAPHS '

' )*

4 * 4 6 & 4 A# 4 $ 6 ; * T 44 2 47 (

4.5.1 History F = -2 H9/0> ! F + ! * 341 * # # *

. 349 6 ,

6 ! & ! 34/ ! , ! * % 4 * , B " -J 470 344 6 J & B ' 2 ! , => # 6 J & & B * N J & ! ( -8 :+ 370 DEFINITIONS

' ' '

= >

4.5.2 The Classic Attacks 6 @ 6 & Æ U #

© 2004 by CRC Press LLC

262

Chapter 4

CONNECTIVITY and TRAVERSABILITY

6 # ( Degrees

6 # Æ = >

Æ =

>

DEFINITIONS

' + = > ! 6 = >

' 6 ! = > 5 5 !

' ( ' ? = 9 > = ? 9 > 5 5 ( 3 9 ! E

6 ! '

: =

>?

/ (

(½ (

FACTS

' '

-. 40 " :

-;710 " : = >

Æ = >

: = >

-;70 " := > E

EXAMPLE

'

! ! # , = ( 4> 6 * Æ = > ? =* > := > ? * . % 6 ;% 6 =( > 6 ! Æ = > ? * : = > ? * ! ; = ( 4>

Figure 4.5.1

' -H 310 :

2$$" ! " " & ;"& "$

© 2004 by CRC Press LLC

!

Section 4.5

263

Hamiltonian Graphs

' -$$70 " ? = 9 > = > ! => E => E 5 9

'

-8970 :

! =

6

> ! ,

!½=

!=

>

>

' -*/0 REMARK

'

6 # 6 => & , *!

! " #

1 6 ! ! ! =! >

Other Counts DEFINITION

' 6 # ( - =(>

5 ( - =+ > + 5 # +

! "

+ ! ,

' 6 = > !

#

'

= >

EXAMPLE

' 6 =5 *> ! * # + ! + ? ? 5 ? * 5 ! ! 5 + * + Y ¾ ( 4 ! ! = 7> = 4> © 2004 by CRC Press LLC

264

Chapter 4

Figure 4.5.2

+"

= 7>

CONNECTIVITY and TRAVERSABILITY

= 4>

FACTS

'

-;70 " ½ E ( ! # E

= > = 4> "

'

-( 3 0 "

/ / (=> => = > ?

' -8 8L: 3/0 " := > E = >

'

-A90 :

=

>

= > = > " = > = > " = > = > E ' -+930 " + - =+> ' -(370 : " # & + ! & ! - =+ > "

'

-8L/0 -( 2H :/0 "

- =+ > + ! REMARK

'

6 #

1

Powers and Line Graphs

! & = >

DEFINITIONS

' 6 ;= > ! ! ! ! ;= >

5 5 = >

© 2004 by CRC Press LLC

Section 4.5

265

Hamiltonian Graphs

' !

#

!

' + 5 = ½ > 5 ' 6 ! = > ? = > ! = > = >

'

= # 5 > = >

"

FACTS

' -* F+740 : ! 6 ;= > ½

' -2*//0 : ! 6 ;= >

! = >

'

' '

-+ 90 " -(9 0 "

" -8930>

! Æ = >

; =

> ? ;=;= >>

= > =

Planar Graphs FACTS

'

-630 A = -6470>

' -2730 : ! ! " 5 ! 5¼ # ! = >=5 5¼ > ? 1

4.5.3 Extending the Classics Adding Toughness DEFINITION

' " # + + ! & &

, & =+ > + ! =+ > & 6 #

FACTS

'

-H930 :

© 2004 by CRC Press LLC

:= >

6

266

Chapter 4

'

-8 $L/10 :

CONNECTIVITY and TRAVERSABILITY

'

-8L/10 : 6

:= > 6

! Æ = >

REMARK

'

G 5 & & ( & ? *! -8 8: L110 # =/ 6> 6 $ 1 !

More Than Hamiltonian DEFINITIONS

' '

2 2

' ! / # / E =! > ( # #

2

' = > B = > B

FACTS

'

=> Æ = > => := > !

-8( 2:/90 "

EXAMPLE

' 6 ! 6 FACTS

'

-8990 "

'

! = > ¾

-*/10 " := > # ! : = > = 4> # ( Æ = > = E > #

'

-*/0 " ? = 9 >

5 ( 3 9 ! =(> E =3> E

© 2004 by CRC Press LLC

Section 4.5

267

Hamiltonian Graphs

' -*/0 : / " ? = 9 > Æ = > / = > $ ¾ / E /¾

' -J /70 -J /30 6 # Æ = > = E > ! ' -J //0 : " => E # ' -( 2J: 0 : ! " => E => E = /> 5 REMARK

' 8 !

N# . ! ;

4.5.4 More Than One Hamiltonian Cycle? A Second Hamiltonian Cycle FACTS

' A 6 = -6 70>

' -6/30 "

/ ! / 11

! ! : # ! ! '* =! > = - => = =! >>> 6 ! ¼ ! ¼ ? ! # ! ! ¼ ! !

' -6/90 :

' -*110 ( # =>

! Æ = > => Æ = > Æ E " =Æ = >Æ = >>

' -$ 970 -2$ 970 6 #

! 5

!

Æ =

>

' -Q 970 -3/0 6 # , # 4

= > !

© 2004 by CRC Press LLC

268

Chapter 4

CONNECTIVITY and TRAVERSABILITY

REMARK

'

( 6 -6930 # % 5 ! 5 = ! > 6 # =( >

Many Hamiltonian Cycles FACTS

'

-6/70 :

! ' (½ 3½ (¾ 3¾ ( 3 (½

= > " 3½ 3 (½3½

=> " 3½ 3 $ ½ ¾ =1 ' > ! 3 ½ (3 ·½ = '>V ½

'

71¾ :¾ = > E

-( 340 :

= > "

5 => " 7 ¾ ½ E 5

' -A/0 : ! = > " ! :¾= > Æ = > 5

Uniquely Hamiltonian Graphs DEFINITION

'

#

FACTS

' -A!310 6 # , B !

'

-H +3/0 B # = E /> B #

'

-8H /30 A B #

¾ =3 > E ! ? = ¾ > ½ ( B !

© 2004 by CRC Press LLC

Section 4.5

269

Hamiltonian Graphs

Products and Hamiltonian Decompositions DEFINITIONS

' = E >

'

A ! & # = ½> = ¾> 6 ? ½ ¾

=

> ? =½ ¾>=½ ¾> ½ ? ½ ¾¾

=

6

=

> ? =½ ¾>=½ ¾> ½½

?

=

¾> ½ ½>

¾ ? ¾

¾

½½ =

?

¾ ¾

=

¾>

¾ = > ? =½ ¾>=½ ¾> ½ ? ½ ¾¾ ¾ ¾ ? ¾ ½½ = ½ > ½ ½ = ½ > ¾¾ =

6

½>

½

¾>

6 = ! > ? ½- ¾0

=

> ? =½ ¾>=½ ¾ > ½ ½

=

½ ? ½ ¾¾ =

½>

¾>

REMARK

'

H & -H 9/0 5 E B ' " ½ ¾ ½ ¾ T

FACTS

' -/0 : ½ ¾ ! 7 & ! & 7 6 ½ ¾ ! '

7 & => & =>

=>

¾

= >

½

'

77&

" ½ ¾ * ½

¾

'

-8/10 -Q3/0 ½ ¾ " ½ ¾

© 2004 by CRC Press LLC

270

Chapter 4

CONNECTIVITY and TRAVERSABILITY

'

-( : /30 6 ½

'

-8 30 6 # !

' -J/90 = > "

=> =>

= > - 0 " = > - 0 " = > - 0

=> " =>

= > - 0

= > - 0 " = > - 0 " = E > = > E - 0 " # = > - 0

=> " =>

#

4.5.5 Random Graphs + 5= > ! - ? DEFINITIONS

'

=3 > 1 * : - ! * ' =3 . ( > 4 ? 4 = > ! &

6 7 ? @ ! 4 # + ! 7

' ! @ B = >

& & ? 1 -

' " Z ! Z " 5="> F B

"

' 6 4 2 %

# 2 6 %

© 2004 by CRC Press LLC

Section 4.5

271

Hamiltonian Graphs

FACTS

'

- 970 -J970 6 # ¾ ' -J970 -J30 <= > * ? E E <= > 4 = > ? E E <= > 6

'

4 = > ? = E

1 E >' 5= > ? ' ' ' -+/ +/ 0 ( 5 5 ' -(/ 0 % ' -(110 % " % ; % % -J30 (

REMARKS

'

" - V =! B > ! C D = *> AI ! , 5 ! & 6 ! , 8G

' " & ! # !

, 8G

( ( -8((340

'

B

4.5.6 Forbidden Subgraphs DEFINITION

'

= > 6 - # 5 ! => 6 => 6

; ! # 5 5 # # = &>

Figure 4.5.3

© 2004 by CRC Press LLC

!" - ;

272

Chapter 4

CONNECTIVITY and TRAVERSABILITY

FACTS

'

-.2H 30 "

½ -

= >

=>

'

U

-8.J110 6 # ,

-

'

-8L/10 "

'

-2H 30 "

'

-8/0 "

'

-( 2/40 "

-

-

-

1

Other Forbidden Pairs B ' T 6 ! -8/0 -( 2/90 1 + !

FACTS

'

,

?

+

+

6

, +

-

, +

?

=

-

>

-

=

-

!

1>

'

-( 2/90 :

, +

,

'

?

+ 9

?

1 6

-2: 0 :

9

?

= > A

, + ? > , + - -

=

=>

,

-8/0 -( 2/90 :

9

'

9

?

!

>

?

;

-( 2/90 "

!

Claw-Free Graphs " !

=

6 ! B

'

9 - - - - ?

6 B '

!

" !

T 6 ! ! -( 2H :10 ! !

!

&

Æ

( -( 2H 0

Æ ! 8& -810 ! ! " -( 2H 0

!

© 2004 by CRC Press LLC

Section 4.5

273

Hamiltonian Graphs

DEFINITIONS

' ( # ( -- =(>0

( -- =(>0 =- =(>> =; ! ! >

' 6 ! 2= > , # (

' 6

5/=

>

FACTS

'

-( 2/90 : , + =, + ? > 1 6 , + # , ? + - -

-

' -/90 "

'

-/90 :

2=

> ! ,

=>

5/=

! 6

= > =>

> ? 5/=2= >>

2=

> ?

REMARKS

'

6 ! @ = -8970> ( -8110

' 8 ( 73

!

2=

>

'

6

! ( -8930 -8930 -+ 2 3 0 -8/40 -2 /70 -2/0 -210

References -8 30 Q 8 2 * # + % , =/3> 4K7 -8 8: L110 . 8 * H 8 : * H L F

$ // =111> 9K -8 8L: 3/0 . 8 * H 8 * H L :

* & F + % , 9 =/3/> F 9K

© 2004 by CRC Press LLC

274

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-8 $L/10 . 8 $ A * H L : ! $ 9/ =/3/M/1> 4/K91 -8970 H 8 L G

=/97> K4

$

4

-8( 2:/90 8 2 H ( H 2 : : & ; + =//9> 74K9 -8/0 8 - . 6 $ N //

-8930 H 8 * % 8 & + A : =/93> -8 :+ 370 F : 8 A J : H + N ;# =/37>

:

DE" !E" ;#

-8((340 8 8G 6 " ( $ ( ; $ % F! <& /34 1K / -83 0 8 8G

6 37 =/3 > 49K9

$ %

-8H /30 H 8 8 H & L B + % , 9 =//3> 74K94 -8990 H 8

+ % , =/99> 31K3

-8930 H 8 * ! % *

F SS" =/93> K3 -8/40 H 8 8 K ( A =//4> 4K1 -8/10 H 8 &

J! =//1>

-810 H 8& ( $ 4 =11> 9K97 -8.J110 8 I ( ( . A JI : = ! > %($ + 1 =111> 77K 799 -8110 * 8 Q 5G [& " ' 7 =111> 9K 3 -8L/10 * H 8 * H L ½¿ $ 8 & A 8" + L $ + Q =//1> 3K /

© 2004 by CRC Press LLC

Section 4.5

275

Hamiltonian Graphs

-8L/0 * H 8 * H L : ! + 4 =//> /K3 -+ 90 2 A + ; # % $ 3 =/9> K 3 -A90 L G AI =/9> K

%

$

-(/ 0 $ ( * + % , 7 =// > 4K7 -(110 $ ( * ) % 7 =111> 7/K 1 -2 /70 H H 2 *

$ 47 =//7> K3 -. 40 2 . =/4> 7/K3

0 $ %

-.2H 30 . .@ H 2 $ H (

2 H . 2 : : & . : & =/3> /9K7 -A/0 < A ! A 5 ; / =//> 4K41

%7 + $

-A!310 A * ! +

% , / =/31> 1K1/ -( 3 0 2 * ( F! Æ , 9 =/3 > K9

+ %

-( : /30 ( H : * / =//3> 4K44

+

-( 2/90 H ( H 2 $ 9 =//9> 4K71 -( 2H 0 H ( H 2 $ H ( ' -( 2H 0 H ( H 2 $ H ( ' Æ -( 340 H ( A 5 .

% =J $ /3 > + F! <& =/34> K / -( 2H :/0 H ( H 2 $ H : : & ;

. % $ 14 =//> 7K9

© 2004 by CRC Press LLC

276

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-( 2H :10 H ( H 2 $ H : : & ! $ / =11> 9K3 -( 2J: 10 H ( H 2 J& : : & "

. + F =11> //K1 -( 2/40 H ( H 2 Q 5G [& " ( ' 1/ =//4> K -(9 0 * ( 6 B ! + % , 7 =/9 > /K -(370 ( ! Æ + 1 =/37> 14K 1/ -2 H9/0 $ 2 . H ( 1 ' ( F! <& =/9/> -2: 0 H 2 6 : \ & ( ' -2/0 H 2 N K + 4 =//> K49 -210 H 2

K / F =11> 9K4 -2*//0 H 2 A * ,

( 7 =///> 7K 3 -2H 30 H 2 $ H ( $ =/3> 3/K/7 -2730 A H 2 ! 08 $ F =/73> 4K43 -2$ 970 8 2 H $ & 5 $ =/97> /K/7 -* F+740 ( * H F + ; $ , =/74> 91K91 -*/10 2 * A# $ 34 =//1> 4/K9 -*/0 2 * A# + % , 4 =//> /K -*110 * & : ! $ =111> 94K31

© 2004 by CRC Press LLC

Section 4.5

277

Hamiltonian Graphs

-H 9/0 8 H & A 5 +

0 $ % => / =/9/> K7 -H 310 8 H & * + % , / =/31> 9K 7 -H +3/0 8 H & + + ! B + =/3/> 499K431 -H930 * H ; # , $ =/93> /K -J //0 * J 2 F G &I &! ; + =///> 9K4 -J 470 6 J & ; ) % =:> 7 =347> K 3 -J30 H JG A G : # $ =/3> 44K7 -J /70 H JG 2 F G &I A G ; B ) / =//7> /K -J /30 H JG 2 F G &I A G 5 =//3> K71 -J970 . J AI G * %8 $ 9 =/97> 971K97 -J/90 $ J # +

0 =//9> 4K3 -$ 970 $ $ =/97> 9K 1

#

-$$70 H $ : $ ; ( +

$ =/7> 7K74 -;710 ; ; $ $ 79 =/71> 44 -;70 ; ; * + $

=/7> K9 - 970 : G * $ =/97> 4/K7 -+/0 + F + ) % =//> 9K4 -+/ 0 + F + ) % 4 =// > 7K9

© 2004 by CRC Press LLC

278

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-3/0 $ 5 4 $ 3 =/3/> 41K44 -/90 Q 5G

[& ; ! + % , 91 =//9> 9K -/90 ( 8 * ! + % , 4 =//> 9K/ -/0 * $ /1 =//> 7/K/1 -6930 2 6 * B

$ =/93> 4/K73 -630 6 + 9 =/3> 7/K97 -6/70 6 ; 4 =//7> 9K -6/90 6 + %

, 9 =//9> K -6/30 6 " + % , 9 =//3> 1 K1/ -6 70 + 6 6 ; + 0 $ % =/ 7> /3K1 -6470 + 6 6 $ % 3 =/47> //K7 -+ 2 3 0 . + H 2 K * $ 4 =/3 > /K1 -+930 . + Æ + % , 4 =/93> 3 K37 -Q 970 H Q & 4 +

% , =/97> 7K -Q3/0 $ Q . 3 =/3/> 43K73

© 2004 by CRC Press LLC

Section 4.6

4.6

279

Traveling Salesman Problems

TRAVELING SALESMAN PROBLEMS

& +' $

7 6 6 7 A# 7 * 7 " * 74 6 2 6 77 6 L

Introduction 6 6 =6 > " 6 Æ 6

# - 10 " ! 6 2 6 L

4.6.1 The Traveling Salesman Problem J $ -$0 ! , 6 =6 > * # ! ! & " ! ! ! ! ! =( ! 6 -*+340> " 6 Symmetric and Asymmetric TSP DEFINITIONS

' # 3#& =#3#&>' 2 = > ! ! , = > !

'

3#& =3#&>'

2

!

© 2004 by CRC Press LLC

! ! ,

280

Chapter 4

CONNECTIVITY and TRAVERSABILITY

' 6 / 3#& 6 ! A ! A !

'

6

#

8 3#& ! 6 6 Matrix Representation of TSP

A 6 ! # ! # 6

6 DEFINITIONS

' 6 = > 6 # % ? - 0 ! ! ! 6

6 # % ? - 0 ! !

' 6 E EXAMPLES

'

6 ! 1 % ? 9

# 7 4 1 1 9 4

1 / 3 1

! ( 7 6 V ? 7 ! / 9 1 9 6 !

Figure 4.6.1

'

,7

6 ! # 1 1 9 1 1 / % ? 9 / 1 9 7 / 4 1

© 2004 by CRC Press LLC

9 7 4 / 1 1 7 7 1

Section 4.6

281

Traveling Salesman Problems

! ( 7 4 V ? 6 4 !

Figure 4.6.2

,,7

Algorithmic Complexity FACTS

' 6 # 6 ! !' ! 1 U ! 6

'

( 6 F B

8 ! 1 ! 5 ( ! !

'

- 2970 ( 5 ?F

! ! 5

Exact and Approximate Algorithms DEFINITIONS

' '

!

= > &

: ! # 6 = 6 == > == > == > ! ! =

' 6 * -Q30 ! #! =!> = == > == >>= == > == >> & 6 = ! == > ? == > FACT

'

-* J10 6 #

#! =!> 6 ! #! =!> © 2004 by CRC Press LLC

! 6 !

282

Chapter 4

CONNECTIVITY and TRAVERSABILITY

The Euclidean TSP

. ( 4 ! ! A 6 ! 6 6 ! , -/30 //7 = ( 7> $ -$ //0 ! = -10> FACTS

' - 992 2H970 A 6 F 6 $ 1 !" A 6 , E 6

' -10 (

' ! =6>> - /30

!" #

1= E

6

!" ! 6 -10

' -6/90 6 # 5 $ A 6 1= > A , 5 F REMARKS

' % =( 7> A

*! ( 3 ' A# B

= # >

# ! ( # !

' 6 ! ' 4 * * 4 $ ! 6 -2340 -H2$<QQ 10 -H$10

4.6.2 Exact Algorithms 6 F * Æ 6 F

! ! = > -8 /30 ( 6 ! # -: 10

© 2004 by CRC Press LLC

Section 4.6

283

Traveling Salesman Problems

FACT

' 6 - # # 6 @

=

>V @

°

Integer Programming Approaches

L 6 6 + , =-8:110> '* ** =- 30> -'' -'- = -8 6340 -( :610 -F 10> 6 ! &! =-+/30> 6 = > 6 . (& H -. (H4 0

' .,

( ?

(

= > 1 !

: ! = > 6 6 # ' > ?

( 5 ( ? ? ( ? ?

( + 1 + ¾ ¾ ( ? 1 ?

FACTS

' 6 , #

# # # 6 ! ! 5 # *! " ! # 5 = - >

' 6 - * B

+

+

' 6 ! * ! * 1= > -+/30 ! # 5

! !

© 2004 by CRC Press LLC

°

"

&

? 6

284

Chapter 4

CONNECTIVITY and TRAVERSABILITY

;! ! ( # ! B 6 6

4.6.3 Construction Heuristics #

!

Greedy-Type Algorithms

6 6 ! # #

Algorithm 4.6.1: < " <! " 3<<4

#' # - 0 ,# # ½ 5' 6 ½ ¾ ½ " + '? ½ ( ? ½ ? ½ ¾ + '? +

# 5

°

= > #

°

= >

Algorithm 4.6.2: +" #" 3+4

#' # - 0 5' 6 =6 >

+

=>

+ ? / ? = > = 6 > / ? = > = 6 > => ½ ¾ ! ( ? / " + => # 5 => + '? +

EXAMPLE

' + >> 6 A# !

( 7 # >> # # # 6 ! 1 =! ! >

© 2004 by CRC Press LLC

Section 4.6

285

Traveling Salesman Problems

Figure 4.6.3

,7

# -H2$<QQ 10 ! 6 >> U ! 111] # 6 -H$10 ! >> ! A 6 >> 6 Insertion Algorithms

, 6 ( 6 !

! # ( 6 ! 6 # !

6 ! 5 B ! 6 DEFINITION

' : ! ? ½ ¾ ½ # B # ! ( = > ! = > = > ! = > = > = ( 7 > 6 ! = > 6 = > ? = ·½> / ! = > ? ½ ¾ ½ = > ? = ½> ! = > ? ½ ¾ ½

Figure 4.6.4

2" /"9 " =

>

REMARK

'

'* , , , ! , ! @ # A ! -^0 #

© 2004 by CRC Press LLC

286

Chapter 4

Algorithm 4.6.3:

CONNECTIVITY and TRAVERSABILITY

1"9 2" 3124

#' # - 0 5' 6 ½ : ! " ! ? ( 7 ? : # ! -^0 " # = > ! ? ! ! = > ! = > = > ! ! '? ! = >

2 # ! = ! >

!

= ! > ?

DEFINITIONS

' '

6 567 # !

6 )67 # 6 = ! > ? = ! >

' 6 ,67 # ! # 6 = ! > ? # = ! >

6 # B

! A 6 = -H$10> # ! # 6 -22<Q 10 ! # A Minimum Spanning Tree Heuristics

6 6 6 -H2$<QQ 1H$10 ( 6

% ** 4% : <' = 7 7 -2<//0> + ! , , % # !

DEFINITION

' 6 6

" ! A 6 -H$10 *! = -H$10> , ! A 6

© 2004 by CRC Press LLC

Section 4.6

287

Traveling Salesman Problems

Algorithm 4.6.4: 6"= #" 36#4

#' # - 0 5' 6 ½ ¾ ½

( ( ! 4 ! = > ? = > 4 : ? ½ ¾ ½ =! B > ( 7 ? " ? & 7 ·½

FACT

' -H 340 1= ¿ > Worst Case Analysis of Heuristics

+ #

6 Æ $ # ( # ! 2 6 = 44> 6 6 !

6 FACTS

6 6 ! B ! >> B ! ' -2<Q 1 0 (

' -90 : # = 6 !

6 ! = >V ! ! = >

= >V

' -2<Q 1 0 ( ? 7 6 !

# ! = >V ' ( 6 ! B % ! ! ! ! 4 = -H 340> *! ! % B ! ! ! V - $ J 10 REMARKS

' , ( -2<Q 1 0 "

= -2<1 0>

© 2004 by CRC Press LLC

288

Chapter 4

CONNECTIVITY and TRAVERSABILITY

'

( 7 ( 4 @ ! , L "

- 970 2

< -2<1 0 6

! # ? 7 -6 310

= -* 7/0>

4.6.4 Improvement Heuristics # = > 6 &! ' , ! = > ! = >

( ! &! *' * B # 6 6 !

# : J =+-$ > =-H2$<QQ 10 -H$10> 6 #

" , : J -210 : J 6 6 6 = -H2$<QQ 10> DEFINITIONS

'

( 6 8 ! 5 ! ! = ( 74> ; ! 6 = #>

Figure 4.6.5

!

2

" "$

2

' ( #

= >

' 6 , U @ => => =

> 6

© 2004 by CRC Press LLC

Section 4.6

289

Traveling Salesman Problems

FACT

'

- $ J 10 & # ,

= > !

1= >

6

Exponential Neighborhoods 8

_=

>

# #

= >

'

6

( 6

! # !

=

>

" # 6

!

1= > =1= >>

6

-A; 10 -.+110 -2<Q 1 0

+ 6

# ! 6 ! 6 # ! # ! # ! = ! &>

4.6.5 The Generalized TSP 6

1' !

# 6 -( 610

DEFINITIONS

'

6

1 ( 3 # & 13#&'

2 !

,

! # = > #

?

'

6

# = >

#

'

6

1 ( # 3 # & 1#3#&

!

REMARK

'

; B W % W# % 26

26 ! B

6 W# %

26 26

© 2004 by CRC Press LLC

290

Chapter 4

CONNECTIVITY and TRAVERSABILITY

Transforming Generalized TSP to TSP ; ! 2 6 6

6 Æ 26 6

26 6 -F8/0 -: //0

FACTS

'

" -F8/0 26 6

+ , 6

# 6

4

! " ! #

'

" -: //0 26 6

! , Æ ! ! ( #

!

4 !

!

4

¼

6 !

!

!

?

!

½ ½ ½ 4 ¼

¼

¼

"

!

26

6 ! ! !

! 26

6 ! !

4

=

4

> !

! 8 #

¼

!

26

'

( -: //0 -F8/0 5 !

*

&

#

Exact Algorithms FACTS

'

# =-82 <Q 10 -: //0> !

-F8/0 -: //0 2 6 *! B 6 % = > ! =>

'

26

-( 610 : 26 -F8/0 6 # 5 &

6 26

'

% ? = > ½ ¾ % #

# ½ ¾ F ? = # B > -2<10 :

6 & !

© 2004 by CRC Press LLC

Section 4.6

291

Traveling Salesman Problems

Approximate Algorithms 6 # 2 6 , 2 6 6 6

FACTS

'

( -: //0 -F8/0

& #

'

6! -( 610 -82 <

Q 10 # = # >

6

26 26 6 ! 26 !

½

=> :

=

>

B

8

8

8

! ½ ¾

# #

# 26 -82 <Q 10

! @ !

!

½ ¾ ½ # = # > 6

!

! >

=> ( !

!

=" -( 610

-82 <Q 10 => :

8

B

*! -82 <Q 10 ! !

! !

4.6.6 The Vehicle Routing Problem 6

! * 4!:

! .

-. 4/0 6 = ! > 6 L L ) 6 B

"

DEFINITIONS

'

2 !

1

! ! ! " ¾ ?

# 1

© 2004 by CRC Press LLC

"

!

1

65&

! !

?

292

Chapter 4

'

6

CONNECTIVITY and TRAVERSABILITY

6 5 & 65&'

E

2 !

1 ?

! " , L ! !

REMARKS

'

; '

! = #> !

* &'&

! @

@ - J + //0

'

" L =

½ >

F

= W %> L -L /70

Exact Algorithms FACTS

'

6 Æ # L

=-8*110 -F 10 - J 60>

'

( L #

-6L 16L 10

'

L 6

3 ! " -

L ! -8 10

6 # ! L

6 ! 11

# ! L ! ! 4 - J 66L 10 ; L

& #

6

L

Heuristics for CVRP L ! '

L

B &U

6 &

@ ! B

=-A;0 -2: 10 -6 /0 -6L /30>

( L

B & )# # ! L '

&-

+

REMARK

'

6 ! L

L ! W % W % W % W%

© 2004 by CRC Press LLC

Section 4.6

293

Traveling Salesman Problems

Savings Heuristics

6 & + -+7 0 &! L * ! ! -2 /0 -+7 0 -: 10

= > ( # + &=+ > = # > !

6 + => 6 # + =+ > ?

¾

DEFINITIONS

' !½ !¾ =!½ !¾> ! # B =!½> =!¾> 6 !½ !¾ # " = = =!½> =!¾>> ">

' 2 !½ !¾ =!½ !¾> 7=!½ !¾> 7=!½ !¾> ? &= =!½>> E &= =!¾>> &= =!½> =!¾>>

°

'

: , ? !½ !¾ ! / ! ! # 1 6 %=,> ! /

!½ !¾ ! =! ! > # = =!½> =!¾>> "

! =! ! > 7=! ! > REMARKS

'

" "-( * =-+7 0-: 10> !

! &= =! >> 6 #

&= =! >> ! 6 =! > !

' 6 ! !½ !¾ & +

" = 1> !½ # 1 =1 > !¾ # 1 ! = > ! = ( 77>

Figure 4.6.6

6$ "0>"! '"! $ !½ !¾

REMARKS

' 6 ! 4 =! ! > ! # 7=! ! > 4 " ! 4 -: 10

© 2004 by CRC Press LLC

294

Chapter 4

CONNECTIVITY and TRAVERSABILITY

Algorithm 4.6.5: , /! #" 3,#4

#' # - 0 ? "

5' L , ? !½ !

" / ? ! ? 1 1 ? / " , ? !½ ! + / $

%=,>

4 %=,> ! 4 / ( =! ! > 4 , '? =, ! ! > =! ! >

/ '? /

' ; , ,

6 = 7 > ( # -: 10 L & + Insertion Heuristics

" L -: 10 ! ! ? 1 1 6 B 6 ! W % # ! ! , " B !

! ! 6 # ! &= =! > > &= =! >> REMARKS

' # , $ 6 -$ 69/0 ' ( H & -( H 30 6 1' * ! * , W % Two-phase Heuristics

6 &- ½ 6 1 ? + * -+*90 ! B A L ! 1 A A !

+ !

! =1 > 6 # ? =8 > ! 8 ! 1 1 6 ! !

© 2004 by CRC Press LLC

Section 4.6

295

Traveling Salesman Problems

Algorithm 4.6.6: ,:! #"

#' # - 0 8 ? " 5' L

!

?

½ ¾ 8

" + ? ?

?

(

7

?

" =+

>

'?

8 ½

7

?

$ "

E

'? + ?

+

(

:

6 +

!

1

REMARK

' # L -8 /40 # ! -*$ 69/0

References -A; 10 J 5 ; A H 8 ;

B

$ =11> 94K1 -2 /0 J & 8 2 5 ) / =//> 47K 7/ -8 /30 . A 8 # L G

+ & ; 6 $ + $/ *6 / ($ , !!9 ((( =//3> 7 4K747 6 M4M///

-/30 # A 6 +$ 4 =//3> 94K93 9 "AAA ( //7 3 "AAA ( //9 -10 # 6 " 8 % / =2 2 A> J! 11 -8 6340 A 8 6 8 8 $ " 8 % 1 5 > =A : : ! H J : * 2 J . 8 A> + /34

© 2004 by CRC Press LLC

296

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-8:110 8 : # ( % $ 111 -82 <Q 10 . 8 2 2 $ < Q 6 2 6 6 ' # 5 ) 0 =11> 49K74 -8*110 N 8 + *I

G H . ( JI 6 111K37 -8 /40 H 8 . : 5 ) =//4> 7 /K771 -8 10 H 8 . : 8 L / ) = 6 . L A> "$ 11 -+7 0 2 & H + + 5 ) =/7 > 473K43 -$ 69/0 F , $ 6 6 L " 5 > = $ 6 A> + /9/ -. (H4 0 2 8 . . (& $ H 5 ) =/4 > /K 1 -. 4/0 2 8 . * 6 & $

% 7 =/4/> 31K/ -.+110 L 2 . & 2 H + # B $

% 39 =111> 4/K4 -A;0 ; A H 8 ; ( ' $ 11 -( :610 $ ( : 6 A# $ 6 8 % / =2 2 A> J! 11 -( 610 $ ( H H 2G

6 6 2 6 ; 8 % / =2 2 A> J! 11 -( H 30 $ : ( H & '. =/3> 1/K -2 2H970 $ 2 : 2 . H F 9 $ % =/97> 1K

© 2004 by CRC Press LLC

Section 4.6

Traveling Salesman Problems

297

-2: 10 $ 2 2 : H < $ L " / ) = 6 . L A> "$ 11 -22<Q 10 ( 2 2 2 < Q 6 * + 5 ) / =11> 444K473 -2340 8 : 2 + ! A * 8 % 1 5 > =A : : ! H J : *2 J .8 A> + /34 -2<//0 H : 2 H < (

/// -2<1 0 2 2 < 5 ) 0 1 =11> /9K// -2<1 0 2 2 < # 6

O !

$ / =11> 19K7 -2<10 2 2 < 2 6 + 9 =11> /K4 -2<Q 1 0 2 2 < Q 6 ' 6

$ 9 =11> 3K37 -2<Q 1 0 2 2 < Q A# F . 6 8 % / =2 2 A> J! 11 -* J10 * J > # + =11> /K -* 7/0 ( * + /7/ -*+340 H *@ + * 8 % 1 5 > =A : : ! H J : *2 J .8 A> + /34 -H2$<QQ 10 . H 2 2 : $2 < + Q

Q A# * 6 8 % / =2 2 A> J! . 11 -H$10 . H : $2 A# * 6 8 % / =2 2 A> J! 11

© 2004 by CRC Press LLC

298

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-H 340 . H * 8 % 1 5 > =A : : ! H J : *2 J .8 A> + /34 -: //0 2 : (

('-5) 9 =///> K1 -: 10 2 : ( * L / ) = 6 . L A> "$ 11 -: 10 : 6 ! " 8 % / =2 2 A> J! 11 -$0 J $ . * * $ ; =/> K -$ //0 H 8 $ 2 # ' # 6 $6 %($ + 3 =///> /3K1/ -F 10 . F 6 8 6 8 % / =2 2 A> J! 11 -F 10 . F 2 8 L / ) = 6 . L A> "$ 11 -F8/0 A F H 8 : 5 ) / =//> 7K7 -F8/0 A F H 8 Æ ('-5) =//> /K - 990 * 6 A F % =/99> 9K - 30 * J 5 > * /3 - J + //0 2 J $ + A N A " A //3M - 10 6 6 ' ( L 8 % / =2 2 A> J! 11 - $ J 10 ( $ J 6 '

# 4 =11> K9

© 2004 by CRC Press LLC

Section 4.6

299

Traveling Salesman Problems

- J 60 6 J : J + & : A 6 ; $

- /30 + # C D C D EA $ % =//3> 4 1K441 -210 ( 2 : $ 8 % / =2 2 A> J! 11 -90 L " & A 6 ' $ =/9> 3K = > - 2970 6 2 # +$ =/97> 444K474 - 970 L "

; / '8 ,%%) % - > $ '8 =/97> 9K = > -6 /0 A 6 ' . =//> 77K79 -6 310 6 + 6 * % , / =/31> 73K9

¾ / 3 +

-6L 10 6 . L 8 8 L " / ) = 6 . L A> "$ 11 -6L 10 6 . L $ # #

$ =11> 39K4 -6/90 : 6 + * A ' # 6 $6 =! $ % =//9> K/ -L /70 . L * + 5 ) 3/ =//7> 13K7 -+/30 : + ( + //3 -+*90 + * 5 ) I =/9> K -Q30 A Q $ B # $ 5 ) 7 =/3> /K

© 2004 by CRC Press LLC

300

4.7

Chapter 4

CONNECTIVITY and TRAVERSABILITY

FURTHER TOPICS IN CONNECTIVITY

9 * 9 8 9 9 2

Introduction ! =Æ > ! ( ! # = # > F# ! ! = !> ( C D ( #

4.7.1 High Connectivity ! C D ! # , ! C D . @ ! ' = > L ! => = > => = > => : = > 6 ! # # Minimum Degree and Diameter

: ? = > ! Æ = # > " Æ Æ Æ ` DEFINITIONS

' 6 ! ,

© 2004 by CRC Press LLC

Section 4.7

' '

301

Further Topics in Connectivity

6

# 7& = > ¾

6

<=

> #

* # *

' = > = > FACTS

' '

-:9

? Æ

' ' ' ' '

Æ B Æ 0 " 5 => E =>

-770 "

% ? ? Æ -L330 " Æ E ? Æ -L3/0 " * =* > ½ Æ ? Æ -6L/0 " * =* > Æ ? Æ -. L/40 " B < * Æ ? Æ - 940 "

!

REMARKS

' ' '

( ( ( 4 * ?

! ( (

" ( 9 -. L/40 Æ B ! - Q3/0 Degree Sequence

( # # ! B ? Æ ( # - => 5 FACTS

' -2+930 " # = > = C D # > = > E = > ? ? Æ

'

'

, 4 => 9 4

-2A9/0 " #

? Æ

¾

-89/0 : ! " B ? Æ , = E > ! Æ ? Æ

© 2004 by CRC Press LLC

302

Chapter 4

'

-. L/90 " Æ Æ Æ ! Æ ? Æ

CONNECTIVITY and TRAVERSABILITY

= E Æ

'

-L10 7 ! B ! ? ? 1 ! " Æ Æ Æ # Æ ? Æ =Æ E > #

> = > E

< * : ? ?

REMARKS

'

F ( 3 ( ! ( / ( $ ! # - Q3/0 ( / ( (

' ( 1 ( ! ! - Q3/0 ( /

'

(

S -S/ 0 ! = ( > ! ( ( 1

' ( -L330 -L3/0 ! ( 9 (

! -*L10 ( # U => ? => => Distance DEFINITIONS

' 6 7& = > ! ! 7& = > ! >

= > =+

' 6 ; ! 5 5 = > FACTS

'

: ? ? 6 ! ; , $= > ? = > E ; %=; > %= > E

'

- Q3/0 :

'

-8 ( ( /70 :

# #

! , 7&= > 6 = ? Æ > 6

; " ;

!

= > "

=>

!

© 2004 by CRC Press LLC

Æ

;

# = ? Æ >

# = ? Æ >

Section 4.7

303

Further Topics in Connectivity

REMARKS

'

6 Æ ( # ( ( Æ B ! ( 4= >

'

( & ( 4= > ( ( = &% >

Super Edge-Connectivity

* ! DEFINITION

' # U # EXAMPLE

'

( 9 ! #

6 e

g

Figure 4.7.1

f

' 9' $$ ! "

FACTS

'

-:9 0 :

=>

'

¾ ¾ " 5 ?

" 5

' '

-J90 "

Æ E

=> E

=> E => E

-( /0 " ! Æ ! => ? Æ = >

'

-/0 :

! # ` "

$ Æ E `

REMARKS

'

( 9 3 ! ( B ( 7

© 2004 by CRC Press LLC

304

Chapter 4

CONNECTIVITY and TRAVERSABILITY

'

( / , ( =! B > ( 1

Digraphs

** ' =

B

>

DEFINITIONS

' 6 = >

# !

' 6 = >

!

= > => -' 5 * # => -'

5 # U Æ => ? => => => Æ ? ¾ => Æ ? ¾ => => Æ ? ¾ Æ => ? Æ Æ = > ( #

! ` #

=

(

5

>

= >

#

FACTS

' '

% ? ? Æ -S/ 0 : " = @>

= > Æ = > E Æ = > ? ? Æ ' -*L10 : ! ! 6 => ? => => ' -*L1 0 : ! Æ " # # 9 # # 9 ? Æ ' -*L10 : * Æ ! * " =*Æ >=* > => ? => =>

' -*L10 : ¼ ¼¼ Æ ! " =(> E =3> = E > ( 3 ¼ ( 3 ¼¼ => ? => => -H90 "

!

REMARKS

'

F Ga&% =( > B H =( > ( ( 3

© 2004 by CRC Press LLC

Section 4.7

305

Further Topics in Connectivity

'

( ! . & L& ! B -. L/9. L110 ! !

' ( ! ! # $ H % =( > -(;!110

'

B ( (

Oriented Graphs DEFINITIONS

' # ! 5 # !

' = > ! ! = > == > = >> EXAMPLE

'

( 9 ! # " ? ( 3 = - 0 > =

5 # ! >

u

x

v

Figure 4.7.2

y

' 9' $$ "

FACTS

'

-(910 :

!

Æ = E > ? Æ

Æ "

' -( /0 " !

'

-( /0 "

! !

© 2004 by CRC Press LLC

Æ E

306

Chapter 4

CONNECTIVITY and TRAVERSABILITY

REMARKS

'

( 9 3 ( ! ( / (

B ( /

' " Æ -(910 -( /0 =( 9 3> ! Æ E Æ Æ E Æ E ( ! ( 3 /

'

* * !

Semigirth

6 H % =( > !

! " # = > DEFINITIONS

' -( ( 3/( ( A/10 ( ? = > ! % @= > @ ! % = > 7&= > @ ! & B ! & 7&= > E

7&= > ? @ ! & ' ( * ! #

=>

* 5 # 6

EXAMPLE

'

( 9 ! ! @ ? % ?

Figure 4.7.3

© 2004 by CRC Press LLC

,'!" @ ? % ?

@ B

Section 4.7

307

Further Topics in Connectivity

FACTS

'

-( ( 3/0 :

'

-( ( /7 8 210 :

!

@ = > " % @ ? Æ => " % @ ? Æ => " % @

Æ $ %

* =* >

% @ E * ? Æ " % @ E * ? Æ " % @ E *

= > " => =>

'

! #

REMARKS

' 6 ( 1 @ ! ! ( = > ! Æ

7&= > 7&= > @ * @ #

' '

@ ( ( 1= >

( H % =( > " = > Æ E % ( (

Line Digraphs DEFINITION

' 6 ; ; =; > = >

# = > 5 # = > > ? = ' = > = > > > 6 ; , ; ? ;; ½ FACTS

'

6 ; B =; > ? = > Æ =; > ? Æ = > ? Æ $ =; > ? = >

'

" $ % @ ;

%=; > ? %= > E @=; > ? @= > E - 790-( ( 3/0-( <3 0-J*:30

© 2004 by CRC Press LLC

308

Chapter 4

'

-( ( 3/0 : @

CONNECTIVITY and TRAVERSABILITY

!

Æ$

%

% @ ; = > # " % @ E ; = > # " % @ E ; = >

= > " => =>

REMARK

'

! ( 4 ( ( 1 Girth

( M ! ! DEFINITION

' 6 , =., > = ! &> "

@ ? @=

> ? @= > B = > !

? =

>

FACTS

'

-F "34F " 39( ( 3/0 : ! %

% " % " %

= > " => =>

'

? Æ ? Æ

-8 ( ( /7 ( //0 :

: 6

%=; > " %=; > " %=; >

= > " => =>

'

;

! Æ $ ! %=; >

? Æ ? Æ

-( ( /7 0 ! #

REMARKS

'

Æ $

( 7 B ., 4 ( 1

© 2004 by CRC Press LLC

Section 4.7

309

Further Topics in Connectivity

' ( 3 ( ! Ga&%

Cages DEFINITIONS

' = > !

' ? = > 9

# ? # # ! EXAMPLE

'

6 : ! ( 9 = 7> !

Figure 4.7.4

# : !"

FACTS

' -(*/9S+ + 10 = > = > #

' '

-. //H $/30 A = > !

' '

-$ 8 ( 1T0 A = > !

1 -+ S+ 10 A = > ! #

' '

-$ 8 1T0 A = > !

-$ 8 1T$ 8 10 A = >

B

-$ 8 1T0 = 7> = 3> #

CONJECTURE

-(*/90 A = > #

© 2004 by CRC Press LLC

310

Chapter 4

CONNECTIVITY and TRAVERSABILITY

Large Digraphs

6 ! = > DEFINITION

%

' ( ! # ` $ =` %> =` %> ? E ` E `¾ E E `% FACTS

' '

# ! # ` % =` %> -+ 790 6 = > ! $ % ,

=% > E

'

-";&340

Æ - =` % > E ` E 0 " Æ - =` % > E `0

= > " =>

'

$ =Æ >- =` % > E ` E 0 ? Æ " $ =Æ > - =` % > E `0 ? Æ

= > " =>

'

-( /0

Æ - =` % > E 0 E ` " Æ - =` % > 0 E ` E

= > " =>

' = > =>

-( /0 :

" $ % ½ E ? " $ % E ?

'

-/( / 0 :

= > "

$

=> "

$

% ½ E %

¾ E % ½ E %

% ? ¾ E E ¾ E % % ? ¾ E E ¿ E ¾ E %

EXAMPLE

'

( 94 ! ! ? 7 ` ? Æ ? ? % ? $ % ½ E $ % E ( 4 # = ? ? >

© 2004 by CRC Press LLC

Section 4.7

311

Further Topics in Connectivity

Figure 4.7.5

? ??

REMARKS

'

6 &! ( 9 + & ! , ! % ! = > " ! 5 ! % $%

' ! / = > ! / % *! Æ =6

%' ? ! % ! ? ! ! C D %>

'

" ! & $ , " ! ! & C&D ! ( 3 =

( /> ( 41

' ( 41 ( 3 F ! ( 41=> ! $ =` %>

? `

Large Graphs

! A -A340 &

-F " 390 -/0 -( /0 -( / 0 DEFINITION

' 6 $ ! # `

% =` %> ? E ` E `=` > E E `=` >% FACTS

' = > " =>

-F " 390 $

=Æ >- =` % > E 0 E `

" $ =Æ >=` >%

© 2004 by CRC Press LLC

E

? Æ

? Æ

312

'

Chapter 4

-/( / 0

%

= > :

=> :

Æ

"

CONNECTIVITY and TRAVERSABILITY

$ Æ - =` % > E 0 E =` >%

½

% Æ 4 " $ =Æ >- =` % > E `0 ? Æ

4.7.2 Bounded Connectivity 6 B = > " ! #

A-Semigirth 6 ! , =., > DEFINITION

' -( ( 3/0 : ? = > ! Æ %

A 1 A Æ 6 A @& = > @& ! %

7&= > @& B A ! & 7&= > E 7&= > ? @& ! &

= > =>

FACT

' -( ( 3/0 : ! Æ $ % A @& 1 A Æ ! ; 6 = > " % @& Æ A => " % @& Æ A => " % @& =; > Æ A => " % @& E =; > Æ A REMARKS

'

A @&

'

A Æ !

F @ @ $ ! , ! 6 , @&

Imbeddings

* ! ; & - Q/30 - Q10

© 2004 by CRC Press LLC

Section 4.7

313

Further Topics in Connectivity

DEFINITION

'

+ !

+

!

FACT

' -90 : $ 1 =! " ' - 90> 6 =4 E E 3> Adjacency Spectrum

2 = > ! # ! 6 5 5 = 74 #& -8 / 0 - . /40> DEFINITIONS

' 2 ? = > # # ! ? 5

? 1 !

' 6 & # =

, & ? + = + > ! + + > +

FACTS

'

-/48/40 : #

6 & $ %¾ # ! # ! ! =6 -8 ( ( /70>

2

'

-( 2 </90 : Æ ! %¾ $

= 5 # > =? Æ > $ $ $ : =(> '? =( >= > ¾ Æ 6 = > Æ ' ' ¾ Æ REMARKS

' 8 ( 49 8! -8/70 #

' ( !& -8* J 30 -: 10

' F ( 4=> ! ( 43

© 2004 by CRC Press LLC

% ?

# ;

314

Chapter 4

CONNECTIVITY and TRAVERSABILITY

Laplacian Spectrum DEFINITION

' 2 0 , ; ? % ! % # # 5 # = -8 / 0> 6 0

: #

6 :

B½ ?

= > !

, FACTS

'

; ,

! , B " Æ ! = >

=? Æ > $ $ :

B B B ! B ? Æ ? 5

'

-( 90 :

! :

1 ? 1 => ( ! = > = > => ( # ! = > = > ' : Æ ! % $ :

B =? 1> B B B " Æ (((½ (¾½¾(½ ((½( ? Æ = >

REMARK

'

( 7 5 B ( 43 :

4.7.3 Symmetry and Regularity Boundaries, Fragments, and Atoms

6 = > ! ( ! + & -+ 910 $ -$ 90 6 ! -970 , # * -* 990 -* 310 -* 30 8 !

! , =( , > DEFINITIONS

'

6 # C 5 * C 5

© 2004 by CRC Press LLC

Section 4.7

'

315

Further Topics in Connectivity

6

< < < ? = > ' < ? = > '

' : ! # C ? = C > ? C ? =C > ?

'

:

! # < ? D < ?

D

' # - 0 D 5 -0 < -< 0

'

= >

EXAMPLE

' ( ( 97 ? = > ! = > = > &> < = > ? =( > =3 > < = > ? => (> =& 3> " # F

z

x y t

u

v

Figure 4.7.6

" !'

FACT

'

" C ? - C ? 0 C -C 0 # => < -< 0 N ! ! , '

? C ' C ? ?

? < '

Fragments and Atoms in Undirected Graphs FACTS

'

-+ 910 " ! 5

© 2004 by CRC Press LLC

316

Chapter 4

CONNECTIVITY and TRAVERSABILITY

'

-$ 90 : ! : ½ ! 6 ½ ¾ ?

'

:

¾

! " ! ? $

REMARKS

'

6 B & + & ' C" $ " C D ! " -+ 910 " C D ! $ -$ 90 C D C D " B ! D

'

+ & -+ 910

$ -$ 90 F ( 7 ( 7

'

, -H+ 990 -* 3/0 Fragments and Atoms in Digraphs

6 B ! * FACTS

'

-* 990 : ! = > = > 6 ? " ! = > 5

'

-* 310 " !

= > D #

Æ · = Æ

>

REMARKS

' ! !

!

'

( 79 H % =( >

Graphs with Symmetry

2 ! CD

" ! ! 2

7 7 DEFINITION

'

= >

= > ! = > = > >

>

© 2004 by CRC Press LLC

Section 4.7

317

Further Topics in Connectivity

FACTS

' '

-$ 9+ 910 :

'

-$ 910 :

A # # ! 6 ?

? E (

?

! 6

?

'

-+ 910 : # ! 6 -0 # $

= >

'

-* 990 : # ! = > 6 -0 # ( = > = >

' -* 30 : # ! = > 6 ? $ REMARKS

'

6 , ! ( 7/ B ( 7 6 $

' ( ( 9 9 = > B # = > ! # = ? ? Æ > " &! = > = >

'

8 ( 77 * -* 990 J % # -J 9 0 * # Cayley Graphs

6 # = > ! = 7 7> " & ! & ! &! . ' " *

! . E ? . E E DEFINITIONS

' : b , ! + b 6 ? =b + > ! b = > ! + " ! + ? + =! + ? ( ' ( + > !

+

' ' ¼

" b

+ b ! + b

© 2004 by CRC Press LLC

318

Chapter 4

CONNECTIVITY and TRAVERSABILITY

' 6 c

' : ( !

6

( ? (EE

FACTS

'

-"9/0 : + b ? c ! 4 (+( ½ ? + ( b 6 =b + > # = ? + >

'

-* 3 0 : b , ! + : = > =b + > 6 b + = > =b + > b !

'

-* 3 0 : b , ! + : 6 =b + + > #

+ + ½

'

-* /70 : b + b + 1 : % ? =b + > 6

# = = >>% ! ? $ ! B =3+ > ½½ = >

REMARKS

' '

( 94 * ( 9

( ( 97 ! 2 -230 B & J -&J390 * : G -* : /0 -/0 = > ! = ! , , E > Circulant Graphs

8 !& ! & " " # !& = -8*/40-86 3 0> FACTS

' -* 3 0 : b . : + B 7 = ?>½ ¾ = > ·½ ½ ? 7 6 =b + > # = ? 7>

' -* 3 0 : b . : + B 7 = ?>½ ¾ = > ·½ ½ ? 7 + + ! + = > + 6 =b + + > # = ? + + >

© 2004 by CRC Press LLC

Section 4.7

319

Further Topics in Connectivity

'

6 . B

+ . * =. + > # = ? + >

REMARKS

'

6 + ¼ ? + ( 9/ = > ! -8(910 C # D ½ ½ = -86 3 0>

' ( 31 * ! =( 5 ! -* /70> Distance-Regular Graphs

6 ! 8 /91 . ! ! ! " # DEFINITIONS

'

: ! % ! % 2 ! ! 7&= > ? 5 ! E ' # = U > !

5 ! 5

FACTS

' : 6 !

' '

A # -8$340 A #

CONJECTURE

-8/70 A # REMARKS

' '

( 3 B ( 3

6 5 '' 1 = = >

5 , >U -2/0 " 5 ! 5 2 -230 #

#

© 2004 by CRC Press LLC

320

Chapter 4

CONNECTIVITY and TRAVERSABILITY

4.7.4 Generalizations of the Connectivity Parameters 6 @ ! -8 8: 390 -8; 10 -* 30 -+90 * ! ! # !

!& Conditional Connectivity

6 # ! , DEFINITIONS

'

2 ? = > 7 # ¼ C = > ! 7¼ 7¼ 7 ¼ 7 <= > ! 7¼ 7¼ 7

' 6

7 6 7 FACT

' = > => => =>

-( ( 3/( ( A/1( ( / 0 :

!

% : @ ? ) ½ 6 " % @ ? Æ " % @ ? Æ Æ " % @ Æ Æ " % @ Æ

Æ $

CONJECTURE -( ( / 0

% @ 7 =7 E >Æ 7 " % @ 7 =7 E >Æ 7

= > " =>

REMARKS

' * -* 30 " # ! ! ,

' F = ( 3 ( 7> " $ Æ

$ Æ

'

6 5 ! 7 = Æ $ @ $ =7 E>> -( ( /70 $ 7 ! Æ Æ ! -8 ( ( /90 -8 //0

© 2004 by CRC Press LLC

Section 4.7

321

Further Topics in Connectivity

Distance Connectivity

* ! = > -( ( / 0 -8 ( /70 ! & ! DEFINITIONS

' : ? = > 2 = > =

> + => + => => => => => ' : ? = > ! % 2 & & % & =&U > ? =&> , =&> ? => ' 7&= > & & => ? ! & =&U > ?

=&> ? => ' 7&= > & & FACTS

' = > =>

? => ? => => =%>

? => => =%>

' : > 6

!

Æ $ @ = .,

Æ % @ E ? =@ E > " Æ % @ ? =@>

= > " =>

'

? Æ % @ =@ E > Æ => ? Æ % @ =@> Æ ' A ! => Æ # = >

'

:

!

&

=&U

> ? =&U >

' = > =>

=&U > ? =&U >

Æ $ % ? => " Æ % ? = > % ? = > " Æ % ? = > :

!

© 2004 by CRC Press LLC

322

Chapter 4

' = > =>

=> Æ

? Æ % % = > Æ = > Æ ? Æ % % = > Æ

'

CONNECTIVITY and TRAVERSABILITY

! => Æ #

REMARKS

' " ( 39 =&> =&> , & % ! Æ

'

( 37= > ( 33 ! H % =( >

'

( /1 ! ( 37 ( 3/ @= > ? = >

High Distance Connectivity DEFINITIONS

' 2 # => ? #Ú¾ 7&= > => ? # ¾ 7&= > ' ( & & % & Æ =&> ? Æ =&> Æ =&> ! Æ =&> ? ¾ => ' => & Æ =&> ? ¾ => ' => & ' ! % 7 7 % ! 5 B 7 " 7 ? % = -8JQ730- Q9 0> FACTS

' Æ ? Æ=> ? ? Æ=5> Æ=5 E > Æ=%> ' ( & & % =&> =&> Æ=&>

& ! =&> ?

& ! =&> ? Æ =&>

=&> ? Æ =&>

'

" & 5

'

#

# &

7 6 = > =&> ? Æ =&> =@ E > & 7 E => =&> ? Æ =&> =@> & 7 ' -8 ( ( /9 0 : 7 = > # & & 7 % @ => # & & 7 E % @ -8 ( ( /9 0 :

© 2004 by CRC Press LLC

Section 4.7

' = > =>

323

Further Topics in Connectivity

% 6 & % % # & % % # & % :

!

Maximal Connectivity

" & ! = > ! # 6 !

. # = # > FACTS

' ' - : *370 (

! # ( 2 ` ' - : *370 : # ! Æ = > " Æ Æ => " Æ = E > ! = E > => " Æ = E > ! = E >

` ? !

Hamiltonian Connectivity DEFINITIONS

! ' = > $ + = > ! ? + # ! + = ' *

> FACTS

'

"

'

-2+ 370 : 5 ! =>E=> E

= >E

-2+ 370 ( ! = E > = > REMARK

'

6 =., 9> $ F 2 2

© 2004 by CRC Press LLC

324

Chapter 4

CONNECTIVITY and TRAVERSABILITY

References - 790 $ ; $ G 1 =/79> 47K7 -&J390 8 & 8 J ; (*** 7 =/39> 334K333 -/40 F 6 ! +

=//4> 3/K/4 -/0 8 ! (*** =//> 9K/ -(910 HF "6 ( ; (*** 6 9 =/91> /K41 -8 //0 $ 8 A# 7 $ /4 =///> /K4 -8 8: 390 J 8 :+ 8 & $H : A ; ' 71 =/39> K -8 ( ( /70 $ 8 H (P $ ( ; '. 3 =//7> /9K14 -8 ( ( /9 0 $ 8 H (P $ ( ;

7 ! $ 9 =//9> /K9 -8 ( ( /90 $ 8 H (P

$ ( A# ! $ 79M73 =//9> 34K11 -8 ( /70 $ 8 $ ( .

+ =//7> 3K/ -8*/40 H 8 ( .( * . !&' + =//4> K1 -8; 10 :+ 8 & ; ; A 6 $ 4 =11> K 4 -8 / 0 F 8 N // -89/0 8 8G ; ! B $ 3 =/9/> K -8(910 ( 8 ( '. =/91> 7K3

© 2004 by CRC Press LLC

Section 4.7

Further Topics in Connectivity

325

-8* J 30 ( 8 ( * H J 2 !& ' '. =/3> 49K7 [ QG +

-8JQ730 H 8G & J 4 =/73> 91K97 -86 3 0 ( 8 6 + 3 =/3 > 39K // -8/40 A 8! 6 0

7 3 =//4> 79K9 -8/70 A 8! <( I / =//7> 9K 1 -8$340 A 8! .$ $ 6 * + 7 =/34> 4K7 - ( //0 $ 8 H (P

; '. =///> /9K14 -770 2 %($ +

$ =/77> 993K93 -970 2 ; # = > ;> =/97> /K1 -: 10 A $H : L '

$ 3 =11> 7K9/ -90 H & * !% $ 0 1 =/9> 1K19 - . /40 . & G $ . d * %

6 H 8 L : //4 -. //0 $ . = > $ // =///> 19K4 -. L/40 . & : L& F! Æ B 1 =//4> 91K93 -. L/90 . & : L& . B # + 7 =//9> 9K -. L110 . & : L& . B # B $ =111> 9K -A340 * A :! + / =/34> 41K4 -( ( 3/0 H (P $ ( $ # + =/3/> 749K773

© 2004 by CRC Press LLC

326

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-( ( / 0 H (P $ ( A# ! $ 9 =// > 7K91 -( ( /7 0 H (P $ ( 8 ! #

$ 7/ =//7> 7/K99 -( ( /70 H (P

$ ( ; # $

44 =//7> /K49 -( 90 $ ( > $ + =/9> /3K14 -( /0 $ ( ; + 7 =//> 4 4K444 -( /0 $ ( 6 + 9 =//> K 4 -( / 0 $ ( 6 $

=// > 79K93 -( ( / 0 $ ( H (P ; $ 4 =// > 7/K97 -( ( A/10 $ ( H (P $ A /8 =//1> 9K -( 2 </90 $ ( A 2 H: < 6 !

$ 79M73 =//9> /9K19 -( <3 0 $ ( H: < " : = > (*** =/3 > 11K 1 -(;!110 2 ( & ; ; * ! 6 =111> -(*/90 *: ( J * + =//9> 39K/ -2/0 2 !& 1 1

$ 9M3 =//> 14K 7 -230 . 2 AB =/3> 7K79 -2A9/0 .: 2 A Æ B + =/9/> 4K 44 -2+930 .: 2 6 + ; ! B

$ =/93> K7

© 2004 by CRC Press LLC

Section 4.7

Further Topics in Connectivity

327

-2+ 370 $ 2 < + & ; +

% , =/37> K7 -* 990 <; * % G ) % %4 3 =/99> 4K47 -* 310 <; * D $ =/31> 14K17 -* 30 <; * OB P # G G +

% , 1 =/3> K1 -* 30 ( * '. =/3> 9K49 -* 3 0 <; * ; * +

4 =/3 > 1/K -* 3/0 <; * % , $

9 =/3/> /9K11 -* /70 <; * !& K/ . Q . =A> .

5 J! . //7 -* :/0 <; * : G ; 6 B

$ 9M3 =//> 94K31 -* /70 <; * ; ; ! B $ % / =//7> 19K 1/ -*L1 0 *! : L& $ #

+ 9 =11> K -*L10 *! : L& $ # -"9/0 + " ; + % , 7 =/9/> K7 -";&340 $ " 6 & J ;& ! (*** =/34> 79K9 -H $/30 6 H . $ + / =//3> 4K -H90 H: H # G G ) % 9 =/9> 3K41 -H+ 990 * H $A + & ; , , $ $ 3 =/99> K -J 9 0 6 J F * % +

% , 9 =/9 > K

© 2004 by CRC Press LLC

328

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-J90 J J

9 =/9> K4 -:9 0 : : & $ 3 =/9 > 4K4 I -$ 910 + $ N Q I

2 $

,# =/91> K7 -$ 90 + $ A A 2 $ ,# =/9> K7 -$ 8 1T0 S $ 8 A -$ 8 1T0 S $ 8 " ; ! , # -$ 8 ( 1T0 S $ 8 " H (P

=Æ > ! 1 -$ 8 10 S $ " 8 A B $ - 8 210 " 8 H 2G ; * 43 =11> 4K - : *370 J : & * $ # M =/37> 4/K91 - 940 H Ga& - ) ' 7 8

$ 1 =/94> 9K/ [ QG - Q9 0 H Ga&

- ) ' 7 8 $ / =/9 > /K [ QG ; B - Q3/0 H Ga& $ , # 4 =/3/> /K4 - Q/30 $ S Q ; + % , 9 =//3> 1K3 - Q10 $ S Q ; "" * + / F 3 9 =11> -J*:30 $ H2 J * * * : ; ! # =A =/3> 13K17 -/0 6 &

$ 9M3 =//> 4K4

© 2004 by CRC Press LLC

Section 4.7

329

Further Topics in Connectivity

-F "340 6 & * F & $ " Æ # (%%9: =/34> 3K3 -F " 390 6 & * F & $ " Æ # $ 7 =/39> 4K77 -6L/0 H 6 : L& Æ B

+ 9 =//> 7/4K911 -L330 : L& 8& * J 2 - & * 0 7 8 , $ 9 =/33> 94K9/ -L3/0 : L& A * + =/3/> K7 -L10 : L& . B B B + =11> K 4 -+ 790 $A + & !

$ $ 9 =/79> /9 -+ 910 $A + & + 3 =/91> K/ -+ S+ 10 + 8 S H + * + 1 =11> e -+90 . + 6 +

% , 4 =/9> 4K44 -S/ 0 H $ S Æ B $ =// > 4K3 -S+ + 10 8 S + H + ; = 7 =11> 3K/

© 2004 by CRC Press LLC

>

330

Chapter 4

CONNECTIVITY and TRAVERSABILITY

GLOSSARY FOR CHAPTER 4

'

'

. ' "9

K = >

!

?

'

#

? =

> #

= >

? 1

!

$! " / K #

'

:

''" !" 3" " !" 4'

!

"9' 3" "9' 4 $!"''

&

" "' '

''" !" ' '

''" ,7 3,74' !

' K '

$ K #

+

=

>

+

'

+

=

> @ !

5

-". !" "" B

' !

U #

+

5 #

=

>

+ + +

=

>

!

8 5 U

, # ! ! # !

$ !" ' '

$ " K

= # >'

= # >

$ /"9 K ' # ! B " !" $"' 5 5

(

3 9

!

E

" !" ' ! ! # = > " 9 K ' !

$0 K ' # " K # ' ! " !' ' "! K ' ! U

-'

© 2004 by CRC Press LLC

331

Chapter 4 Glossary

!

>

=

'

!

6 $ 34!"

+

K b !

' = > !

, ! b

6 7' 7" $'

½

!

b

' ,

! U

$ :" $"

'

*

$? ' "

K

' #

'½ ' !" " $$

K ' # K ' # ' ! # ! & !

=

>

'

'$ ! " !

=

>

#

' ! !

' !

' !

' , = >

'

,

5

!"

-

?

>

' ,

" $ ! " $ !"

'

=&

!

'

!

/"! : $0

=

* >'

K

' ! &

" $$ !" ! ' /"9 617 "

'

'

'

K ' # ! K ! ! #

1 '

! ! ! "

!

1

# 1

$ /"

K

'

$ '

>

?

!

"

! !

+

+

¾

?

K = >

'

=

= >

$ $ /" 36 64 # !

© 2004 by CRC Press LLC

+

'

+

332

Chapter 4

! / = / E =! >

$ 9 $'

CONNECTIVITY and TRAVERSABILITY

> #

$$' # #

$ 0! K ' 5 $ K ½ ¾ ½¾ ' ? ½ -". K ½ ¾ ½¾ ' ? ½ $'

!" $"' 5 5 ! !" ? K

' = >

' " ' ., ' K ' B #

=

'" K = > ' # ! !" ' ! U ' '

>U

''" K ' ! = > = > ' $ ' ! U * '

*-

' ! U &"

' " $$ ' = >

= >

"!$ ' !

'

#

! & U

3"!$4 ' ! U - - '

''"½' ! = > # ''"¾' = #

> B

! !½ K

' !

! !¾ K

' !

! 3/"94 K

' !

=> K ' = # > !

@

© 2004 by CRC Press LLC

333

Chapter 4 Glossary

! 3/"94 K

' !

& / 7& = > !

=

>

K = >

&

' "9

6 ' #

!

½

K !

5

¾

"!$ " !" '

' ! "'

!

! =

2

>

!

? - 0 !

= >

7&

%

!

U ,

K ! #

!

'

'

2

'

! & !

1

#

$ " !' ! & # ! " $'

!

"!' ! ' ! 5 ? = >

=

>

! " ' ! " K #

?

E

?

' #

!/½ K '

U

!/¾

!

=

>

K '

!

U

!' ' ' - !. ' !''" !" ' !

$ ,7' ! $" !" 3" !" " '9 !" 4' $" " K = >' ! & = > ,

# ! &

#

/ !" ' ! 9 $!"''

½ ¾ "'

"

K

'

>U

"

© 2004 by CRC Press LLC

*

=

' = # 5 >

> ?

=

> =

=

>

334

Chapter 4

" !' K

CONNECTIVITY and TRAVERSABILITY

'

!" $% *$' = > ! # * ! = *>

!" $% ,7' ! 7! !" ' 9 ., 4 !" K ' $ !" K # ' '$ ' ! ! '$ $' '$ ''

= E>

'$ !" ' ""' B

B

?$' #

' ! K + ' ! + ! " K = >' ! U

= >

K # ' = > ! =

? U >

" '' ., 7 !" K # ' 5 &' # ! & ' "' K ' ! 5 /"' ! ! !" 3" !" 4 ' # = >

! #

U = > - 0

" K # ' # 7&= >

"" /"9 K ' # ! 5 " $$. ' ! #

"' ! !" 3" !" 4' = > ! ! !" 3" !" 4' = > !

© 2004 by CRC Press LLC

Chapter 4 Glossary

335

" $'' ½ ¾ ½ ¾ ! # 5 ½ ¾ ? ' ' ' 5 =! > 5 0 " "' '

U U 7 @ $ ' "9 K ' # ; ? =2 > ! 2 ? => 5 2 ? 1 ! $ !" K ' !

# = > 5 # = > " K ' $ !" K ' ! ! ! ! ;= > 5 $0 !" ' B @ ½ ¾ ½ ¾ # ? # 5 "$0 !" ' ! , 5 ! $ $ '$ K # ( -- =(>0 ' -- =(>0 =- =(>> ' !' ! ! # " K ' # ' 9' $$ !" 3" !" 4' = > ! B ' 9' $$ ! !" 3" !" 4' = > ! B ' 9'' !" K = > ' # = > '' $$ !" 3" !" 4' = > = > = > '' $$ ! !" 3" !" 4' = > = > = > ''' 3&4!" K = > ' & = > ''' !" K = > ' = >

''' !" K = > ' = >

© 2004 by CRC Press LLC

336

Chapter 4

'9 !" ' 8" K

CONNECTIVITY and TRAVERSABILITY

' #

#

! /

D" !'

K

'

!

! / ' K ' ! / " K # !

! / ! " K

! / " !' K

!

#

! , #

'

'

' !

! "½ K # (' 5 ( ! "¾ K + ' 5 # + " K = >' !

· =

U

>

!" K # ' 5 &' # ! & " K # ' #

7& =

>

"' ' $' = > * " !" ' ! # * #

" K

!

'

? =

> !

?

=

=

>

>' U

=

! & ! & ! ,

'

/

K

D" !'

#

'

5

K

'

#

!

/ ' K ' ! / " K # '

/ ! " K # ! #

/ " !' K

© 2004 by CRC Press LLC

U ,

>

'

' !

337

Chapter 4 Glossary

' " $'' , = ! >

"' '9' # =!

> "" $' # = #> ! 0" " ' # ! "' ! :' ! = >

' " = &">' K ' ! &

:" K ' ! = = > = >

> ?

=

> !

? !" '

# ! ? # # !

"" K ! &

·½

? ·½ =

' ! & ½ ½ >

·½ ?

·½

"" " : $0' ! & " "' # 5 # 5 ; -

'!" @ K ' @& A ? 1 '!") A K '

' = > 7&= > @ ! & B A ! & 7&= > E U => 7&= > ? @ ! &U @&

" ! K #

'

3$4 " K

½¾

'

? ½ ¾ ¿ ? ·½ ?

! !" 3" !" 4 K

' = >

$! " ' ., 0" " " $'' # !

"!$ !" ' ' "!$ "!$ " !" ' ! ! !" " '

© 2004 by CRC Press LLC

338

Chapter 4

CONNECTIVITY and TRAVERSABILITY

" !" ' # ! 5 # U ,

" !" ' # ! # U ,

''" !" ½' ' ''" !" ¾' ' ''" !" ¿' ! #

9

U

''" !" ' ! # ''" ,7 3,,74 K = > ! ' !

$ !" K # #

,

'

$$ " ! ' 2 # ! K ' # & & =+ > + ! =+ > + " $' " '' ., 9 " /$! $' " $'' , ! ! ! U ** !

" !$ ? $ K ! = >' E ! ! #

# = ! >

,7' * * ''" 3,74' , ! !

* '

= >

$ '

!" $% K !

6 ! A ! A !

½ ' , ! # = > # ?

=>3!4 ' '

' # #

/$ "! " $') 36174 K ! E ! 1 ?

! ! " ' , L ! !

© 2004 by CRC Press LLC

339

Chapter 4 Glossary

3/"94/½ K '

!

U

3/"94/¾

Ú

K '

!

U

Ú

/"9 $!'

! #

! 5

5

!

# U

: $0'

/"9''" !" '

! 5

#

B

B

: 0$ $0 !" '

=

> #

5

: 0$ !" '

:$'

! U

' '

#

!

© 2004 by CRC Press LLC

!

# 5

Chapter

5

COLORINGS and RELATED TOPICS 5.1

GRAPH COLORING

5.2

FURTHER TOPICS in GRAPH COLORING

5.3

INDEPENDENT SETS and CLIQUES

5.4

FACTORS and FACTORIZATION

!" # $ 5.5

PERFECT GRAPHS

% &' ( % 5.6

APPLICATIONS to TIMETABLING

) ( % & " *+ ,

) #"+ - ./ GLOSSARY

© 2004 by CRC Press LLC

Section 5.1

5.1

341

Graph Coloring

GRAPH COLORING ! !" # $ % & ' () *'

Introduction ' " $ ' " $ + ,

- # - ' + ' , - .+ !' / 0 ,

' 1

2 341 5 6 , ' ' + , / 77

% + - - ' 315&6 381 556

9 " , '!

- - + + , + )

5.1.1 General Concepts , , '

' + ' , ) - .+ 0 % ' ' + - , + , , 2 , # 1 . 0+ - ' ' ' Proper Vertex-Coloring and Chromatic Number DEFINITIONS

:

; '

< . 0 ' :

' ' - ' = >

© 2004 by CRC Press LLC

342

Chapter 5

COLORINGS and RELATED TOPICS

' , ? - : ; - ! , (' - + , . )

, 0

< :

1

:

;

! -, 0 ½

, ?

! 0

:

<

½ .0

<

-

;

½ .0

.@

'

' - ! ,

' !

. 0! : 1 ' +

.0+ - ! :

;

EXAMPLES

:

1

:

;

-

.

0 0 < .0 .

A ' '

' '

( + ' - ! + , ' !

FACTS

:

; ' )

0

.

' -

:

+ , .0

3" BB6 1 ,

"

.

0

´µ

.

1 02 $

' +

)

.

0 <

C

0

.0

.7 0 , .0

¼ · : D : Ê Ê + ' , .70 < (' < . 0 + ' - + - ' - .0+ .0 .0 ¼

' - )

© 2004 by CRC Press LLC

Section 5.1

343

Graph Coloring

List Coloring and Choice Number E ' ,

DEFINITIONS

:

' , ' , ' -

; .- 0

-

:

'

1

.

:

('

<

:

;

:

1

'

1 = > '

,

.0 '

+

0 ' - !

'

+

' ! ' - !

!

+

'

+

" .0+

-

.( ' +

.0

'

" .00

EXAMPLES

-

:

1

:

1 ( !

!

!

, . 0 . 0 . 0 . 0 . 0 . 0 ! ! + ! +

. ' , . 0 . 0 . 0

+

-0 ( + ,-+ ' - !

+ , '

- . '

-'0

FACTS

:

:

3

*1&56 1

:

:

" . 0

.7 0 , .0

38776 ('

. 0 .0

.4 8 3;56A 31 560 ;

-

. C .00

'

7+

<

. 0

< . C

< . C

.

0 '

. 0 7

.00 . 0 ,

" . 0

.00

" . 0 < . . 00 ' . 0 A . . 0 . . 000 ' , +

+ '

3;8@55+ 556 1

' '

. - ' +

© 2004 by CRC Press LLC

0 + !

344

Chapter 5

COLORINGS and RELATED TOPICS

The Hajós Construction FACTS

:

3F$6

- '

'

' ' ' , :

.0 ( , - G

+ '

.0 F- - !?

. < 0+ -

,

+ + ,

.0 ( ' ? -

:

35$6

- '

'

'

' .0

.0 -+ ' , :

< . 0 ! , ? - - + ' , ¼

. 0 F-

,

'

Lovász’s Topological Lower Bound DEFINITION

< . 0 + ' - :

.0

1

'

, - - '

FACT

:

3D &B6 ('

+

0 C

.0

'

!

!

.

Alon and Tarsi’s Graph Polynomial Characterization DEFINITION

:

1

+

< . 0 , < -

< . 0 :< -

'

<

<

+

¾

+ , ) < . 0 :<

¾

.

.

'

0

0

FACT

: 3;156 ; ! ' '

© 2004 by CRC Press LLC

Section 5.1

345

Graph Coloring

List Reduction FACT

:

31 5&6 D < . 0 + ! . 0+ + ' , - - (' . 0 !

+ , .0 + ' + ,

' - ( + ".0 ". 0 C

5.1.2 Vertex Degrees DEFINITIONS

: 1 . H

IF?0 ' +

.0 .

0 - ' , ' - + - ', ,

: ; ' '

. .0 . 0! 0 FACTS

:

# -

+

.0 .0 ".0 .0 J.0 C ,

:

.0

+ $ '

. I 1 0 3 $6 # < . 0 ! + .0 C .0 C

:

3 *1&56 1 I ' : ".0 C ".0 C

.0 C .0 C

:

3F@/&76 - < . 0 , J .J C 0!

:

."

2K 1 0 3"6 ('

+

.0 J.0

:

J.0 < 3&$+ *1&56 . "

2K ' 0 ('

+

".0 J.0

J.0 < / 0

© 2004 by CRC Press LLC

.@

'

346

Chapter 5

COLORINGS and RELATED TOPICS

REMARK

:

' " J C + "

2K

/ < J C " < J C 1

+ < ' ' . 0 @ '+ 315&6 ' ) ' < " < " + ' " < FACTS

:

3 *15&6 ; !

' ' -

' ! - - + - + - ! - , , ,

:

3;776 1 ' , 7 +

".0 . 0 ,- - ' - + ".0 . .00 .0 .0

5.1.3 Critical Graphs and Uniquely Colorable Graphs 1 , : ! . 0! + ! , ? ! DEFINITION

:

#

+ ! < . 0 ' . !0 <

' -

! A ' . 0 <

' - -

FACTS

: - !- ! + , - + !

:

31 &B6 ; - ! ' ' ' 2 - !

: # < < + =! > ' ' =!- ! > 1 ! + !

:

. "

2K 1 0 - ! + ' . 0!+ < <

: 3E56

- !

© 2004 by CRC Press LLC

< .

0 '

Section 5.1

347

Graph Coloring

: 3$6 (' ! + - '

, - 2 -+ ' " !' ' , 2 + ! , - ' " .1 " 0

: 3@ B 6 # + ' + ' 2 1 ) ! , - ' ,

-

. 0!

: 3D &6 ; " ' ! ' '

" ' "! .' ! ." 00 . 0!

: 3@&+ 1 &6 1 ) ' ! -

, .(

2 , , # ! ! 0

: 3 &6 ; ! .

0

-

: 3$6 ; ! . 0 -

C

C

¾

: 3D &6 # !

- + L + ' ) -

' + : 3*M 1B 6 D $. %0

' , - % .+ ' 1N + ' B0 (' . C 0! $. %0 - +

%! + $. %0 1

.# . C 0! + $. C % C 00 : 3*M 1B 6 D + , - & @

' -+ .0 - .0 C + .0 .0 C 1+ ! ' + - .0 & .0 .(

2 , , - G -

! , - ' - G0

: . *M 31 B 60 1 ' ! ¾

# + ' #

OPEN PROBLEMS

:

' ! - . )+ 0 : .4 O O *M 0 # + ' . 0 - . C 0! , , - P

© 2004 by CRC Press LLC

. 0 - !

348

Chapter 5

COLORINGS and RELATED TOPICS

FACT

: 388 6 1 ' . 0 + + - QK ! Q K EXAMPLES

: 1 ? ' , ? ' $ C $!

< $ C - , C .3 60 E + ' ! - . # 0 + , # 7 : % < B$ C - + . 0 ? - ' $ C + ' + ? + ' ' 1 ! + , $ .31 &760

Uniquely Colorable Graphs DEFINITION

: ; < . 0 ' ? ! , < .0 . ' ' 0

1 ! + ' , ,

EXAMPLES

: :

1 !

(' ! + - ' , ' + ! ( + ! '

' , ! - ' ., ' , - - ' '0

FACTS

: 3 $56 - ! . 0! : 3" &B6 (' ! - + ,

+ ! E -+ , ' )

: 3" &B6 D ! - + , ! ,

' , (' + ! E -+ , ' )

5.1.4 Girth and Clique Number 1 , , '

2 ! A , + ' - -

© 2004 by CRC Press LLC

Section 5.1

349

Graph Coloring

FACT

: 3 56 # #+ ' - ( !! ' ( # - E -+ ' +

' ! - , , R. 0

REMARK

: 1

' 3 56 ' /

H K # ( + ,A ' (+ + + 3D $B+ *M &56 ; ' + - - *? . 0 - + ' 3DS@BB6

EXAMPLES

: 3T56 (' ! !' +

' - ! ? ' . < 0+ , , ' / . 0 1 ! . 0 . 0 . 0+ ? 1 . C 0! !'

: 3B6 (' ! ' $+ 2

' ) < . 0 C+ , & < - !?

' . < &0 # ' & . 0 ! * + , ' , * . 0 1 . C0!

' $

: 3E 6 # - ' ! !' + 2 , - ? ' + ? , - 1 . C 0! !'

+ '

: .8 0 3D &B+ "N&B6 1 - ' 8 . 0 ! ' . 0+ , - ? ' ' , ! ? 1 . . 00 < C + . 0 !' ' E -+ - ! ' . 0

- ! - ' -

.3@&B60

CONJECTURE

: 3*5&6 # - + .0

.J.0 C .00 C

FACTS

: 3*5&6 # - & J + ' J.0

J .0 J.0 C &+ .0 J.0 C & E -+ 7

' J.0 + .0 . 0.J.0 C 0 C .0

: # , 2 , 3;?8 @/B76 *

,.) $0+ ' , ' .0

$+ ' # < #.$0+ , - .0

© 2004 by CRC Press LLC

#

350

Chapter 5

COLORINGS and RELATED TOPICS

(' .0 $ J.0+ $ .J.0 C 0 .0 $C

: 3" 8 &&+ &B+ D&B6 D $

: 3 $&6 1

.0 <

.0

, , R. 0

: 34 5$6 # - -

!'+

# ' # J.0 J.0 " .0 J.0

: 34 5$6 (' !'+ " .0 # ' J.0

# J.0 J.0+ '

: 385 6 (' + " .0

,

7 J

. C J.0J.0 0

: 38 E&&6 # - J + !' J .0

,

: 3"76 D < . 0 !' ! + ,

1 :

A -+ ' - + .0 38 &B6 # - J ( < (.J0 .0 J C ,- .0

( J

: 3E@5&6 (' - -

C C .1 ' 0 3*@76 (' - ' + .0 .1

+ .0

:

/ ' 3@B6 - !' , ' ! 0 ; + ' , - , ! / .3*760

CONJECTURE

: . H I#ID -N/ ? 0 1 '

!? '

FACT

: 3856 (' !? ' ' - + .0 < C . 0 " .0 < C . 0

© 2004 by CRC Press LLC

Section 5.1

351

Graph Coloring

The Conjectures of Hadwiger and Hajós CONJECTURES

: .F,K ? 0 3F6 # - ! + ' - G !

:

.F?N K ? 0 # $+ - ! -

' + + - ? ?

FACTS

: :

3 6 "

F,K F?N K ? '

<

3 &56 (' .0 < + - ' , ' ' ' - .(

2 , , - ' , ' '

' - 0

:

3T5B6 1 !

, ! - ' : 3 &56 # - & !

- '

:

39&6 1 ! ' ! '

F+ ' < + F,K ? .

9K 0 - # 1 .' 0

:

3* @156 # < $+ - ' F,K ? ' # 1

:

- + F,K ?

:

38 B6 D ..0

' -

1 ..0 C ..0 $ ' - - +

3" B7+ 8 B6 # - , .0

,

@

'

5.1.5 Edge-Coloring and -Binding Functions

( , , , DEFINITIONS

: ; ' '

' + , -

:

1 ' +

' ! '

© 2004 by CRC Press LLC

¼

.0+ !

352

Chapter 5

:

COLORINGS and RELATED TOPICS

) 5 7 ' ! / !

. 0 !/ !

,+ ' - ! 1

! '

:

:

0A

.

.

+

¼

00

.

< . 0 '

+ ! ,

'

.!0 < .0 ,-

1 '

'

' / '

¼¼

00

" ¼¼.0

.

< . 0 1 .0 ' - ' + , ' ? .0 ' - 1 / .0 - + .0+ -A -+ ?

! / .0 ' ' ! 1 - A , - ? '

:

D

:

0

.

¼

;

- '

¼¼

A

" .0 .

;

.

.

00 '

' '

:

REMARKS

:

+ ¼

0 <

.

0 <

.

.

.00

" ¼ .0

/ .00

"

<

" ..00

+ ¼¼

.

¼¼

.

0 < " ./ .00

( + ,-+ - '

¼

" ¼

'

" +

'

:

1

.0 .

/ .00

., 0 U ! U . 0

1+

J.

E -+

" ¼ .0

¼

0

.

0

.

.

00

J.

0

0

J.

/ .0 ' < . 0 ' 0 .0 , + , ! ? 0 ' ' ! :

1

0

<

© 2004 by CRC Press LLC

Section 5.1

353

Graph Coloring

FACTS

: ./K 1 0 3$6 (' + ¼

.0

J.0 C

(' 1.0+ ¼

.0

J.0 C 1.0

# ¼.0 J.0 ' , .) - 3@560 1+ .0 < C ! ' ' ' + .0 < '

: 38@B6 ('

.0

:

.0 C

.8M

K 1 0 38M

$6 ('

.0

!+

+ .0 < J.0 ¼

: 3E *5B6 1 '

¼¼

.0

J.0C ' ' -

: 35 6 (' + " ¼.0 < J.0 : 3 56 @ ! ,

' ! + '

: # - + " ¼¼.0

"

.0 < (' + " .0 <

¼

.0 C

: 385$6 # - ' J+ " ¼ .0

.0

7 J

¼

¼

. C .00J+ ,

CONJECTURES

: ./ 3$6A "/0 # - , 1.0+

J.0 C 1.0 C ( + ' + .0 J.0 C : !"#$ .; 4 9 F 0 (' .0 < J.0 C J.0 .0 + = -'> + + " ,

." 0 J." 0 ." 0

¼¼

.0

¼¼

¼

: %& ./A A ; A " N F0 # - + , - " ¼ .0 < ¼.0 : .- EL0 E ? &+ ' - ,!'

, - " .0 < .0

O : ." + 8 2 9

A 4-+ E @22 -2A F 4 0 # -

+ " .0 < ¼¼

¼¼

.0

: .8 2 9

0 E ? +

' ' -

© 2004 by CRC Press LLC

"

.0 <

. 0

354

Chapter 5

COLORINGS and RELATED TOPICS

Snarks DEFINITIONS

: ; ! ' - , +

: ;

. ' = - 2>0 !+ !! ' + !! REMARK

:

; , - /K + ' ' J.0 J.0C+ - F+ 2 - ' ' 1 L 2 , - ) @- ' , ' ' ' + + 38 76 3 E*@ 5B6

FACTS

: 3 B 6 (' - ? . &$0 '+ 2 ' B

:

3 B6 (' !V , ? . 0 '+ !! 2 ' &

:

38 5$6 # - (

( !! 2 '

Uniquely Edge-Colorable Graphs DEFINITION

: ; < . 0 ' ? ! , < ¼ .0 . ' ' 0 EXAMPLES

:

1 ! 1 ! - 1 !! . / - ' < 0

:

31&$6 1 ' , 5! 2 2 2 3 3 3 ' 23 . 5+ 50 !! 1 2 , !' ! ' <

FACTS

:

31&B6 1 ! + '

: (' ! ! + ' , =YIJ > ! ! D

© 2004 by CRC Press LLC

Section 5.1

355

Graph Coloring

* - ½ ¾ ½¿ ¾¿ ' <

' - ,

, -

½ ¾ ¿

Further -Bound Graph Classes CONJECTURES

:

:

/+

.N 'N A @0 # -

'

3B&6 1

/

! '

! ' ' ' , -

FACTS

:

38S56 ;

! ' '

'

:

/

3@5&6 # -

/ + '

/ + ! ' ' /

- '

5.1.6 Coloring and Orientation Paths and Cycles FACTS

:

3$B+ * $&6 ;

:

3E$6 ;

:

'

'

.0 ' ' ' ' '

-

,

0 ' -

.

,

'

3156 ( Æ E K - ? ' ' .

0

.1 I* 1 +

0

E - -+ '

! ' ' ( ' +

-+ + '

:

!

3" &$6 ( - '

'

! . 0+

Eulerian Subgraphs DEFINITION

4 < . 4 0 4 < . 4 0 ' 4 , - + , .0 < .0 ' " 4 A + 4 < , - 4 .@ - ' " :

;

0

© 2004 by CRC Press LLC

'

·

356

Chapter 5

COLORINGS and RELATED TOPICS

FACT

4 3;156 (' ' ' , - ' L ' ' , ' 4 ! ,- ) .0 ' + .1 - 0

:

Choosability and Orientations with Kernels FACTS

4 < . 4 0 - "4 ! @

* ' - * ' "4

* (' , .0 ' +

:

:

1 - - ; , * . % 0 ' ! % 3 " . 0 ( + Æ , !

: (' - ' - + , ! + ' + - 8M IF ( + ' ) -! + " .0 C E - ! . 0 Acyclic Orientations DEFINITION

: . ' 3E1 B6 3;BB+ 60 1 , ( + S ; .Q;K0 ! ! ' + S @ .Q@ K0 +

1 - , ? ' 1 ' 5

) ', + , 6 2

9

#.0 ' ,

5 6

FACTS

:

3;115 6 (' < . 0 $ - , #.0 < +

& ' ' .(

2 ,+ ,-+ , #.0 C . 0 '

- 0 : 3;115 6 # - (+ < . 0 , ( #.0

+ '

© 2004 by CRC Press LLC

Section 5.1

357

Graph Coloring

: 3;15 6 # , - + #. 0 < R. 0 , .0

,- 7 ) # + #. 0 < . 0

5.1.7 Colorings of Infinite Graphs FACTS

' ' - : 34 56 # + ) " .0 ' ' - ) ' !

: 3" &&6 (' .0 < + ' - ) 5 + :

3" 6 # + ) ) ' !

,

:

.0

5

38 56 1 .0 ,!) ' - + - !

9!% 1 A - + - ; '

:

38 BB6 1 ' , : 1 < + .0 < + ." 0 ,- ." 0

< . 0

Coloring Euclidean Spaces DEFINITIONS

:

1 ' - A

' ,

: - .) ) 0 = > 7 < + .70 - A , - ? ' ' 7 FACTS

: : : : : : :

&

3F 6 ( + . 0

3E E $6 " ! - - + . 0

3 76 ( ! + . 0

; ! 5 - . 0 3D* &6 ;

+ . 0 . C .00

3#9B6 ; + . 0 . C .00 3@5 6 1 " . 0 ) ' ' <

< .() , ) - 341 5 60

: # - ) 7+ ..700 7 C : # - 7 , 7 < + ..700 .3T760 # +

,-+ ' !

/ '

© 2004 by CRC Press LLC

358

Chapter 5

COLORINGS and RELATED TOPICS

: 3 @2B 6 (' 7 ' + ..700 < E -+ ..7 00 < ..7 00 <

: .W 8 / A 3*1 76A W S 9 @0 D 7

' - + ' ' 7 < ' (' 7+ ..700 ) '

7A ' < 7+ 7 ..700 <

References 3;BB6 E ;+ 4$ " + 9I1+ 5BB 3;115 6 E ;+ 1+ T 1/+ @ ' ' + * !"5 .55 0+ I7 3;?8 @/B76 E ;? + 4 8 N

+ @/N+ ;

* + .5 4$5 " 5 5 .5B70+ I$7 3;56 ; + * ' + I 8 92 . 0+ 4$ + S " '+ D E @ D

@ B&+ - S+ 55 3;776 ; + + 5 5 $ .7770+ $I$B 3;8@556 ; + E 8--+ " @2 -+ D ' ! + 4$ 5 .5550+ I& 3;156 ; E 1+ ' + 4$ .550+ I 3;15 6 ; T 1/+ 1 + 3 5 5 $ .55 0+ $I$B 3"N &B6 ( "NN+ ;

' ' 8K ? + .5 4$5 " 5 .5&B0+ I$ 3" &&6 " " N+ + (5 !"5 5 5 .5&&0+ 5&I5B 3" &B6 " " N + + .5 4$5 " 5 ( .5&B0+ I$ 3" BB6 " " N + 1 ' + 4$ B 5BB+ 5I 3" B76 " " N+ S ; + S H + F,K ? ' - + ) 5 .5 4$5 .5B70+ 5 I55 3" &$6 4 ; " + ? ' D + .5 !"5 5 .0 .5&$0+ &&IB

© 2004 by CRC Press LLC

Section 5.1

359

Graph Coloring

3" 8 &&6 % " ; 8 2+ % ' K + K + .5 4$5 " 5 ( .5&&0+ &I 7 3"76 @ " + ; ! ' !' + * !"5 .770+ I$ 3"6 * D "

2+ % ' , 2+ # 5 4$ #"5 5 & 5+ 5I5& 3" 6 "? S H + ; ' ) ' + & 5 %5 ,"5 # 5 5 .5 0+ &I& 3 &B6 S ; + ; ' + * !"5 .5&B0+ BIB 3 &56 S ; + F?N

K ! ? : - + .5 4$5 " 5 ( $ .5&50+ $BI& 3 E*@ 5B6 ; - + E E+ " *+ # @ + ; - 2 , : + + .5 " " B .55B0+ &IB$ 3 B6 ; + %

- !! + S 1+ - ' 9

+ + 5B 3 $56 S + % + .5 43 $5 " $ .5$50+ &I&B 3 76 + ; ! ' !

+ * !"5 $ .770+ BI57 3B6 " + @

2 5+ ) % 7 .5B0

3 6 ; + ; ' ! 2 + .5 !"5 5 & .5 0+ B I5 3 &6 ; + ; ' * D "

2 ? ' F F,+ # 5 !"5 5 162 & .5 &0+ $I5 3 @2B 6 * "

+ S H

+ 8 @2 + + .5 5 .5B 0+ B$I77

4$5 " 5 (

3 5$6 E D + D ' !' / + 4$ $ .55$0+ I 3 56 S H + + 45 .5 !"5 .5 50+ IB 3 $&6 S H + @ 2 + 4+5 !"5 $ .5$&0+ I $ 3 F$$6 S

H

; F?+ % ' ! +

& .5$$0+ $I55

© 2004 by CRC Press LLC

360

Chapter 5

COLORINGS and RELATED TOPICS

3 *1&56 S H + ; D *+ F 1 +

+ S 9 ! ' + 1 + ; + ! ' + 4 5 & 5 XX( .5&50+ I & 3#9B6 S #2 * E 9 + ( , ! + 4$ .5B0+ &I$B 3$6 1 + 8 (+ #$5 !"5 75 5 5 5 B .5$0+ $ I5 3$B6 1 + % + IB S H

% F 8 . 0+ " " + E @ 4 " + 1 .F0+ 5$$+ ; S+ @ + 5$B 35 6 # -+ 1 ' + .5 4$5 " 5 ( $ .55 0+ I B 38 56 # - S 8 ?N + ' + # !"5 5 .550+ &I& 3 B 6 D + ; ' - ? + I$ + !F E @ + 5B 35$6 @ -+ ; F?N

!2 ' + * !"5 .55$0+ 55I7 3D &6 , D D -N/+ ; ' + !"5 5 5 5 .5&0+ I7 3B&6 ; N'N+ S ' , ' + X(X .5B&0+ I M 3F6 F F,+ 8)2 @ 22 + 8" "5 & "5 5 9 " BB .50+ I M 3F 6 F F,+ 2 2 * 2 E+ # 5 !"5 .5 0+ BI 3F@/&76 ; F? @/N+ S

' ' ? ' H

+ $7I$ S H + ; *N+ 1 @N . 0+ 4$ " 5 77 + E @ 4 " + !F + 5&7 M 3F$6 F?N

+ 8 2 !'M + ,5 5 ! 3 " 35 3,$ !"53& 5 " + 7 .5$0+ $I& 341 5 6 1 * 4 " 1 ' + " 4 # $+ 9!( + 55 34 5$6 ; 4 + ; - ' ' + E + 4 55$ 34 5$6 ; 4 + 1 ' + S - + ; 55$

© 2004 by CRC Press LLC

Section 5.1

Graph Coloring

361

34 56 S 4 + 1 ' + $ .550+ I B 3856 4 8+ !? , C . 0 + .5 4$5 " 5 5 .550+ I5 385$6 4 8+ ;

! + .5 4$5 " 5 & .55$0+ I 5 388 6 4 " 8 D E 8+ S + &$ .5 0+ &B$I&5

38S56 F ; 8 @ S+ * , ' ! + .5 " " B .550+ 5I5 38@B6 F ; 8 4 F @+ @ ' /K

- ' + * !"5 .5B0+ &&IB 385 6 4 F 8+ % "

2K ' + 4$5 # $$5 45 .55 0+ 5&I 38 5$6 E 8 + @2 , + .5 4$5 " 5 ( $& .55$0+ I& 38 76 E 8 + @ ' , ,!/

!V ,+ ) 5 .5 4$5 .770+ BI7$ 38 BB6 S 8 ?N + ) + 7 .5 !"5 $ .5BB0+ B I5 M 38M

$6 8M + ;, ' E+ !"5 5 && .5$0+ I$ 38 &B6 ; 8 2+ + + $&5I$5$ ; F?! 1 @N . 0+ 4$ + E @ 4 " B+ 8/ .F0+ 5&$+ !F + 5&B 38 B6 ; 8 2+ 1 F, ' , - ' - + ! *% 5 5 B .5B0+ &I B . *0 38 B6 ; 8 2+ % F, ' + &I ; F?+ D D -N/+ 1 @N . 0+ - 7: + E @ 4 " &+ .F0 5B+ !F + 5B 38 E&&6 ; 8 2 S E/ -+ ; ' + ! *% 5 5 7 .5&&0+ I5 . *0 381 556 4 8 -NY+ T 1/+ E + , ' :

+ BI5& * D . 0+ 4 * !" + (E; @ @ E 1 @ 5+ ; E @ + 555

© 2004 by CRC Press LLC

362

Chapter 5

COLORINGS and RELATED TOPICS

38776 E 8--+ 1 ' + 4$5 # $$5 45 5 .7770+ 5I$ 3D* &6 D ; * + 1 / ' , + !"% 5 .5&0+ I 3D&B6 4 D,+ - - ' , ' + * !"5 .5&B0+ $I$B 3D $B6 D D -N /+ % ' ) ! + !"5 5 5

5 5 .5$B0+ 5I$& 3D &6 D D -N /+ ( + 5 !"5 3 5 B .5&0+ $ I$B 3D &B6 D D -N/+ 8K ? + +

.5 4$5 " 5 .5&B0+ 5I 3DS@BB6 ; D

/2+ * S + S @2+ *? + 4$ B .5BB0+ $I&& 3E1 B6 E E 1 + 1 L ' ' F

' - + # 5 ;<" -=4 .5B0+ 7I& 3E56 S EN

2+ ; ' "

2K + * !"5 .550+ I$ 3E@5&6 S EN 2 ( @+ ' ' , + + 55& 3E$6 4 E + ; ! ' + 5 !"5 !" $& .5$0+ $I$ 3E *5B6 E E " *+ ;

+ 4$3 B .55B0+ IB7 3E E $6 D E 9 E + S S7+ 45 .5 !"5 .5$0+ B&IB5 3E 6 4 E2+ @ + 4+5 !"5 .5

0+ $I$

3*M &56 4 O O *M

+ ;

' ' ' , + .5 4$5 " 5 ( & .5&50+ I& 3 $6 ; 4 9 + % + 5 !"5 !" $ .5 $0+ & I&& 3*76 " * + ! ' (: / + * !"5+ 3*@76 " * ( @+ , + *5 !"5 " " .770+ $&IB

© 2004 by CRC Press LLC

Section 5.1

Graph Coloring

363

3*5&6 " *+ + J+ + .5 " " & .55&0+ &&I 3* @156 * + S @ + * 1 + F,K ? ' !' + 4$ .550+ &5I$ 3*M 1B 6 *M

T 1/+ % + .5 4$5 " 5 ( B .5B 0+ 7I 3* $&6 * * + K + -7= .5$&0+ &I 3*1 76 ( T */+ T 1/+ E + , ) + .5 4$5 " 5 ( B .770+ BIB& 3@5 6 4 F @+ 1 ! ' + $ .55 0+ $ I$B 3@&B6 ; @?-+ ! ' 8 + , ; 92 $ .5&B0+ I$ 3@5&6 ; @

+ ( ' + .5 " " .55&0+ 5&I 3@B6 S @ + ,!/ $!V ,+ .5 4$5 " 5 ( 7 .5B0+ 7I 3@56 @ + ; ' , 2+ .5 !"5 #"5 B .550+ BI 3@&6 E @ - + % ! + 5 !"5 5 & .5&0+ $&IB 3@ B 6 E @ /+ ! , ? !- + $5IB F @ . 0+ " " .S ' 1 + + *+ 5B0+ 1!1 / E 2 &+ 1+ 5B 3@B6 S @+ @ ' + &I &$ + S ' 8/

.E0+ 9+ 5B 31&B6 ; 1 + F + 3 * !"5 .5&B0+ 5I$B 31 &76 " 1 ' + % ' ' ! + 5 !"5 5 .5&70+ $I&7 31 &6 " 1 ' + 1, ! + 5 !"5 5 & .5&0+ BIB5 31 &B6 " 1 ' + ; - ' ! , ' , ! - + * !"5 .5&B0+ &5IB&

© 2004 by CRC Press LLC

364

Chapter 5

COLORINGS and RELATED TOPICS

31 B 6 " 1 ' + @ ' + &BIB$ * " 2+ F @+ 9 . 0+ " - "

"> -" 5 , + " #/2 + 5B 31 6 9 1 1

+ ; ' + 45 .5 !"5 $ .5 0+ B7I5 31&76 9 1 1

+ E + 5I * 8 . 0+ 4$ " + "+ 5&7 31&$6 9 1 1

+ F + 5I55 .* + 5&0+ 1 (+ ;

- D+ &+ ; / D+ * + 5&$ 3156 T 1/+ + .5 4$5 " 5 ( $I

.550+

315&6 T 1/+ , U ; -+ *5 !"5 " " & .55&0+ $IB 315&6 T 1/+ ! ' F + + 55& 31 56 T 1/ E + '0+ F+ 55

+

$ %

.

31 5&6 T 1/ E + D + * !"5 &5 .55&0+ &I $

3$6 /+ % ' ' ! + ! *% 5 5 .5$0+ 5I& . *0 3&$6 /+ - ' + ! " #" + 5 .5&$0+ I7 . *0

3556 F + %

. 0 ' .5 " " .5550+ 7I$ M 39&6 8 9+ ' 8 + !"5 5 .5&0+ &7I 57

3T5B6 9 T+ , ! + * !"5 B .55B0+ 7 I 3T76 X T+ ' , ' + .5 " " .770+ 5 I7& 3T56 ; ; T2 -+ % ' + !5 $ % .570+ $IBB . *0

© 2004 by CRC Press LLC

Section 5.2

5.2

365

Further Topics in Graph Coloring

FURTHER TOPICS IN GRAPH COLORING E # @' @ # 1 ' S ' F ; *'

Introduction ( , - ' + ' @ ' .+ ' ' 0 -, - ' - ! ' + - ' , - @

/ - ,

1 - @ ' .S 0

5.2.1 Multicoloring and Fractional Coloring DEFINITIONS

:

Ê

;

' ' : ' - < . 0 '

.6 0

1 )

' .+ ' ! ! 0+ , ' - ' ' -

:

6

1

'

.0 :<

. 0

, 2 - ' - ! ' .1 " 0 .0 ) 0 - + ' ' < . 0 ) - ' 6 6 6 .

0+ , ' (

.0

: # , ' ( ' Æ+ , (.0 .0 ' + < . 0 . (0 ' ' - , < .0

© 2004 by CRC Press LLC

366

Chapter 5

COLORINGS and RELATED TOPICS

' ' < (.0 ' ' ' < ' 1 ' G ( + + =.2 30!

> .0 < 2 (.0 < 3 ' (' (.0 < ' - - + , EXAMPLE

: # - $ + ' .$ C $0!

.3;1 5&60 ' C FACTS

: # - + , -

.0 ( + '

' +

: # - < . 0+ , -

. 0 . 0

.0 < .0

.0 .0

: # - - + -

.0 <

'

.3 4 &B60+ ' .' $50 ) ' , ' .3#5 60

: 3*@5B6 # - $+ '

.0 $+ ' . ' F,K ? 0

: 3D & 6 # - + .0 . C .00 ; + - 0

." 0

." 0

- C .,

'

: 3*576 # ' ' + 1

? .'

0 + +

.0 J.0 C ' -

: 38776 1

' + + ' - 7 < .0 .0 " .0 . C 0 .0+ ' - : 3;1 5&6 # - + - ' 23 .2 30!

.0

: ."

2K 1 ' 0 3 *1&56 @ +

' 2 (' .0 ' ' - - + !

: $ % 31 5$6 # 5+ .& &0!

' & : 1 D * . 31 5&6

+

0 ' .& &0!

: 31 5$6 - !

.& &0!

+ ' - &

© 2004 by CRC Press LLC

Section 5.2

367

Further Topics in Graph Coloring

OPEN PROBLEMS

: : " .

3 *1&56 ( - .2 30!

.2& 3&0!

' 3 *1&56 - '

" 0 " .0" ." 0P

& P

" ' - +

REMARK

:

; Æ - , S , Æ - , S ! ' ' - 31 5$6

5.2.2 Graphs on Surfaces F + ' ' + ' , ' ' + , - - ! ' + ,

' ' ' ' ;?

' ' ' A ' -

, ' ,2 DEFINITIONS

: :

; , -

; . '0 ' ' '

: ;

' - ,2 ' ' FACTS

: : :

.#- 1 0 3F:B576 - ! .# 1 0 3; F&&+ ; F8 &&6 - !

31:BB76 ;

! ' ' !!

:

3" &56 - ! '

, '

: :

.M

/K 1 0 3 56 -

¿ !' !

3F:B5B6 ; ! ' ' ' - - -

: - !

.31560+ !!

! .3 560

: :

38156 -

¿ !' !

315 6 ; ' )- !

© 2004 by CRC Press LLC

368

Chapter 5

: : :

COLORINGS and RELATED TOPICS

3;156 ; !

3 5 6 1 !!

¿ !'

3 F&6 - - , ,

: 3F56 1 ' , REMARK

:

;

' # 1 - 3* @@15&6A '

' , - ' 2 ,

Heawood Number and the Empire Problem DEFINITION

:

1 ! " ' '

". 0 <

&C

E + ' -

" . &0 <

6 '

5

& , ,

$& C C

.$& C 0¾

FACTS

:

3F:B576 (' ' 6 +

' , 6 U - + 6 U , " . 0

: :

3#6 - , 8

$!

3*W $B6 %

' 6 8

+ ' 6 " . 0

:

3 6 . < < 0+ 3;F&56 . < 0 (' 6 ' ' ! +

8

+ - " . 0! 6 ´µ

:

3"M

E @ 556 9 ' < + - ' 6 ' , " .0 < " . 0 ´µ

:

3F:B576 (' ' ' '

& .& 0+ , " . &0

, L .# $& 0

: 34*B6 # - & + , ' + , $&

© 2004 by CRC Press LLC

&

Section 5.2

369

Further Topics in Graph Coloring

: " . &0 ' - & .F 1 3B760+

? - .34*B60 8

.34*B 6 ' 3" B56 ' & < 0

&

OPEN PROBLEM

:

. S 0 # , ' 6 ' , - ' & 6 , & . &3 0+ " . &0 ' ' P Nowhere-Zero Flows DEFINITION

4 < . 4 0 ; " D #" ' 8 :

:

8 8 + .

' - -

0<

.

0

FACTS

:

31 6 ; ! ' ' , ,!/ !V , .1 ' '0 ( + # 1 - - , ! ,!/ !V ,A M

/K . ' 0 - !! ,!/ !V ,

:

31 7+ E$&6 ; ! ' ' ,! / !V ,A !! ' ' ,!/ !V , ( + ' / ' % .+ 0 ' - !

:

3@B6 - , ! ,!/ $!V ,

CONJECTURES

: '() !V ,

:

31 6

- , ! ,!/

31 6 - !! ,!/ !V ,

Chromatic Polynomials DEFINITION

: 1 . 90+ 9 + ' < . 0 ' , - 9 .9 < 0 ' : 9

' , 9 F+ , L - '

© 2004 by CRC Press LLC

370

Chapter 5

COLORINGS and RELATED TOPICS

EXAMPLE

: 1 '

- + -+

.

90 < 9

.9 C 0

9 . 90 <

Z < 9.9 0

FACTS

:

!

. ! # 0 # -

< . 0 -

+ , - . 90 < . ! 90 .! 90+ , Q!K Q!K

' !+ -

: 3"6 (' - + . 90 '

9+ , Æ + .0 9 . 90 < 7

:

,

. ( 0 31&76 ('

:<

C

- +

. : C 0 < : ¿

½¼ ¾. :

C 0

REMARK

: ; ' ( . : C 0 7 ' - .: C < $B0+

!'

' ' # 1 .;

' ' + . : C 0 / ' - 0

5.2.3 Some Further Types of Coloring Problems Variants of Proper Coloring

9 V ' + , ', ' DEFINITIONS

: ; $ ' - ! : - - ' ' .0

: 1 $ ' '

: ; ' - ! ' ?

: 1 '

"

: ; ' - ! - ? - -

© 2004 by CRC Press LLC

Section 5.2

:

371

Further Topics in Graph Coloring

1

"

' '

3! : ; 9 . . 0 0 - ! : 7 .+ C 0 ' .0 .0 + ' - - .0 < .0 : - / 7 + / : .0 .0 / ' : ; ' .0 ., 7 .0 ' 0

- + , ' .. 0 .0 ' 0

:

;

'

' - !

' ,

FACTS

:

0

.

1 ' '

?

%

+ ' )

I ' I

'

@+ + 3 @&56+

3FF"B6A 3@B6 , L

:

(

NP!

+ - ' .3

! - ' ' .3

0

, '

7 .38 8760

5&60+

5B60 (

+ '

REMARKS

:

9 ' 3(E556 381 76 ' -

:

3! .+

0

%

.+ ,

:

J C

9! 2

,

# ' '

:

9! +

'

( - '

:

Æ

3W56+ 3 85$6+

/ ! + 7 /

@+ + 3156+ 395$6+ 3 DT556

< J

0 ' - , J

3" 81D776

1 ' / - '

, 9 ' - 3T76

+ 3" #48E 76 - '

:

1 ' - / ' -

@ 3

5&6 ' -

Graph Homomorphisms DEFINITION

:

;

' < . 0 "

:

© 2004 by CRC Press LLC

.0.0

'

< .

0 -

372

Chapter 5

COLORINGS and RELATED TOPICS

; -, ! - - E -+ " "

1

" " ! + "

" ! 1 , .+ , .0.0 0 FACT

:

31556 D .0 1 , # ' " ! + Coloring with Costs DEFINITIONS

: ; ' < # # # 7 , ( , ' 7 # # + ' .0

: - < . 0 ' + : ¾ # 9

[ .0 ' - ! ' (' ' < +

) [.0 :< ¾ .0 '

: 1 ' , ' ' ! (' ' < + ' ' FACTS

:

31 ;E@B56 ('

:

& +

B&

[.0

3EE 5&6 # - ) '

.& C 0

' / ,

:

31576 ' - ' ) # - ) + .. C 0 . 0 0 E -+ ' - ) +

, ' )+ , - ' ) '

:

3EE 5&6 - ' J

:

' <

J C

1 # $ ' - J , .3EE @5&60 , ' < .3495560 REMARK

: " + -

, -

- ( ' + - ' -

- # ' + + + 3E76

© 2004 by CRC Press LLC

Section 5.2

373

Further Topics in Graph Coloring

Vertex Ranking DEFINITION

: ; ' < . 0 . 0 : , ' , - ' + - I - ; , .; 0 .0 1 ' + + ,

.0 (

' +

EXAMPLE

:

3" 488EM 15B6 1 '

.

0 2

.

C

(. 00+ , ' (. 0 ) - , (.0 < + (.0 < (.0+ (. C 0 < (. C 0 C ' - . ' -0

FACTS

+ # <.0 .0A ' 2 + & &0!

' - & .31 5$60 : 3" 488EM15B6 (' .0 < .0+ .0 < .0 E -+ ." 0 < ." 0 ' - " ' ' '

' :

# -

.

Partial Colorings and Extensions ; + , , ', '

DEFINITIONS

:

' < . 0 - =

;

! : =

( % + - % + , - + + ! 1 , ! ! ' : # - $+ $% - ' S , - $ :

.F+ 7!S

- ' ' 2 ,

! 0 : (

' - + , ' , ; , '

? +

-

2 , 9

FACTS

< . 0 ! + $ - + ' - : 3;F776 (' ! , - +

' - $!

- :

('

© 2004 by CRC Press LLC

374

Chapter 5

COLORINGS and RELATED TOPICS

:

(' !

, - + - + ' - $! . $ 0 3 556 E -+ ' ' + $ - ' 3476 .( ? 3;F776

, - ' $! 0

:

. ' 0 (' / ' + ! ! ' '

/ .38S1560A ' ! !'

! ! .3D560 REMARKS

: ( - + Æ Æ - ' ' ' 3F15$6

:

* ! - 385B6

Partitions with Weaker Requirements REMARKS

: 1 , - ! !

+ ' ,2

) " 3" + + ' ' " ' .0 ' " ; ' - 3" "#E@5&6

: @ - # ' 2 ' + + + 39 76

5.2.4 Colorings of Hypergraphs " ' ) . 0+ ,

) ', ' ' ' !- ' = > # + ' - 3 76 DEFINITIONS

:

; ! < . " 0 - A " ' 9 " < + ' " : ; ' ! : ' ! - + - : ; ' !

'

- !?

:

1 .!0 ¼ .!0 ' - ! + - & + + ) '

© 2004 by CRC Press LLC

Section 5.2

: :

375

Further Topics in Graph Coloring

; - ' -

- - 1 - ' . -0+

# +

< + ' -! ' : ; ! < . 0 - + , ' :

' 7 + - ( ' < + ' ' < + 7 ' < 9

' + - .' ! 7!0 , -

: ; ' - ! , + - ' ! , - , - 7! , - , L .( ,+ 7! ? 0

*

: ; ' + ' K

: (' ! + ' " +

.!0 .!0+ -

: ; ! ' ! . ' 0

.!0 < .!0

EXAMPLES

: (' ! ' ! + .!0 < + - , A ' ! 7! + .!0 < + -

:

1 .-, 7! 0 .# 0 < A , -, ' ! + .# 0 < - (' < # + < # +

. 0. 0+ !

: 3 D & 6 D < + , < ' -+ -! ' ' ' - - , - 1 -!'

! + ' - : 31 776 1 ' , ! < . 0 : ! ' ! < . 0+ < + < + ! * ' ! ' ' * FACTS

:

3 $6 # - - + !! -!' , ', -

: 3*@776 # Æ - ' -+ - -!' , 7& ! + Æ

) ! .1 -

:

¿½

- 3"&B60

.@ 3D $B+ *M &56 0 # - ' - ( + -!' , (

© 2004 by CRC Press LLC

376

Chapter 5

:

3876 # - -

,

Æ

; + '

,

:

¾

COLORINGS and RELATED TOPICS

- ' - !- ' < . 0

7 5

.

7 )0

-!' , - < . 0 " .+0 < . C .00 .+ 0 , .0 ' ' Æ ' + " .+0 .+0 < . C Æ 0 - 7

38 @ 776 ('

- .

C 0! +

!

:

31 776 1 - '

:

< .

0 ,

<

-

.

0

.

0

+

! + ' .

$

¿

0

!

31 776 1 D

< . 0 ' -

+ . 0 C ¾ - ' ' !

' + ' , # -

!

!

:

31 T76

!

-

'

E + -

, $ ¼ , $¼ ¼ ¼ ) ' .+ - ' ¼ ¼ ¼ 0 < $¼ 0+ , . 0 < $ . + - ? , $ , $¼ + , ¼

, ' +

!

!

!

:

!

34E1 976 # - )

:

!

!

! 6

!

6

! ' '

!

.388S 776A ; 0 (' '

) + ' - '

' ,

!

!

REMARK

:

@ , 3E15&6+ - ' @ !

. 2 0 # -+ 3E1 76

Clique Hypergraphs DEFINITION

:

"

1

'

< .

0

-

( ,

OPEN PROBLEM

:

A

' , '

3@@9 56

P

© 2004 by CRC Press LLC

0

.

.

' -

0

Section 5.2

377

Further Topics in Graph Coloring

FACTS

.0 ' - :

+

< ' + . - 0 : .0 ' .3@@9 560+ ,!'

: 3"S@6 1 ' , -

, ' .3"S@60

: 38156 # ' +

.0

.0 + ? -

: 3E @2556 (' + !

'

5.2.5 Algorithmic Complexity FACTS

: " .+ ! 0 / ! : 1 !

/ . ! / 3 *1&560

: 3E$B+ #@$56 1 : 38&6 # - + NP! , .0 ; +

NP! , ' ! .34 @ &$60

: 3F B6 . < 0+ 3DB6 . !

0 ( NP! , !

: 3"D 5B6 # - ) + ,

- ! + ' ,

: 3D @B6 1 ' ' !

: 1 ! ' , +

! ' ' + .3*@760 %

+ ! ' !' ! ' !' NP! .3@9 760

: 38176 # - + NP! , ! '

, .0 < !

: 38176 (' + ' ! .

' 0 ' ! -

: S ! + ! + !

%

.39560+ #P!

' ! '

© 2004 by CRC Press LLC

378

Chapter 5

COLORINGS and RELATED TOPICS

: 3156 - , ! - +

! ' REMARK

: 1 ! ! ! Æ 1

' '

# 1 ! + ' Q K ! 2 , FACTS

: 38819 76 # ) " + -3." 0 '

" .0 ' -3." 0 ' " ' ' .

' , - 0+ NP! '

"

: 3F576 # - ! " !

" NP! ,

: 3776 1 ' " ., " )

0 ' ' " ,

' - + ,

+ - A #P!

, : 3FT5$6 @ ' , ' " : ;

/

" ! ' ' /

/ " 1 " !

: 399 56 1 , /

: # -

+ \! , ".0 .315&60 (

\ !

.35$60

/ NP!

'

< ' < !'

: 34@5&6 ( NP! , , - - : 38156 1 D NP! ! ' ' , . ' 0: - + - + - - %

+

- ' - ' - + - + - - : 3D &6 # -

+ NP! ,

! ( NP! ' < !'

: 31 T76 ( NP! , A - , + !NP! ,

!

!

: 3E76 # -

+ ' [!

\ !

! +

, - - .1 ,

' - ! + \ ! 0

© 2004 by CRC Press LLC

Section 5.2

379

Further Topics in Graph Coloring

:

% NP! + .38@B560 ' .387760

: 345&6 (' ' + NP! '

:

3F156 S ' + + - , Æ . 0

" E A NP! ( NP! - ' ? .38560

:

3"F156 % - + !S - + !S NP! .% - + S 2 , NP! 3E760

:

!S ! - .3E760 NP!

.345&60

: 3" 488EM 15B6 ( NP! 2 ' !

+ ' '

: :

3DW5B6 1 2 ' NP!

1 2 ' '

.31 @5 60+ ' ' / .3" 488EM 15B60+ - .3;F560+ , ' .3"88EM 760

: 381556 ( NP! , + 2 ., ' + ' 2 / 3" 488EM 15B60 Approximation DEFINITIONS

:

D -. 0 : ' ; -. 0 ' '+ ' - - , - .0 -. 0 .0 ; / + + " .0

: ; ) , - . +% 0+ U U ' +

? .+ . 0 .¼ ¼ 0 , ¼ ¼ 0+

C C '

$

$

FACTS

%. 0! ! : 3#85B6 ZPP < NP+ ! %. 0!

:

3" @5B6 P < NP+ ! ' .0+ , 7

' .0+ , .38@5B60

© 2004 by CRC Press LLC

7 1 - ' '

380

Chapter 5

COLORINGS and RELATED TOPICS

: 38D@776 P < NP+ !

! , C

: 1 ,

%. . 0. 00 .3F560+ , %. 0 %.J 0

' )

& .3S760

: 1 !

%. 0! !

.3"85&60 E -+ / ! ,

%.J J 0 %. 0 + , < .0 J < J.0 .38E @5B60 : # .0 ".0 .0+

! ' , + , -

: 3#M5&6 1 L

.0 , $7B5

: 3" F85 6 1 2 ,

%. 0

: 385B6 # - ! -

%

! , . 0 # < < - . 0 .

0+ -

%

% : 3"B6 # - + !

' )

, ' ! ' ; ' ' , " " + ,- " ) ! .3 5B60 # " + Æ

' .0

2 ,

References 3;F776 E % ; + @ + * F+ S + + .7770+ I7

3;F&56 E % ; 4 S F + 1 ' K ! + &' (! + B .5&50+ &I& 3;156 ; E 1+ ' + .550+ I

+

3;1 5&6 ; + T 1/+ E +

' + + $ G$$ .55&0+ IB

© 2004 by CRC Press LLC

Section 5.2

381

Further Topics in Graph Coloring

3; F&&6 8 ; 9 F2+ - ' S (: ! + + .5&&0+ 5I57 3; F8 &&6 8 ; + 9 F2+ 4 8 + - ' S ((: * + + .5&&0+ 5I $& 3;F56 " ; S F+ # ' - + ) .550+ BI 75 3"S@6 "N + @ -+ ; N 'N + E S+ ; @H +

' + + 3"&B6 4 "2+ % ! +

+ .5&B0+ &I&

3" @5B6 E "+ % + E @+ # + S S ! ! ! , + & .55B0+ B7I5 3"6 "2 L+ ; ' ' ' , ' + + .50+ I$ 3"F156 E "N

+ E F? + T 1/+ S ( ( - + + 77 .550+ $&I&5

. ¿ 0! ' ! + 3"85&6 ; " 8+ ; % * + + $ .55&0+ 5I 3" 488EM 15B6 F D " + 4 @ + 8 4+ 1 8 2+ 8 + F EM + T 1/+ *2 ' + + .55B0+ $BI B 3" F85 6 F D " + 4 * + F F' + 1 8 2+ ; ! , + , + + B .55 0+ BI 3" 81D776 F D " + 1 8 2+ * " 1+ 4 - D,+ ; !

' ! ' + S @1; @ 777+ D

@ &&7+ 777+ 5 I7$ . - 0

9

3"M

E @ 556 1 "M + " E + E @ /+ K ! '

! + + .5550+ &I5 3" #48E 76 " 2+ #?-O/+ E 4-+ S E 8+ " E + 1 ' + + 77 . 0

+

.5&50+

3" B56 % " + * ! 8

+ ' + 5 .5B50+ I

,

3" &56 % " + % ' + I$

3" "#E@5&6 E " ,2+ ( " + E #2+ S EN

2+ @O+ @-

' ' + + & 55&+ I 7

© 2004 by CRC Press LLC

382

Chapter 5

COLORINGS and RELATED TOPICS

3"88EM 76 F 4 " + 1 8 2+ 8 + F EM + ; !

' ;1!' ' - + .770+ 5I$7 3"D 5B6 8 "N T D + ' + + B .55B0+ I5

3"B6 @ ; "+ @ - - ! - ! ' + + 7 .5B0+ &I&& 3 5&6 8 ,+ @ + + $ .55&0+ 5I$ 3 5B6 8 ,+ 1 ' + + BB .55B0+ B&I5& 3 85$6 4 8 + 1 + 5 .55$0+ 75I$

. 0! +

3 DT556 4 + !# D+ X T+ ) + & .5550+ 5I$5 3 556 + ; , ' + .5550+ 57I5

/

! +

+

3 F&6 + S + @ 1 F + , ' + + 7 .5&0+ I 3 @&56 ; @ E @2 ,+ @ ' ' + ) + & .5&50+5I 5 3 4 &B6 -N + E * + @ 4 + 1, !

+ + .5&B0+ I 31 @5 6 S 1 + * ,+ ; ; @ML+ % 2 '

+ .55 0+ 5I$B 3 6 ; + E +

+ .5

0+ B7I57

38156 L+ F ; 8 + 9 1 1

+ # + + B .550+ BI$ 3@@9 56 L+ " @+ E @+ * 9

,+ 1, !

, ! + + & .550+ 75I$ 3#M 5&6 * E #M + ; ' ! - ! /!

+ -. / + S ; ; E @ 1 ' + ; E+ 55&+ $I$ 3 5B6 S 2+ F + E-+ / + + $ .55B0+ 5&I

© 2004 by CRC Press LLC

Section 5.2

383

Further Topics in Graph Coloring

3776 E + 1 ' + , + & .7770+ $7IB5 3 5&6 8 ,+ 1 + I& " + D E @ D

@ + + D E @ + 55& 3 $6 S H

+ % ((+ .5$0+ I&

+

3 D & 6 S H

D D -N/+ S ! + $75I$& ; F?+ * * 1 @N . 0+ 0 $ + E @ 4 " 7+ !F + 5& 3 *1&56 S H

+ ; D *+ F 1 +

+ S 9 ! ' + 1 + ; + ! ' + &+ XX( .5&50+ I & 3#85B6 # 4 8+ T 2 , + + & .55B0+ B&I55 M 3#@$56 F!4 #2 F @+ - F @ 9' @2 'M T + &+ 5 .5$50+ &IB$ 3#5 6 #+ # , + .55 0+ 7I75 3#6 S #2+ ; ! +

+ 7

*+ .50+ $I$5

3B76 E + E +

+ .0 .5B70+ I

34 @ &$6 E * + @ 4 + D 4 @ 2+ @ ) S! + + .5&$0+ &I$& 38776 8 E 8+ ! ' + * + + & .7770+ $ I$5 3W56 4 * * 8 W+ D , + + .550+ B$I 5 3D @B6 E M

+ D D -N /+ ; @?-+ S ' ' + I $ * + !F E @ + BB+ !F + ; + 5B 3 56 F M

/+

' / 'M 2' / ' 8+ 1 50+ 75I7

2+23 " 21 2& , B .5

399 56 9 ?+

9/+ 9 + S + .550+ 5I$

+

35$6 @ + 1 '

+ .55$0+ 5I7

© 2004 by CRC Press LLC

+

5

384

Chapter 5

COLORINGS and RELATED TOPICS

315&6 @ E 1+ @ 2 .2 : 30!

+ + 55& 3D56 ; N'N 4 D+ + .550+ BIB

L - ! ' !' +

2

3F56 E E FN + ;

' ' + * + + .550+ 5I 3F56 D @ F + , , + ; EI@(;E @ ; .@ # + 550+ ; E+ 55+ 5 I 7 3F:B576 S 4 F,

+ E + IB

4 * + .B570+

3F:B5B6 S 4 F,

+ % ' ! + 4 * 5 .B5B0+ &7I$

+

3FF"B6 @ E F + @ 1 F + 1 "+ ; '

. 0 ' + S @! ' + 1 + & $ .5B0+ I$ 3F576 S F 4 O O+ % ' ) + B .5570+ 5I7

" ! +

3FT5$6 S F+ 4 O O+ X T+ ' + + B .55$0+ BI5& 3F B6 ( F + 1 S! ' ! + &BI&7

+ 7 .5B0+

3F156 E F? T 1/+ S ((

+ 3 " $ .550+ I 3F15$6 E F? T 1/+ S ((( ' ' + * + .55$0+ I $ 3(E556 * 9 (- # E -+ 1 ! ' + + 5 .5550+ &I 34*B6 " 42 *+ E ' + $ .5B0+ I7

&!

? - +

34*B6 " 42 + *+ @ ' F,

K + , ' + & .5B0+ $I 34*B 6 " 42 *+ F,

K + B .5B 0+ $BI&B 345&6 8 4 % + 5 + D

@ 7+ I$

© 2004 by CRC Press LLC

) +

2

Section 5.2

385

Further Topics in Graph Coloring

34@5&6 8 4 S @]+ / ' !2 + + & .55&0+ I

3476 4 E 4+ ; ' + S @! ( ' + 1 + & .770+ & I&5 34E1 976 1 4+ E+ T 1/+ + " 9 + 1 ' + + B .770+ 75IB 349556 1 4 " 9 + ' , ' + + .5550+ I B 38776 4 8+ ;

' ! ' + + & .7770+ &I $

,

38E @5B6 8+ * E

,+ E @+ ; ) + + .55B0+ $I$ 38&6 * 8 + * + B I7 * E 4 9 1 . 0+ 5 + S S+ 5& 38D@776 @ 8+ D+ @ @'+ % ' + + 7 .7770+ 5I 385B6 F ; 8 + %! ! + 5I7 385B6 F ; 8 + *- ! + + + !F + 55B+ I$5

+ 7

.55B0+

! , "

38S156 F ; 8 + @ S+ 9 1 1

+ %! - + + & .550+ &IB5 38 876 8 / * 8 + % + + .770+ 7BI 38 @ 776 ; 8 2 E @ /+ % ' ! + + 7 .7770+ I 7 388S 776 8NK+ 4 8 -NY+ ; S 2 ,2+ F!4 + + &5IB5 2 + D!

@ 5B+ @ + 777 38819 76 8NK+ 4 8 -NY+ T 1/+ 4 9 + ! ' , ' + I$ ; " M " D+ 2 + D

@ 7+ @ + 77 3856 4 8 -NY+ S , ) + $ .550+ 5I

© 2004 by CRC Press LLC

3 "

386

Chapter 5

COLORINGS and RELATED TOPICS

38156 4 8 -NY T 1/+ ; ' + + 7 .550+ 5&I7 381556 4 8 -NY T 1/+ *2 ' + + .5550+ &IB

38176 4 8 -NY T 1/+ % '

' + + .770+ 7I 381 76 4 8 -NY+ T 1/+ E + % ! ' + 7I7 D 8O+ + 2 + D

@ &+ 77 38@5B6 E 8-- " @2 -+ ; ' ' + * 6 7 + @;K5B+ 55B+ &&IB5 3876 E 8-- F +

+ ) + B .770+ I & 38@B56 82 ; 4 @,2+ ; + 5I * 89 + 5B5 3DW5B6 1 9 D # D W+ + B .55B0+ &IB$

2 ' +

3DB6 D- T + S! ' ) ' + + .5B0+ I 3D &6 D D -N/+ - ' + S @I ' + 1 + &+ B .5&0+ I 3D & 6 D D -N/+ % ' ' -+ .5& 0+ BI57

+

3E76 E+ 1 ' + * $ -::- + D

@ 7+ I 3E76 E+ 1 ? + + 77 3E76 E+ 1 ' + + 77 3E76 E+ S + + 77 3E$B6 9 E + ; ! ' , + ," ' + 7 .5$B0+ BIB 3E15&6 D E// T 1/+ ' @ + + & .55&0+ &I 5 3E1 76 D E// + T 1/+ + @ ' @ : -+ + $ .770+ 55I

© 2004 by CRC Press LLC

Section 5.2

387

Further Topics in Graph Coloring

3E$&6 4 E + ; ! ' - + + .5$&0+ $I$& 3EE 5&6 4 E S E + % ! ' + + & .55&0+ 7I

3EE @5&6 4 E + S E + @+ %

' + + + + & .55&0+ 5I O 3E @2556 " E * @22 -2+ 1 M

/ ' ' + 4 $ .5550+ *$+ 31556 4 O O 1'+ + 5I * D . 0+ + (E; @ @ E 1 @ 5+ ; E @ + 555 3S76 1 S + ;

' ! + + $ .770+ $&I&

$

3*@776 4 *2 ; @-+ ( - ' ! + , + $ .7770+ I 3*@76 " * ( @+ ! +

&

'

!' + 2

3*@5B6 " * S @ + # F,K ? + ) + & .55B0+ &I

3*W $B6 * 4 9 1 W + @ ' F,

! ! + * & 3 + $7 .5$B0+ BI 3* @@15&6 * + S @+ S @ + * 1 + 1 ' ! + ) + &7 .55&0+ I

+ & .5570+ B&I5 3@B6 S @ + ,!/ $!V ,+ ) + 7 .5B0+ 7I

3*576 4 *+ #

+

3@9 76 4 @ 4 9 + 1 ' , + .770+ 7&I 3@B6 4 @ + % ' + S @! ' + 1 + & 7 .5B0+ 5I$$ 31:BB76 S 1 + % ' + * .B&BIBB70+ 7I 7 3156 " ; 1+ D &&IB5 3156 1 + .550+ B7IB

© 2004 by CRC Press LLC

/ !

' +

, 7

+

- !

+

+ 7

.550+

) +

$

388

Chapter 5

COLORINGS and RELATED TOPICS

315 6 1 + ! ! ' + .55 0+ 7I7&

) + $

31 ;E@B56 1 + S H

+ W ;-+ S 4 E+ ; 4 @,2+ 1 ' + + .5B50+ I & 31 76 9 1 1

+ % ' '+ ;-<+ .5 70+ &IB

* +

31 6 9 1 1

+ ; ' + + $ .5 0+ B7I5

31&76 9 1 1

+ E + 5I * 8 . 0+ + "+ 5&7 31576 T 1/+ ! + I 5 31 5$6 T 1/ E + % ? ' *+ 5 .55$0+ $5IB

+ $ .5570+

H

+ * 1 +

31 5$6 T 1/ E + - !

.& &0!

+ + .55$0+ I 31 776 T 1/ + + + 55 .7770+ 75I&

31 T76 T 1/+ + F T + + + B .770+ I$ 3956 D 4 ; 9+ 1 ' 1

: + * + .550+ BIB& 3 56 E + D ' +

+ 7 .550+

3 5 6 E + ;

!

, !+ .55 0+ IB 3 76 ( + 5 % + ; E @ + S -+ 77 395$6 ; % 9+ ; ' .55$0+ &I

/

I5

+ $

= 2

! +

) + + B

39 76 * 9

+ D ' + * ) 2 + D E @ D

@ BB+ - S+ 77 3T76 X T+ : -+ 7

© 2004 by CRC Press LLC

+ 5 .770+ &I

Section 5.3

5.3

389

Independent Sets and Cliques

INDEPENDENT SETS AND CLIQUES

" ) ; ( S # ; " ( ; $ F *'

Introduction # 9 - --, '

5.3.1 Basic Definitions and Applications ( + + +

- '!

! Some Combinatorial Optimization Problems DEFINITIONS

: # ?

:

+ 6 ' - ' , - 6

1 ' - !/ ' .0

'

: ; ' ? - ' 1 +

.0+ ' - '

: ; 6 ' -

' - ' 6

:

;

' !? '

REMARKS

:

1 . .0 ' - 0 . .00 %-

!

© 2004 by CRC Press LLC

.0 < .0 ' + ,

390

Chapter 5

COLORINGS and RELATED TOPICS

:

; ' )! - - '

:

@ , , , ! - , -

EXAMPLE

:

( -' ' S , # .0 < + - - /

Figure 5.3.1

.0 < +

$ & #$,

Vertex-Weighted Graphs DEFINITIONS

: ; " ' - - ! - , +.0A

. +0 : 1 " 6 - !, ' , ' - 6 1 , ' !,

. +0 1 , ' !,

. +0 REMARKS

:

1 " . . +00 / , : , - - ' ; ! 2 ' "

:

1 ) ' ' , ' ' )

FACTS

: # - - !, . +0+ , ,

. +0 < . +0

1+ -

:

< . 0 - - ' ' 6

:

/ '

; 6 ' -

1 / ' - '

: 38 + 6 (' + / ' ' - - '

© 2004 by CRC Press LLC

Section 5.3

391

Independent Sets and Cliques

Applications Involving Hamming Distance

1 - ' / - S ' DEFINITIONS

: 1 ! , < . 0 - ' ' , <

< . 0 '

:

1 - ' ! " . 0 ' - , ; ' - " . 0 ? ' F ,

REMARKS

:

1 F ' ' ! : ; ! ' ' - + ' , - F . 0 3E@&56

:

; : F , - ,

, , - F P ( - ' ) ' " . 0

: # ' ' + + ' + 3" "! SS556 3%76 % ' 3"S57+FS56+ 3F @2B5+F4576+ , 2 ! 3EE77+ET-6

5.3.2 Integer Programming Formulations 1 ' ' !, ! '

*'"&* -

: D < . 0 - !, , < , + < +. 0 1 , ' ' - ' , : + < ?

+

C ' < 7 <

FACTS

:

( ! ' + - ' - 6 ' ' ' ,: < ' ' 6 : 31&+1& 6 D .7 0!- ! ' ' + > < : < 1 ! ' < ' - >

© 2004 by CRC Press LLC

392

Chapter 5

COLORINGS and RELATED TOPICS

REMARKS

: ' + # $ - ' - !

- ' ' ! ' ! - ' + ,

' ! '

3" "SS556

: 1 - ' ! ' ! ' ' !, ! ! + , -! ' , / ) 3@576

!$ -:

+

+ < ?

< 7 ' < 7 <

@ 3@576 -

'

Two More Formulations of the Maximum-Clique Problem

9 ' - - ' ,+ ' , E

/2 @ 3E @ $ 6 - DEFINITIONS

:

1 J J < , :

#

5 "

,

) ' ,:

7

<

, ? 5

<

-

,

+ (.0 <

: D < . 0 , - 6 1 , ) ' ,: < . 0+ , < ' 6 < 7

, FACTS

: ./0'!& $- 3E @ $ 6: D < 2-(&2(.0 : J .+ J ' , (.0 0 1

.0 < (. 0 (.0 ' J

E -+

6 ' - ' ' ' < 2-(&2(.0 : J

# , ? 5 + .0 < " 5 C ." 0

© 2004 by CRC Press LLC

Section 5.3

393

Independent Sets and Cliques

: 1-/ $- 3" 5&6: D 6 ' - ' 1

6 ' ' < 2-(&2 .0 : J .0 6 ' ' / ' .0 : J .0 ; / ' .0 : J - .0

REMARK

: % ,2 ' E

/2!@ ' (.0 : J '

' /

- 3SS57+ S45 6 1+ " /K - ' E

/2!@ ' '

5.3.3 Complexity and Approximation 1 ! ' ) , S! 38&6 @

- ' Æ

' .+ + 3; 8ES55+ " "SS5560 % ' ' Æ - # 5

? ' ! ! + ?.0

? , .0 (' @ ' ! + @.0

@ ,

.0 ('

FACTS

: 3F556 D @ ! ' !

S < S+ ' .7 6+ !- ', .0 ½ ¾ -

: 3" F56 1 ! + .0 ?.0 < . ¾ 0

?

@.0

'

!-

: 3#6 1 ! ? ' !- +

.0 ?.0 < . . 0¾ ¿ 0 REMARK

: # 7 . 0

' ' S 3F5B60

H .

Some Results Involving Maximum Degree

@ # - ,2+ ! ' ' , % J.0 ' # --, ' ! , ' ' J.0+ 3D176 # , '

© 2004 by CRC Press LLC

394

Chapter 5

COLORINGS and RELATED TOPICS

FACTS

:

3;#9T5 6 S < S+ 7 ! ? ' , .0 ?.0 < .J.00 ' -

: - . 3F5B60 1 ! ? ' - + .0 ?.0 < .J.0 J.0 J.00

:

3W76 D @ ! ' ! . 0 ' ' S < S+

!- @.0 ', - ´´µµ ½ ' ' 1 ' ' !- ! - REMARKS

: 1 ! ' !- ! - ! - ! .+ , , 0: ) A - - ' A - - ' #

+ 3; 8ES556

:

1 2 # 5 + ,+ ,

1 ! !- ! - + .- ! !

0 - + , '

+

1 = > , - # %- ' ' ! ' -

5.3.4 Bounds on Independence and Clique Numbers - ! ! - = > , ( + , = >

' DEFINITIONS

: :

1 % +

9# .0+ - '

5 1 ' ( ' - +

B .0+ ' - ? + B 36 < B .0 Lower Bounds

9 , - # + ; @ 3;@ 56 -

' ' * + @2+ 1 2+ W! /2 / # - !, .# $0 FACTS

:

3 &5+9B6 D

© 2004 by CRC Press LLC

< . 0 1 .0 ¾ .!(.0 C 0

Section 5.3

395

Independent Sets and Cliques

< . 0 6 ' , ¾ +.0.!(.0 C 0 ¾ +.0 3 $¾% +.C06 E -+ ' : 3@1 W76 - - !, : 3@56 D < . 0 1

.0

!(.0 . C 7 !(.0 C ¾% !(.0 C 0 ¾ !(.0 C

: 1 ? ' + + ! - A + - .+ + 3F 4 B 60 : 39B$6 #

- + .0

, - " 3"60

& .1

Upper bounds FACTS

: 39$&6 # + .0 9# .0 C ' '

# + ^ .0

' - '

5

: 3;F&6 # + .0 ^ .0 C '

'

: 3"6 # - + .0 :

-2 5

' - - ) , 3"6

. 0+

REMARK

: " 3"6 ' &77 ' 77 77 # + ' ) , , ,

5.3.5 Exact Algorithms Clique Enumeration

F * 3F* &6 ' ! ' 1 - ' .+ + 3" $+F* &+S 560 1 ) ) .# 0

M + E

+ E 3E E $ 6 1

' ' ! - # $

© 2004 by CRC Press LLC

396

Chapter 5

COLORINGS and RELATED TOPICS

DEFINITION

: - + ' !? , FACTS

:

3E E $ 6 1 ' !-

¿

' 7 .& 0 ' .& 0 ' .& 0

.@ ' 3@60

:

3 B 6 1 ' ' < . 0

.2.0 1.00+ , 2.0 ' 1.0 '

: 31 11BB6 1 ' ' !-

. 0 REMARKS

:

1 # - .

1.00

' - 31(;-@&&6

: 1 # $ ) ' 2 2 ' " 8 3"8&6 ( ' H !E

!E # + # $ + +

( , '

' 7 7+ D 22 3D B6 , !) - ' ' 3"8&6+ 31(;-@&&6+ 3D 1B6 D 22K Æ Maximum-Clique and Maximum-Weight-Clique Algorithms

+ - . ' ! ) 0 ) F ,-+ !

! ' ' ! ' FACT

: 3* B$6 1 ' - ! , . ' 0+ , ' - REMARKS

:

# & , * 3* B$6 ' -

' 1? 1 ? ,2 311&&6 1 ! 2 , ' !

© 2004 by CRC Press LLC

Section 5.3

:

397

Independent Sets and Cliques

"!! , , ' - !

/ + + +

Ü$

F ,-+ ' ! +

!! + + ' ! '! ! !! 3* @76

:

1 - Æ !! ' !

,

@+ + ,

3%76+ , " 3"576

; '

' ' 3%76

:

"!! ' !, ! !

3" "SS55+%76

( 3%76+ , , ',

:

@ - '

' # ' ' '

! ' 34 15$6 3FS56 M % _ 3%76 ' ' ' !, !

:

1 ' ! !, !

@ - ' ' - - ' L .+ + 3%7+%60

1 ', ! !, !

' - ' .+ + 36 3%760

5.3.6 Heuristics 9 ' !,

' /

., 0 - 2+ E -+ ' '

Construction Heuristics and Local Search DEFINITIONS

:

;

' ,

-

:

;

'&(

' ' + !

+

'

"$ "

'

+ +

:

;

& .

0

,

,

)

© 2004 by CRC Press LLC

398

Chapter 5

COLORINGS and RELATED TOPICS

REMARKS

: - ' + - 2 , F ,-+

'

: 1 ' !

- (

- ' 38 *B&6 ; -+ - ' - 38 *B&6

:

E ' - / + ' + ' ! ' 3" F56 3#6

D -

'

, ' ! 3ET-6 - D !

- V ' ! A V V ,

Tabu Search

1 " -

! L , DEFINITION

: ) , ,

-

' ! $ REMARKS

:

1 , - 3B5+576 F 4 3F4576 =S> ' ! , ' .+ + 3#F957+@ 5$60

1! - )!

-

1 , , -

' ! "

S

3"S76 , ?

: # ' '

! + 3" "SS556

References 3;#9T5 6 ; + #+ ; 9 + T2+ / + 5 .55 0+ $7I& 3;@ 56 ; 4 F @ + * + 9+ 55

© 2004 by CRC Press LLC

Section 5.3

399

Independent Sets and Cliques

3;F&6 ; 1 ; @ D F2+ ' ' + .5&0+ $5I & 3; 8ES556 ; + S /+ + 8+ ; E

! @ + E S

+ 5 5 + @ + 555 3"S76 * "

E S

+ * - ' + 5 .770+ $7I$& 3"S576 S " ; S+ ' ' + * -: $2 ., + 80+ .5570+ 7I$ 3"576 "+ @ ' + .* * 4 9 + 0+ D + 557 3" 5&6 ( E " /+ .55&0+ I$

- , +

/

7

3" "SS556 ( " /+ E "+ S E S + E S + 1 E S + ! / ; > < .!T S E S + 0+ 8,+ 555 3" $6 * " + % + I

) , "

B .5$0+

3"8&6 " 4 8 + ; &: # ' + $ .5&0+ & I && 3"6 E "+ " ' +

+

3 &56 W + , + 1 * + 1!;-- + 5&5 3 B 6 1 /2+ ; .5B 0+ 7I

36 - ' :GG G GG G -G 3 6 -N + % ' + $IB

+! .50+

3#6 #+ ; - + E + 77 3#F9B56 #+ ; F /+ E 9+ @1;"D@: ; ) , .5B50+ I 3W556 4 D 4 W+ 555

© 2004 by CRC Press LLC

+ * S+

400

3B56 # -+ 1 I S (+

Chapter 5

COLORINGS and RELATED TOPICS

/, .5B50+ 57I$7

3576 # -+ 1 I S ((+

/, .5570+ I

3EE776 + + E + ( E + , 2 + / .7770+ I& 3ET-6 + + E + ; T- - + , 2 + +

3W76 + ; + ; W + ; ' % / S + 5 .770+ I 7 3F5B6 E FN

+ ; ' + *,/[email protected]+ .55B0+ I

* *2

3F4576 E!F F 4+ 1 ' - ' ! ' ? + * , .5570+ I 3F4576 S F " 4+ ; ' ' + .5570 &5I7 3FS56 4 F+ S S S + 2 2+ 1 ' + / .550+ $IB 3F* &6 # F ( * + ; ' + 7 .5 &0+ 7 I 3F556 4 F_ + , ½ + .5550+ 7 I 3F 4 B 6 * ; F * 4 + 5 5B

B

+ - S+

3F @2B56 * F 1 @2 + @ ' + 777 * .5B50+ $BIB7 34 15$6 @ 4 + E 12 . 0+ = 0 % 2 + (E; @ $+ ;E@+ 55$ .

:GG G G $ 0 38&6 * E 8 + * + 5 2 .* E 4 9 1 + 0+ S S+ 5& 38 6 8H + /+

+! B .50+ $I5

38 *B&6 * 8 ' *+ ; ' , ' + $ 7 .5B&0+ $&IB7

© 2004 by CRC Press LLC

Section 5.3

401

Independent Sets and Cliques

3D176 F W D F # 1+ 1

: Æ ' + / .770+ I7 3D B6 D 22+ ; , 2 2 ' ' ' ' + 5 .5B0 BI B5 3D 1B6 D 22 1 + ; ) '! ' ' + & .5B0+ 5I$$ 3E@&56 4 E9 4 ; @ + !F + 5&5

7 +

3E E $ 6 4 9 E

D E + % + IB

.5$ 0+

3E @ $ 6 1 @ E

/2 @ + E ' ,

' '

' 1N+ & .5$ 0+ I 7 31&6 D D 1

+ S ' - 2 ! + * $ .5&0+ BI$ 31& 6 D D 1

+ 2: @ + * B .5& 0+ IB M 3%76 @ 2 S * 4 % _ + K + 7+ 2 += ! 3 " = , 1B+ 77+ 1 ' - ' ( M 3%76 S * % _ + ; , ' !, + & B .770+ I$ M 3%76 S * % _ + ; ' ' + 7 .770+ 5&I7&

3SS576 S E S ; 1 S + ; / ' -

+ .5570+ 75I$ 3S 56 E S @ F + E/ ' ) , ' + ,7 7 !B .5 50+ $I$& 3S45 6 E S ; 4

+ # ' ! ' + & &'! .55 0 I7 3* B$6 4 E * + ; ' + .5B$0+ I7

&

3* @76 # * @ @ + ; !! ' ! + / , + B .770+ $I& 3@6 " @ *

+ E , - + S + 77

© 2004 by CRC Press LLC

402

Chapter 5

COLORINGS and RELATED TOPICS

3@1 W76 @ @2+ E 1 2+ 8 W/2+ ;

'

, + $ .770+ I 3@56 @ E @2 ,+ ; , ' + .550+ $I$ 3@576 T @ + "

+ / .S E S 4 " * + 0+ / , .5570+ $I$B 3@ 5$6 S @ E + 1 ' + 34 15$6+ I+ 55$ 311&&6 *

1? ;

.5&&0+

1 ? ,2+ # + &I $

31 11BB6 1 + ; 12 F 12+ 1 , ! ' ) + , 372,28 5BB 31(;-@&&6 @ 12+ E (+ F ;- ( @2,+ ; , ' + $ .5&&0+ 7 I & 39B6 8 9+ ; , ' + " D 1 E + B!&!5+ 5B 39$&6 F @ 9'+ 1 - ' + .5$&0+ 7I

+

39B$6 F @ 9'+ 1 ' ' + ) 7 .5B$0+ I&

© 2004 by CRC Press LLC

Section 5.4

5.4

403

Factors and Factorization

FACTORS AND FACTORIZATION !" # $

S !# # # # / *'

Introduction 1 - ' , 2 ' ' / ,

' (+ ' / ) - ' 3 .' Ü 0+ ' ! ? E -+ $ .' Ü 0 -, ' !' + , , ' ' : - + - , ) ' ' ' '

5.4.1 Preliminaries DEFINITIONS

: - . + 0 + , " ' ' " ' : ; ' !

: ; ' ) ' ' -

: ; ' ' !

-

: (' !? ' ' ½ + , ' ' FACTS

1 , 2 )

' ' / , ! ' 4 S 1 ' "M ' '

)'

: 3SB56 ; ! , !' . !' 0

© 2004 by CRC Press LLC

404

Chapter 5

COLORINGS and RELATED TOPICS

: 3SB56 - ! !' .+ + ' / !' 0

:

3"M B6 - !! . C 0! !'

REMARKS

: 1 ' , ' ' ;2 8 3;28B 6

:

1 , - # + ) !' ) ' ' ,

: ; - ' S K ' / ' ' 3 5 6

5.4.2 1-Factors 1 ' ' , 9 - Ü ( + , ' !' ' ' Conditions for a Graph to Have a 1-Factor DEFINITIONS

:

; . 0 ' ' - !? , .0 : 1 ½ ¿ ' " ; ½ ¿

"

: - ' - ' - - ? : 1 ' +

.0+ ) C ,

,

6 #. 6 0 6 .0 , 2 - 6 .0 #. 6 0

' ' 6

:

1 ' B . 0

<

+

.0+ )

.0 B . 0 < .0

: 9 ' ! - ½ ' ' - ' , -+ + ½

© 2004 by CRC Press LLC

Section 5.4

405

Factors and Factorization

FACTS

;+ V ' !' ? @3 "

: 31&6 2& ' $-: ; !' ' ' ' 6 .0+ #( . 6 0 6 + , #( . 6 0

'

' 6 , - ' - : 3SB56 &2& $-: !'

- !! !

: 3"MB6 - .- 0!! ! , - ' - !' 1 / S K

: 3@&+ @&$6+ 3D& 6 (' ,!' ' - + !'

: 3@&$6 (' ! ' - + ' ! ½ + !' : 3#F E$ 6 ('

!'

-! ' - ! +

! . 0 ' - ' D .0 . 0+ !' .; &+ D .0

. 0

' 0 : (' ' - .0 + !' 1 ' , : 3&B+ &56 ('

' 1

K !' 1

- !' ' '

: 38&6+ 3D &6 1

: . ; 3;&60 D

.0+

' -

B . 0 .0

('+ '

!' 1 : 3DF & 6 (' ' -

' , - .0+ !' -

F ' - - !' REMARKS

: ; ' Æ ' ; -

39 576 ; Æ '

- !' . -+ 32 36!' . ,00 ' 3 % 8BB6 . -+ 385760

: 1 , - ' !'

) F ,-+

' ' !' + , , -

? 3S5+ S5$6

© 2004 by CRC Press LLC

406

Chapter 5

COLORINGS and RELATED TOPICS

The Number of 1-Factors DEFINITION

: ; ' C !' ' - '

, L - C .# ' + 3D SB$60

`.0

' !' FACTS

:

D

- !' 1:

.0 38 56 !' A

.0 3D SB$6 - ' . C 0A .0 3F . 06 .0 . .0 0

:

! !' + .0 Z !' + .0 ('

( , - ( - Æ `.0

:

(' + `.0 .0 C @ ' " ' + A .0+ . 3D SB$60 / -

:

(' ! !' + ' .0 Æ + Z !' REMARKS

:

,+ 8 1? 38155+ 8176 - . 0 , + !' ) + '

: % `.0 , ' #Æ .# + 3D SB$A B60 ( , + S'Æ `.0

:

1 - ' `.0

,

1-Factors in Bipartite Graphs

( ' + ' !' ,

+ S F 3F 6+

8M 38M

+ 8M 6 DEFINITIONS

' .0 - '

/ ' -

: ; ' - '

: 1 ' -

: : .0

© 2004 by CRC Press LLC

Section 5.4

407

Factors and Factorization

: 1 '

: E .0

:

1 '

/ '

5+

5+ -

5<

2

2

2

½) ) )

, -

? '

FACTS

:

3F 6 +2& $-: D , - .0 < 5 0 1 ' 5 0 ' ' B . 0 + ' 5 : 3#6 . $-: D , - .0 < 5 0 1 !' 5 0 ' ' .0 5 < 0 .0 B . 0 + ' 5 1 ' #

' FK 1

: ( + - ! - E .0 : .0

: :

38M

+ 8M 6

342& $-: (' + E .0 < : .0

3FB6 D , .0 < 5 0 + - 5 (' !' + Z !'

:

D

! -

Z `.0 .Z0

1

1 ) - ' , 48 39$6 + , , - 3#B6 3 B7+ B6 1 , - 3"&6

:

3@5B6 ('

! ' + `.0

. 0

REMARKS

: ( ' + , FK 1 8M K 1 - : @ 8M K 1 ' '

'

+ ' ' 0 + ' # , ) ' + ' + 3D SB$A & 6

© 2004 by CRC Press LLC

408

Chapter 5

COLORINGS and RELATED TOPICS

5.4.3 Degree Factors

0-factors DEFINITIONS

: :

; '

!

; . -+ 0 '

- F . -+ 0+ + ' .0 FACTS

: 3 48 @B 6 (' .0 + -N 3 &60

:

!'

.1 , ?

+ - ! - (' F.0 . C 0 . C 0.0

!' ..0 F.0 ! 3B56 D

- + -0

: 3(56 D - ' + - 1 ' .0 C .0 + ' ' !? - + !' .1 = ' % , ) ' , Æ ' ' F 0

:

3(5&6 D ' + - @ ' 5 . 0 C 1 ' B .0 B .0 .0. C 0 ' ' ! ? - + !' .1 Æ "$ " 0

:

356 D '

-+ Æ .0 1 ' .0 .0 + '

' !? - + !'

:

3###DD556 (' ,!' ' , Æ .0 + !' , 3$ .Æ .0 C 06 E -+ !'

. ¿ 0

:

38&6 (' - ½ ½

+ ½ / , ! '

: 38 B6 (' - ! - + ' !' ' .0+ !'

:

3@56 @ , !' ' - 1 ' !' ' + ' !'

:

38 B6 ('

+ !'

© 2004 by CRC Press LLC

Section 5.4

:

Factors and Factorization

409

& ! - 1 .0 ' & & -+ ' .0 & C + ' .0+ .0 &!' A .0 ' + .0 -+ .0 & C ' .0+ .0 .& C 0!' : 38 576 D , .0 < * 3576 D

- 1 ':

< * + .0 Æ .0 + .0 C + , + , -+ .0

!' : 356 (' , .0 - ' Æ ..00 .5 C 0B+ .0 !' : 3856 D - '

, - (' ' ' ? - ' + .0 C .0 +

F ' !' F+ 3 C 6!'

REMARKS

: 38 9 B&+ B56 - # &+

: 389&6 - -

' ' # ' !'

: @ -! ! - 9 ; - ' ' , - !' 2 , 3" @9 B 6 @+ 2 , , 3*5B6

:

F 3FB6 ' , + !' ! ? ' - , - 4 34 776 -factors

9 !' , - - + !' - ' L DEFINITIONS

: D ,

+ ! -+ !- ' .0 1 " ' ' ' .0 < .0+ ' .0

: ; 6 .0 #( . 6 0 6 .* 1

K !' 1 + , !' ! 0

© 2004 by CRC Press LLC

410

Chapter 5

COLORINGS and RELATED TOPICS

D ! .5 0 0

' ? -

0

5

FACTS

: '

31 6

2& ' $-: 1 !' '

.70 .6 0 C * .6 0 .7 6 0 7 ' ? 7 6 .0+ , .7 6 0

' ' ' .7 6 0 ! . .' 0 6 0 C . .' 00 . 0 .70 .6 0 C * .6 0 .7 6 0 . .00. 0+ ' ? 7 6 .0 : 31B6 ; !' ' '

- ! .; ! / ' ! 9 0 : 38 76 D ,

7 .0 D !- ' .0 .0 .0 ' .0 (' -

' ' !' + !' + .0 : 38 1776 D 2 3+ , - @ '

+ .0 + .0 Æ .0 , + + .0 .0 , , .3 C 2 0

1 ' ' ' .0 2 2 C 3 !'

:

.0 -+

349B56 (' !! , !' -

+

REMARK

:

1 ' !' !' ¼ @ 7 ' 3D SB$6 3

"6-factors

DEFINITIONS

: D 2 3 2 3 ; 32 36 ' " 2 .0 3+ ' .0 .1+ !'

2 .0 3+ ' .00

: D ' ' .0 - ; ' , - .0 .0 . 0 ' : ; 3 C 6!' . 0 '

:

;

32 36 ' 2 .0 3+ ' - - .0

© 2004 by CRC Press LLC

Section 5.4

411

Factors and Factorization

FACTS

:

3DD5B6 ('

! ,!' +

3 6!

'

:

3DT 76 ('

:

3

+ 3

:

.

356 ('

0

.

.

.

+ ' .

3D&B+ ;8B6 D

:

.

$

6!'

23

+ '

0 +

+

1

3

.

0

'

6

6!' ' ' .

0

6

0 1 ' 1

K !'

3W8BB6 D

.

C

6!'

- ' '

-

7

6! 3

0 0 C + ,

0 3 :

C

!' ,

C 6!'

38@B6 @

('

! ,!'

- ) $ ) -$ - - ) $ 2 3 Æ 2 3

6 #( 6 ¾ +

0! !' ' '

' '

.

0

.

1 6 .0 1

0

. 0+ '

/ ' 1

K !' .@ 388760

:

3D &7+ 1&B6 ('

:

31B6 ('

.

0

3

!+

- -

3

C 6! +

C 6!' '

3

C 6!' '

+ 7 + 7

-

-

-factors

DEFINITIONS

:

D

) +

-

-

. 0

:

. 0

D

'

0

.

'

!' ' -

. (

+ .

'

.

0 !

(

. 0

0

0

-

(

) +

-

. (

(

; .

(

. 0

. .

( (

'

.

0

. 0 '

.

0 !

' '

.

0

'

0!' .+ . 0!' 0 ? !'

FACTS

:

.

( ' $ ( ' 7 ( 6 * 6 7 6 ( 7 6 7 6 ( ' 7 6 ( ' ( ( ( ' ' ( ( '

3D &76 .

! ' 6 .

0

0C

0

.

: 1

.

00

.

0

.

0 C

' ' ?

:

.

'

3D&B6

-

'

'

0 -

. 0

.

.

.

.

0 7

3

0 <

© 2004 by CRC Press LLC

.

0!' ' '

.

0 < '

a .

0

0+ , a .

7

0

'

. 0 '

, !- ' )

. 0 1

0+ -

0

. 0 <

.

.

0

0!' ' ' '

0 '

6

.

.

. 0

<

. 0 < +

'

412

Chapter 5

:

3

85$6 D

(

( .

0

. ' - ' ? -

(

:

35B6 D

.

7 6

' 76 ! ' 6 '

'

.

.

.

:

0

.

*.

6

0

C

. 0

C

+

(

0C

.

.

00

3XD1 5B6 D

.

(

.

.

:

3;56 D

-

.0

.

0 !

.

(

0

0!'

-

7

' '

1

( ( <

A

<

+

0

'

' .

0 ,

( . 0

. 0

½ ( (

.

0+

.

C

(

!' D

. 0

. 0+ '

-

+

- +

-+ '

-

.

0

0!'

, ! ,

.0 ' - .0 ,

. 0+ '

0

0!' +

. 0

.

7 6 (

0+ ,

(

. 0

0 -

1

.7 6 ( 0

- !- '

0

C

. 6 0 C

0

' ?

('

' '

0!' ' '

(7

(

.0 7 .0

. 0

('

.

-

COLORINGS and RELATED TOPICS

'

!

- +

!

REMARKS

:

' +

@ 3;5B6 ' Æ ' ' . 3;57+ FF8D576 ' )

(

' ' +

0!

38B+ 857+ DB56 ' ' ' - '

:

:

( 2 , , '

(

.

0!'

# ' ' , , 2 V ,

' 3#455+ #455+ #455+ #47+ 8 @ 56

Factors in Random Graphs 1 - ' !

9 ,

' '

DEFINITIONS

:

D

' -

' -+ ,

7 ,

, .

:

; -

'

© 2004 by CRC Press LLC

! +

- ,

1

0 ' '

. * **0+

!

S

!

< (

<

Section 5.4

413

Factors and Factorization

FACTS

:

3 *N$$6 D - !'

1

< . 0. C +. 00+ , +. 0 <

: 3@ B6 D < . 0. C .- 0 C +. 00+ , - +. 0 < @ ' ' .0 - , . 0 - 1 !'

REMARKS

:

# - ' + ' + 38B+ 85 + 4D*77+ " B + " 7+ E *7+ @ B+ @ B6

: 1 ) ' ' ' ' ) @ 3*5+ ;B+ ;B+ ;BB+ ;5+ ;E@5+ ;B+ ;@B+ "&+ F S @ B&+ @ &&+ @ B + @ B + @ B5+ 5+ S 56

5.4.4 Component Factors DEFINITIONS

: ; ' ' ' ,

: ; ,

FACTS

:

3; 88,E76 D ! - ,!' , Æ .0 1 ' , - C -

: 3@ B6 (' !' ' ' B .6 0 6 + ' - 6 .0 .1 -, / ' FK 1 ! 0

:

3EM &5+ F8B6 1 ' ) ,

!'

5.4.5 Graph Factorization * 2+ ' ' / ' ,

2A , , -

© 2004 by CRC Press LLC

414

Chapter 5

COLORINGS and RELATED TOPICS

Edge Partitions DEFINITIONS

: ; ' , '

: 1 ' ' ! ' , .0 CONJECTURES

$ ' / : D ' - (' , J.0 395&A 560

3;2

+

¼

.0 < J.0A +

!' / .@

FB76: 1 ' - ! . C 0

FACTS

:

3 FB56+3 576 (' . & 0 . & 0+ ! # / ? 1

, ?

:

3S156 D ' - 1.0 - 1 ' J.0 -. $ C 0+ ¼ .0 < J.0 1 -,

' # #$

:

- 7+ B < B . 0 ' ' -

B + J . ½ C 0 .0 + !' / 1 - - '- ' ' !# / ? ' => .@ 3S*5&6 FM2- . 00

: 3TT56 - ! ' !? !' + ' 1

!# / ?

:

2 3 7 2 3 1 .0 32 36!' / ' ' 32& 3&6! + ' &A .0 - 3B& C 7 C 6! 3 6 ' / 38B 6 @

:

3WS9 1 776 D ( , ' .0 ! - D & - % , 7 % % &. 0 (' .&( C & C % & & %0 +

.( 0!' / .@ 3W5 6 '

0

:

1 .0 - -! , - 3 C 6!' / A .0 - . C 0! 3 6!' / 3 B$6 D

© 2004 by CRC Press LLC

Section 5.4

415

Factors and Factorization

CONJECTURE

'5$62& 3

Æ ½

½

.

0

C

B6: ('

C

+

,

-

!TN ' - ?

<

½

<

C

-

< 3 F$6 ,

# ' 34 776

? ,

C

!' , -

<

.

0

; 3;5B6 -

Æ

REMARKS

:

(

Æ , '

- -

!

( -

' @ 3;BB+ D9 5B6 ' '

:

@ 3S176 ' ' !' /

Vertex Partitions 9

' -

<

3

.

0 <

½

6 - ) P

9

- '

FACTS

:

3&B6+3D &&6 D

- '

.0 < -

.0

¼

'

½

½

!

@ '

.0

<

<

C

C

1

¼

'

+

.0 ' ' - .0

2 ½

5 2 5 Æ 5

:

3

E5&6 D

'

<

:

.

+

@

0+

.

3

.

60

.

7+ '

0

.

00 ' - 3

6

0

3

<

C

C

1 -

0

.

.

-

0

.

0 <

.0 31B6 # ' - .

) $ ( ) $ ( ) $ 6 / $

2½ 2 Æ 5½ 5 ) $ F ) $ Æ 6/ )

'

,

-

'

, .

.

0+ -

0

.

0 <

0 . -

.

6 - - . - 0

+ -

.0 3FB6 E -+ '

)

$

+

) $ ) .

0

!

0

C

$

REMARK

:

# #& - 2 ,

' 3 8M

B6

© 2004 by CRC Press LLC

416

Chapter 5

COLORINGS and RELATED TOPICS

Factor Algorithms and Complexity DEFINITIONS

: D ; ' " ½

' ' " . 0 - .0

:

5'/ .0: (@1; : ; " b @1(%: " !' P : 1 ' # .0

' # .0 ' ' .0 : ; ' - ' : 1 " %+ - ) " + ' ' " P : (' ' $ + , $ .; - ' - $ ' - .0 0 : 3 BB+ BB6 ; $

' - $ ' $

.0 : 1 $,

, % . - .

%0: - - $+ $! P

$ ' ; ' "

' 1

1 + < .0 & < .0 FACTS

:

1 ) ' , ' ! 3 $ 6 2 , $ " ( . 0

: 1 ' ' .+

0 .& 0 E / 3EB76 .@ 3SD BB60 . +

' ' ' ,

' Z .@ 35600 @ E!/ , + ,

315+ "576 - E!/ -

: # + ,-+ (' ! + ' S 3SB56+ - !' ( + . 0 - 3"" D76 ' ) !' ; . 0 -+ '+ +

© 2004 by CRC Press LLC

Section 5.4

Factors and Factorization

417

: 1 ,+ 8 1? 31855+ 1876 - ) , + !' ) + ' + 2 , - !' +

: ; 3;B 6 -

' '

.( 0!'

!' ' ' ,

+ . ¿ 0

'

' ' ( : ; ' ) !' + ' + , ) '

4 3 4 &76 (' !' !'+ - .@ 3 SB760 (' , ' ! ' $ +

, S! 3F888BB6 1 ,

+ + , ! ' , ! ' + -

: 1 ' ,

F '

) - .38&+ 8& 60 S! 1 S! + - ' ! 34 1&$6

! ! 3S56

: 1 , ' 5'/ . 0 . -0 , => 5'/ . 0 S 5'/ . 0 ? '

' ' " E + ' ' ? ' ' + 5'/ .0 : 38FB6 (' ' , - + 5'/ .0 S!

: 38,76 D ' , Æ .0 1

!' + ,

, -

: 3@94BB6 1 ' # .0 S!+ ' ' !

' ' ' F ,-+ '

' ,

!'

: 3F B6 1 ' ' S! ('

+ ,-+ # B$

: 342& *' $- 38M $+ 8M $6: (' +

¼ .0 < J.0 1

' .& 0 !

: S + ' 8

*// 38*776

' @?- 3@5B6 ' ! + - / ' .0 J.0 (' - - +

- ' .@ 3*760

: 3F B6 @ 1 ' .0 '

S! F - - )-

S!

: 3"D 5 6 (' " , +

" ! S

© 2004 by CRC Press LLC

418

Chapter 5

COLORINGS and RELATED TOPICS

: 3 15&6 1 " ! S S! ,- " , .@ 3; W5B60

: 3"F 5&6 1 ) ' / ' -

! , !' !' , ' /

: 39 B6 (' - $+ -!

' /

$

: " + 2 , ' !' / # #5 , ) - - :

$ .0

: 3 9 BB6 D $ $

1

$!

¼

.0

: (' -! $ - C + $! 1 ' , ' /K

: 3 BB6 D ! ' - ! ! 1 .0 E -+ ' /

: 3@"&B+ F* 9 &B6 - + " !? ' $ ' " ' ' . 0 7. $0 REMARKS

: 3F9&&6 - - , ( # / S - ? ' !' /

' . + , !' 0 ' 2 ' @ - 3@ B+ E* B 6 - 395&6 : 1 ! % ' ' / F ' ' ' ' 2 - + 2 ' ' , ' @ 3D S 576 ' - ' ' CONJECTURES

3"F 5&6: 1 ' /

.0 , ! ' , !' !' + .0 , ! '

!' !'

3FB 6: D ! ' < ½ C C

' (' + ' / !? "½ " + , " ' .1 - ?

S! '

- 0

© 2004 by CRC Press LLC

Section 5.4

419

Factors and Factorization

Subgraph Problems DEFINITIONS

: 1 , %: - + ! P .F ,

! 0 (' < + 2

: / 0 ' (: !

- !

FACTS

:

0 ! - ' !' + .0 < 7 ' !' ' ' + .0 < ' !' ' + .0 < C + , .0

' - . C 0! '

3B6 D .0 .0

.0

:

31B6 "K ? .1

'

- ' ) ! 0

:

3 576 1 ! @ * S S! '

:

3 % 556 (' , .0 C$ Æ .0 C+ , - !? , .+ ' ½ ¿0 .1 ,

0

REMARK

:

1 ' , , - ' = > , = > 1 ' - 3 B+ 57+ * 57+ "5$6

2 3" 57+ 57+ * 556

References 3;5B6 @ ;+ @ ' + S 1+ * - + 55B 3;B6 * ; + ; / ' 1

K !' + ) & .5B0+ 55I75 3;B6 * ; + 8M K ' ) + + 5 .5B0+ I 3;BB6 * ; + E ) + ) .5BB0+ B&I

© 2004 by CRC Press LLC

420

3;56 * ; + () +

Chapter 5

COLORINGS and RELATED TOPICS

5

.550+ I

3;E@56 * ; + E E * @ + % ' 8M

K

' ) + ) .550+ &I57 3;B6 * ; !9+ E ) + + ; S+ 5B+ &I&5

*

3;@B6 * ; + !9 @ @+ ; ' ' -+ * + & .5B0+ I$B 3;2 FB76 4 ;2+

# F+ - 2 (((+ - + " 7 .5B70+ 7 I& 3;28B 6 4 ;2 E 8 + # ' / ' I -+ 5 .5B 0+ I 3;BB6 ; + 1 ' +

$ .5BB0+ I

3; W5B6 ; + W * W + S2 - + $ .55B0+ I& 3;8B6 ; ; E 8 + % ' , - + .5B0+ I$ 3;&6 ( ; + @Æ ' + * .5&0+ 5I$

7

B

3; 88,E76 8 ; + W ,+ ; 82 + 8 8,+ F E ! + S ' ,!' + .770+ 5 I77 3;B 6 * ; + ;

' ' 1

K !' + .5B 0+ I

&

&'

3;576 * ; + @ ) ' .( 0!' + .5570 5IB 3;56 * ; + E : ' + .550+ &I

$

3;5B6 * ; W + E Æ ' - ' + B .55B0+ I M 3"M B6 # "M+ T M @ 2 %+ " 7 .5B0+ & IB& 3"5$6 D "2+ +

&

.55$0+ I$

3"D&B6 " E D + % ' , + & ! 1 7 .5&B0+ $ I&$ 3"F 5&6 " S F N2+ ! !' !' + 7 .55&0+ 75I&

© 2004 by CRC Press LLC

Section 5.4

421

Factors and Factorization

"" D76 1 "+ S " + ; D,+ Æ ' S K + B .770+ 7I 3"576 E "+ ; , + + * .S ' & ( ; 0 .5570+ B$I 5& 3" B 6 " " N +

, + F " 4 - - .5B

3" 76 " " N +

, .

0

0+ - S .770

3" @9 B 6 " " N + ; @ 9 + * ' ' + 5 .5B 0+ 5&I7 3" 576 4 " N2+

+ 8, ; S + 557

3"&6 D "c+ ' - + ! ! &! , .5&0+ &I7 .*0A " ! .5&0+ 5 I55 . 0 3"&6 * "+ @ ) - ' 8M

K + & .5&0+ 7$I7

3"D 5 6 8 "N T D + ; ' F + * B ' '! / + - ' 1, + + 1 + 55 3 56 E + 32 36!' / ' +

.550+ BI7

3 576 # + 1 ' + & .5570+ 5I$B 3 FB56 ; , ; F + !' / , : - + & .5B50+ 7I 3 B6 # * + * + + + & .5B0+ - S+ 7I 3 &6 -N + 1 + IB 3 * 556 ; * +

.5&0+

+ %' - S+ 555

3 SB76 N? 9 S2+ S' !' ! + 2 ;* = 3 " 7 = &' = 8.9.<+ E S @ .5B70+ I& 3 F$6 8 N ; F?+ % ' + .5$0+ I5 3 8M

B6 ( /N 4 8M + : , 2 + 777 & .5B0+ I

© 2004 by CRC Press LLC

422

Chapter 5

3576 * + - S+ 557

COLORINGS and RELATED TOPICS

0 + %'

3576 * + ) + 2 + " ( + E+ 557+ $IB5 3 15&6 E 1+ S! :

'

' F K ? + $ .55&0+ $$IB& 3 $ 6 4 + S + V ,+

& .5$

0+ 5I$&

3 4 &76 4 4 + E : , - ' + + "+ 5&7+ B5I5 3 B$6 W

,+

K ? 3 C 6!' / ' +

.5B$0+ &I7

3 B56 W , F

+ @Æ ' ' !' + , + , ( S .5B50+ 5$I7 3 85$6 W , E 8 + @Æ ' - .( 0!' + .55$0+ B&I57 3 % 556 W , 8 % + !? , + .5550+ I$

5&G5B

3 B76 c-+ @ ' - 9 ' + .5B0+ $ I&+ .*0 3 B6

c-+ 1 ' - 9K ' +

.5B0+ 55I7

3 BB6 E

! "

+ ( ' / ' ' - + .5BB0+ 7I7

3 BB6 E + ( ' / ' -! - + $5 .5BB0+ 5I

3 9 BB6 E 9 + ( ' / ' ! + + & .5BB0+ I 3 B6 E

!TN + % +

7 .5B0+ &I7

3 48 @B 6 F

+ " 42 + S 8 ; @ + 1 ' !' + 5 .5B 0+ B&I5 3 E5&6 F

@ E + , - (((+ .55&0+ I$ 3 % 8BB6 F

+ 8 % E 8 + ; Æ ' - !' + .5BB0+ I

© 2004 by CRC Press LLC

Section 5.4

423

Factors and Factorization

3 *N$$6 S H

; *N+ % ' ' ' ' + & .5$$0+ 5I$B 3#B6 #2+ S

' ' - 9 ? ' + ! 5 .5B0+ 5I5B+ 5 & .*0 & 5 .5B0+ & I&5 3###DD556 * #+ % #- + #+ F D T D+ % !' ,!' + 7$ .5550+ I& 3#4556 # !S 42+ " , 2 V , ( ; ' ', 2 ' ' + &'! .5550+ IB 3#4556 # !S 42+ " , 2 V , (( @ + &'! .5550+ 5I 3#4556 # !S 42+ " , 2 V , ((( @ + &'! .5550+ I $ 3#476 # !S 42+ " , 2 V , ( + &'! & .770+ 5I7 M 3#6 # + E / - + * ! 1 $ .50+ $I&&

#C

3#F E$ 6 #2 + ;F L E E;,+ @ ' , + & .5$ 0+ $$I&& 381556 F ,+ F 8 * 1?+ ! + + ; E , W 2 555+ &7I&B 38176 F ,+ F 8 * 1?+ ! + 7 .770+ 5IB 3156 F , * 1?+ # ' ! + B .55+ B IB 3B6 ; + ! C ! + B .5B0+ 5I77

+

34 1&$6 E + 4 * 1?+ 1 ! S! + .5&$0+ &7I& 3&B6 M

+ % - ' + $ = #= 8.96< + !F + 5&B+ B I5

;*

3FB6 ; F?+ S ' , - + .5B0+ 5 I55 3FB6 E F+ - ' + 5I5$

© 2004 by CRC Press LLC

)

.5B0+

424

3F 6 S F+ % - ' +

Chapter 5

COLORINGS and RELATED TOPICS

+ 7 .5

0+ $I7

3F* 9 &B6 # F+ * * 9 + ( ' / ( + .5&B0+ I$7 3F9&&6 # F 9 9+ ( ' / (( ! + & X(X .5&&0+ IB 3FF8D576 8 F+ S F+ 82 2 D+ : ; ) ' .( 0!' + B .5570+ I& 3F8B6 S F 82 2+ % / + + .5B0+ I

*2

3F888BB6 S F+ 82 2+ 4 8 -N ( 8ONO/+ % , ! ' + .5BB0+ &IB 3FB6 F+ E , !' + & .5B0+ I$

)

3FB 6 ; F + # / ' ' + .5B 0+ 5I5$ 3F B6 ( F + 1 S! ' ! + &BI&7

5

7 .5B0+

3F B6 ( F + 1 S! ' + 7 .5B0+ &I&&

3F S @ B&6 E F /+ 8!S S ,2 8 @ L+ (? - ' + D

E + B+ @ !+ 5B& 3(56 1 ( 1 + ; %! ' ' !' + & .550+ I$ 3(5&6 1 ( 1 +

!' + 7 .55&0+ IB

!

349B56 " 42 * 9

+ ;

, !' + .5B50+ &&I B7 34dD*776 @ 4 + 1 D d /2 ; *2+ , + 9 ( .7770 34 776 S 4 + % ' , !' + .7770+ &I 34 776 * 4 + % ' 5 .7770+ I

!TN K ? +

38B6 E 8 + ' , - + D

E + 7& .5B0 $I$B

© 2004 by CRC Press LLC

= 8.AD+

Section 5.4

425

Factors and Factorization

38B 6 E 8 + 32 36!' / ' +

5 .5B

0+ 5I$

38576 E 8 + ; Æ ' - 32 36!' + $ .5570+ I

B7

38576 E 8 + @Æ ' - ' + .5570+ 5I$

3856 E 8 + ' ' + = = ."?+ 550+ 9 @ ) S! + 55+ 5I5B 38876 E 8 8 +

: % + .770 . 0

38@B6 E 8 ; @ +

: 32 36!' ' + .5B0+ I$ 38*776 ; 8

* *//+

: .7770+ 57I5$

! +

38B6 E 8 N 2+ ; -, ' + 385 6 E 8 N 2+ * + IB7

&

$ .5B0+ 5IB5

! +

- .55 0+

38&6 * 8 + * + 5 + S S+ 5&+ B I7

38& 6 *8 + % ' + &'! .5& 0+ I$B 38 B6 S 8 + @ ' !' ' - + $!" .5B0+ &I&& 38 576 S 8 + E ' ' !' + $ .5570+ I B 38 1776 S 8 12 + E &' 5 .7770+ I7

!' +

38 9 B&6 S 8 9

+ " '

' !' + 4 B .5B&0+ IB 38,76 8 8,+ !' + 5 .770+ IB 38FB6 82 2 S F+ % ' ' ! + .5B0+ $7I$75 389&6 8 9+ ; ' , - - ' + $ .5&0+ &BIBB

© 2004 by CRC Press LLC

426

Chapter 5

COLORINGS and RELATED TOPICS

38 @ 56 9 8 @ + " , 2 V ,+ & .550+ &I

)

M 38M

$6 8M + ;, ' E+ && .5$0+ I$ 38M $6 8M + 2 N 2/N2 N 2 N / 2 N N+ E 7E .5$0+ 7I5 38M 6 8M + + 0

$ +! B .50+ $I5 .F!

M 38M 6 8M +

8

2 . ;, ' E /0+ ; < $ .50+ I&5 38 76 8 8

+ # + .770+ I

38 56 ; 8

/+ % ' ) , ' ((+ F " ! ! > 5 .5 50+ $I 5 38&6 @ 8+ 1 !' ? +

&

2$

$ .5&0+ $&I&$

3D& 6 E D + ;

+ E ;* = 8.9B<+ N * % N+ & .5& 0+ &I$7

3D&B6 E D + ; ' 1

K !' + .5&B0+ I

3DD5B6 D T D+ % ' .55B0+ I&

½¿

!' +

3DT 76 D+ " T + % 3 C 6!' ,!' + $ .770+ 7&I5 3D9 5B6 1 D 9 + # ' '

' + B$ .55B0+ &I$ 3DF & 6 D

+ F + % ' ! + .5& 0+ I 3DBB6 D+ % .( 0! - + BIB 3DB56 D+ % 32 36! - + I

;7 <

B .5BB0+ .5B50+

3D S 576 E D @ S ?2+ @ 2 U -+ ;/'= 8..:<+ S+ F+ 557+ 5I 7 3D &76 D D -N /+ @ , -+ 5I$

© 2004 by CRC Press LLC

B .5&70+

Section 5.4

427

Factors and Factorization

3D &6 D D -N /+ ' , !' + 5I

*

3D &&6 D D -N /+ ; ' ' + 7 .5&&0+ I 3D SB$6 D D -N / E S+

.5&0+

+ !F + 5B$

3E* B 6 E ; * + %!' / ' U -+ 5 .5B 0+ I$ 3EB76 @ E /+ ; . ½ 0 ' ) + * -8 $ ;< .5B70+ &I& 3E *76 E E " *+ @ ! .770

* +

3EM &56 4 EM + !' ' : / + * + B .5&50+ 7&I

5

.550+

356 # + ! ' ' ) + I

3S 56 # 8!S S ,2+ E ) + ) $ .550+ I&

35B6 1 +

: ; / ' - .( 0!' + ) & .55B0+ I $

3*5B6 1 " * + * ' ' ' ! + B .55B0+ B5I7 3 576 1 D 2+ ' - ' + .5570+ I$ 3B56 1 + ( + - ' + .5B50+ &5IB& 3576 1 +

: * ' ' + I5 356 1 + * ' ' ((+ 7I77

3

B .5570+ $ .550+

356 1 + ; ' ' !' + $ .550+ I 356 1 + ' ' +

© 2004 by CRC Press LLC

B .550+ 5I

5

428

Chapter 5

COLORINGS and RELATED TOPICS

3&B6 1 /2+ D , ' ' + * & = E .5&B0+ &I & 3&56 1 /2+ % , ' ' + .5&50+ 5I $ 3SB56 4 S + 1 M + 7

.B50+ 5I

3SD BB6 S S E D + 1 ' E /+ .5BB0+ I 3S56 S +

: ' ! + .550+ $I& 3S156 E S @ 1 + * ' !' /! + * + .550+ 5I77 3S176 E S @ 1 + ; ' - !' + 7 B .770+ e* 3S56 E S+ .550+ &&I5

: -+

&

3S5$6 E S+ .55$0+ I

: +

3

$

3*56 * * + # / ' - + 5I7

4 /5

7 .550+

3*76 * *//+ # !' ! + .770+ BIBB 3* 576 * + + 3@56 ; @ + %!' !' +

; <

.5570+ 5I5

5 .550+ I$

3@"&B6 4 @M

; " 2+ ' ' ! + ;)< .5&BG&50+ 7 I 3@B6 ; @?-+ " + ' !' !' ! / ' + " + D E @ D

@ B .5B0+ 7&I 3@5B6 ; @?-+ " .J&0 + .55B0+ BIB$

3@5B6 ; @?-+ !' + ) & .55B0+ I

© 2004 by CRC Press LLC

B

Section 5.4

429

Factors and Factorization

3@B76 ; @?- 9 + % , ' + 1 .5B70+ I& 3@ B6 @ .5B0+ 5$I7

'+ % ' +

& !

5

3@ B6 @ '+ %!' - + .5B0+ BIB$ 3@94BB6 E @ + 9 9 9 4!D+ 1 ' !

+ @ ' + 1 + ." * + D;+ 5BB0 & $& .5BB0+ 5I$$ 3@ B6 * @ ( + ' / + + + .5B0+ IB 3@ &&6 8 @ L+ E + 3@ B 6 8 @ L+ E + .5B 0+ 5I$

5 .5&&0+ $

I$B

! '

3@ B 6 8 @ L+ #2 + > B& .5B 0+ &I& 3@ B56 8 @ L+ 1 !' ' + - F - .5B50

2

C , *

2 ;9% 8.A8% + = < F+ 5B+ $I&

3@ B6 # @ /+ % ' ' +

3@&6 @+ % 1

K ' / +

E 7$+ @ .5&0+ 7I 3@&$6 @+ !' !' + I 5

+ D!

+

.5&$0+

31B6 12 -+ ! ' ! .*0+ : & $ .5B0+ $I$ 31B6 1 + ; 2 ' ' D -N/ 1

+ .5B0+ I

31B6 1 + , - + & .5B0+ $ I$& 31&6 9 1

+ 1 ' / ' + 7&I

+

31 6 9 1

+ ;

' ' ' ' ) + $ .5 0+ &I

© 2004 by CRC Press LLC

.5&0+

430

Chapter 5

COLORINGS and RELATED TOPICS

" ; = = = 8.99<+ ; E .5&B0+ B5I5

31&B6 9 1

+ 1 + 31B6 9 1

+ ' +

.5B0+ &5I5&

3&56 D + 1 ' + B .5&50+ B5I7

3&56 D + 1 ' + B .5&50+ 7I

356 /+ ; ' ' -

' . 0 + .550+ &I75 3 5 6 D 2+ * + ' + ' S K

+ 2> 5& .55 0+ 5I 39$6 " - 9+ S + &

2>

.5$0+

395&6 9 9+ / 2$ + 8, ; S+ 55& 39 576 9

+ !'

' + + .5570+ B I5

39 B6 9 + ( ' / (( * + B .5B0+ &I 3XD1 5B6 " X+ T D 1 1 2+ ' ½ !' !

.( 0!' + .55B0+ 5I5 3W5 6 W+ @ , .( 0!' / ' + B .55 0+ &&IB

3WS9 1 776 W+ 4 S+ 9 1 1 2+ '

.( 0!' + $ .7770+ &I$ 3W8BB6 W E 8 + @ ' ' + .5BB0+ &I 3TT56 !b T W T+ # / ' + ) $ .550+ &IB5

© 2004 by CRC Press LLC

Section 5.5

5.5

431

Perfect Graphs

PERFECT GRAPHS % &' ( % ( @ S' E

- ; E * ' S' Æ ' S $ ' S' & 1 @ S' 1 *'

Introduction 1 ' ' ' ' ' E -+ ' - ' ' - # + S! ) ' + ' ' 3 B76 - - ' ' ' 3" B6 2 ) ' ' 3;*76 - - ' ; + + -

'!

!

5.5.1 Cliques and Independent Sets 1 ' ' , , ' + DEFINITIONS

: ; ' - ?

. ,+ => - ' ' ? - 0

'

:

1 '

:

; '

+

.0+ / '

' - -

: 1 ' - '

© 2004 by CRC Press LLC

+

G.0+ / '

432

Chapter 5

COLORINGS and RELATED TOPICS

: ; + $ + ' -

!?

: 1 ' +

.0+ / '

: ; ' ' - - ' 1 / ' ! - '

.0

: 1 ' +

.0+ ' - ' @ . Ü 0 + / ' ! -

: 1 . 0 . < . . 0 ' < . 0 - ): . C0 . ' ' . C0

FACTS

: @

. + ' ,

? -

.0 < .. 0 .0 < ..0

:

#

.)0

+ .0 < G.. 0

G.0 < .. 0

.))0

: ( ! - ' - ' + ' L @+ - L 1+

.0 .0 .0 G.0

.) ) )0

EXAMPLE

: 1 ' # ' / + + 1 . ' / + + 3 1 ' / + + 3 1 . ' / + + !? 1 ! - .2 #0+ .3 0+ .! 0 ' + - .2 #0+ .3 0+ .! 0 '

. b

a

c

a f

f

e

c

e

d

d c

G b

Figure 5.5.1

© 2004 by CRC Press LLC

G

#$ * & -#- .,

Section 5.5

433

Perfect Graphs

5.5.2 Graph Perfection " + " 3"$6 ) 3 ' .0 < G.0 ' ' - . ! 0 - ! " ' + ." 0 < G." 0 " , .f0 .ff0 -+ " , ! ' @ "K ' ' ' 3"5&6 ; , ? ! ' ! ' - + " ? ' ' @ S' ? 1 - , , , ' ' + ' .fff0 - - .f0 .ff0 I + + - , I DEFINITIONS

:

; " < . ¼ ¼0 ' < . 0 ' , - C ¼ + . C0 ¼ ' ' . C0

: ; ' ' - . ! 0 - ! " ' . " < 0+ ." 0 < G." 0 : ; ' ' - . ! 0 - ! " ' . " < 0+ ." 0 < ." 0 : :

;

' ! ' ! '

; ' , ! - -

: :

, + +

; ' ; / ,

FACTS

: #$ $-: .D -N/ 3D &60 ; ! ' ' ' ! '

: 1 S' 1 : ; ! ' ' ' ! ' .F+ , ' ' - + , ! ' ! ' 0

: .D -N/0 3D &60 ; ' ' ' ." 0." 0 ." 0 ' - . ! 0 - ! " '

: $ ! #$ $-:

. -2+ * + @ + 1 3 * @1760 ; ' ' ' "

© 2004 by CRC Press LLC

434

Chapter 5

COLORINGS and RELATED TOPICS

EXAMPLE

:

- ' # < .½ 0+ -

.0 < .0 < ! + - - - @

+ - /

5.5.3 Motivating Applications ( + , , ' 1 ) + + ,

- ' "K

) ' ' 1 ' 1 , - ' , .0+ .0+ .0 G.0+ Æ + - DEFINITIONS

: 1 . 0

- ' L - , ,

- ' ' ,

-

: 1 " ' , " ) ' , # - ' - C ' " + - C " + , C ¼C¼ " ' ' ? ¼ C < C¼

C ? C¼ " < ¼ : 1 " ' ' , '

: .@ 3@ $60 ; S ' ' + , - .. 00 < .. 00

: .@ 3@ $60 1

' ) .. 00

EXAMPLE

:

& 4 4" 1 ! ! ' , +

' ' - . 0 ' 1 / ' ! '

+ , ' / .. 00

+ ' -

+ 2 ,

, 2

!

# + )- . 0 ! + ' , .. 00 < ; ' ! - ' + +

( ! + ' 1 , . 0+ !

© 2004 by CRC Press LLC

Section 5.5

435

Perfect Graphs

' L ! ' ! .. 00 %- ' ! , , ! +

' , ' ' L ! # + - ' ! 1 ' . . 00½ + ' .. 00 ! ' ' + '

' . . 00½

' .. 00 1+ , " ' + .. 00½ REMARKS

: D -N/ 3D &56 - @ ' ! - ½ < ( . 0 : 1 .. 00

. . 00

/ ' ! (' S ' + .. 00 < .. 00+ @ .. 00

: % ,

.. 00 . . 00 G. . 00 G.. 00 ' , ' , ' . 0 ' + . . 00 < .. 00

'

: 1 / , !

- 2

' L

1

' @ 3@76 ' EXAMPLE

6 - .- 0 ,2+ , $ % 2 ' ! 2 ' ,2 - ' # + ' ,2

- , 1 Æ

- + ) , ! ,2 ! '

, # )- , 2 - L P 1

' : " 4 " 1 '

B

Tour B Tour C Tour A

Tour E

A

E C D

Tour D

Figure 5.5.2 7 & 0 & * $ &&* #$,

© 2004 by CRC Press LLC

436

Chapter 5

COLORINGS and RELATED TOPICS

- ' + ' , - ' 1, - ? ' , - 9 + L , 2+ - . 0 '

# )+ !

- ,

) ' ' ! 1 ' ' ' Æ + ' , ' + + ) ' , 2 2 '

" ,! - @ S' ? +

2 '

5.5.4 Matrix Representation of Graph Perfection ' ' /!

1 ' ' , , +

7 .

0 '

!

1 .

0 '

9 ' +

7 .

0+ , '

'

@+

1 .

0

DEFINITIONS

:

1

- ' '

'

:

! -

; 7!

7

!- -

8

1

.

0 '

A + .

7! , , 0 '

!

' ' - !- -

+

- - ' :

78 1 E

:

7

8

?

.

0 '

, , - ' '

' '

! -

7!

A + .

0

!

FACTS

7 1 . :

# ' + ' ,

.

0 <

:

.

.

0

G

.

0 <

.

0

8

?

7 8 8 .

0

+

7! -

.0

0 - ' ' , : E

:

.

0 - ' ' , : E

:

1 7 .

9

9" 7 9 8 9 8 9 8 7 8 8 ?

('

' : E E

© 2004 by CRC Press LLC

?

0

+

7! -

.0

-

+

?

9

.

+ .0 .0 '

.

0

+

9" 7 9 .

0

+

.0

.0

Section 5.5

437

Perfect Graphs

:

K ' ! ' - .0 - .0 REMARKS

: ( .0+ 8 - ' ' - ; 8 ' 7.08 ' : 1 ? - ' 8 .0 ' - + E 8 / ' : ( .0+ 9 - ' ' 1 ' 9 , ! ' 7.0 , ' 9 ! - ' 1 9" 7.0 ' - '

C - - ' + +

:

1

% .0+ ! ! ' + .0 1+ .0 - ' : E

7

8 ? 1.08 + 8 7! -

G.0 - ' : E ? " .0 + 7! -

9

91

9

FACT

$62& $- 3 & 6: 7.0 '

:

;

' ' '

REMARKS

:

" -N K 1 + , ' + .0 .0 -

!- %' + ' ' + , 2, .0 < G.0

:

8

1

9

7

@ .0 < .. 0 + S' 1 + . ' , ' + ' , ' ' + .0 '

1

5.5.5 Efficient Computation of Graph Parameters ( ' ' + .0 .0 82K ' + ' .0 + ' , ' .0 ' '

' - '

7

DEFINITION

: 3D &56 1 & 1 H.0 ' ( ) -

© 2004 by CRC Press LLC

438

Chapter 5

COLORINGS and RELATED TOPICS

FACTS

:

D ., 0 - ' , '

:

1 .0 G.0+ / ' - ' + S! '

:

.

+ D -N/+ @?-0 3D @B6 .0 H.0 .0

. 0

.0

, - ' ? - ' .0+ ' .0 ' ) / '

:

1 D -N/ H.0

' - REMARKS

:

@ .0 < .0 ' ' + . 0 .0 < H.0 1+ ' ' + H.0 .0 G.0 . ' .0 < G.00+ +

2 + .0 .0

:

1

- ' ) H.0 ' D -N/K ' @ ' . 0

5.5.6 Classes of Perfect Graphs 1 2 , ' ' F ,-+ ' '

' ' ( , V - ' , '

2 ,

' ' DEFINITIONS

: ; ' - , ½ - ? - , -

: 1 .0 ' - ' ' A , -

' .0 ? ' ' , ' , - FACT

:

1 ' ' '

Interval Graphs

( - , ) F? 3F &6 1 - + ,

© 2004 by CRC Press LLC

Section 5.5

439

Perfect Graphs

DEFINITIONS

: 1 ." 0 ' ' " ' ' - - ' A , - ? ' ' ' - !

: ; ' ' ' ' - '

: ; ' - ' ,

' ' -

: ; ' - , - " ' , - ' " - '

- ' "

: 1 0 ' 7! , , ' - . 0 ' ' ! - ? ! -

: ; 7! 2 ' , K - FACTS

: :

( - '

3" D&$6 ! + , - ' '

: 3* $56 ; - - ' '

½¿ + ,+ - !

:

3* $56 ; - ' ' ? - K

:

3#$ 6 - ' ' . 0 - K

EXAMPLES

:

- * - S 9 - ' + + + - - 9 2 , 6

' , - ' 1 - / ' , 6 ' - (' , 7! , , ' ' . 0 < '

' 6 + / ' '

- K

.

© 2004 by CRC Press LLC

.

440

Chapter 5

COLORINGS and RELATED TOPICS

: # @ 3" 56 "/ - 77 ' ' - 1+ 2 , L ' ' ; .

' ; , , 0 "/ , 2 , , - ' ' , , ) ' 1 + -, - / ; % - ' , - - ' , - - 1 "/K + - P 1 - ' 77C , , - 1 , ) - - ; , - REMARKS

: " 2 ' ( ' + , 2 , + + ; + , , 2 , , ; , '

K ; "/ ' '

- 1 ' '

' ' - L ' 1 L + + , ' L 1 , - ' ;

:

; / ' - ! . 3 B76 ' ' ! 0+ ,

' Chordal Graphs

, ) F? @N 3F@ B6 , - , ! ' 1, ' DEFINITIONS

: ; ' - ' + + ? ! - - , " 3 " " "

: ; ' ? - + + ,- .2 30 .3 #0 .2 #0

: ; , - ' ? ' . 0+ ' + ' ' ' ?. 0 ?.0 : ; - ' ' - '

© 2004 by CRC Press LLC

Section 5.5

441

Perfect Graphs

: ; ' - , ' - , - ?

- FACTS

: : :

; - ! ' '

3F $0 - ' ' .1+ - ' 0

: 3D" $6 - ' '

:

3$6 1 ' ' - . !

0 ' - '

:

3#$ 6 -

,

-

:

3#$ 6 ; ' ) ! - + - - (' -

' +

1 ' 3* 1D&$6

: 3&6 1

' '

' '

5.5.7 The Strong Perfect Graph Theorem ( ' 77+ E -2+ * + S @ + * 1 3 * @176 . 0

' ' @ S' 1 + + ' ' ' " + +

' " - ! ' ' '

' 1 , - - " ' ; ' L - - 7 - -' @ S' 1 ' ' ' - ' -

' ' + +

2 ' , ' 1 '

' '

2 - L

' + ' , ? ' + ? + 2 - 3 76

/ ' " DEFINITIONS

: ; ' ' '

© 2004 by CRC Press LLC

' +

- !

442

Chapter 5

COLORINGS and RELATED TOPICS

: ; 0 < . 0 ' , ½

? ! 5 0 < + ' ' , .0 1 .5 5 0 .0 00 ' +

,

.0 # < + - '

' 5 ' 0

.0 # < + ' 5 0 ' - ' +

: ; 30 < . 0 ' + 5 0 ' 7 ' ' , : .0 - - 5 0 0 + - - .0 1 .5 ' 0+ .5 0+ .0 70 .0 0 ' .0 1 , ' , ' - : .5 70+ .5 0+ .0 ' 0 .0 0 : ; " < . 0 ' , * - ' ' - ' * FACTS

:

@ 1 ' " 3 * @176: ; ' '

"

.0 + ' +

' ' , ' A

:

.0 ' ' , ' - : !? ' .+ E!? ' + 2, '

+ !? '

I 1 @ S' 1 3 * @176: "

- '

REMARK

:

-2 3 76 - E!? - / ' "

References 3;*76 4 D * ;' " ; *+ + * 77

+ 4 9 @ +

3" 56 @ "/+ % ' ) + 3 .5 50+ $7&I$7

© 2004 by CRC Press LLC

* &

Section 5.5

443

Perfect Graphs

3"$6 "+ #M - M /, 8 .T'0+ 1 2+ 3 " 21 = 2& , 7 .5$0+ I 3"5&6 "+ E

- ' ' ? + $ G$$ .55&0+ $I&7 3" B6 " -N + + 5B

2

* + !F + ; !

3" D&$6 8 "

D2+ 1 ' - + - + ! + .5&$0+ I&5 3 76 E -2+ " 1 1 ; + S + S - E + 77 3 * @176 E -2+ * + S @ + * 1 + 1 @ S' 1 + 3 & 6 -N + % , + ) B .5& 0+ BI

3 76 E '

+ N? + 8 2 -+ @ ' + )+ 3$6 ; + + &I&$

3 "

3#$ 6 * #2 % + ( - + .5$ 0+ B IB

.5$0+

* 0

3&6 # -+ 1 ' + ) $ .5&0+ &I $ 3F $6 S ; F L+ ; / '

' - + $ .5$0+ 5I B 3 B76 E + 5B7

* + ; S+

3D @B6 E M

+ D D -N/+ ; @?-+ 1 / + .5B0+ $5I5& M 3F@ B6 ; F? 4 @N + VM

- - M 1 + 3 " ) 7C"C = .5 B0+ I 3D" $6 D2222 4 " + * ' ) ' - + $ .550+ I$ 3D &6 D D -N /+ ,2 ' ? + .5&0+ I$&

© 2004 by CRC Press LLC

444

Chapter 5

COLORINGS and RELATED TOPICS

3D &6 D D -N/+ ; / ' ' + .5&0+ 5 I5B 3D &56 D D -N/+ % @ ' + .5&50+ I&

)

777

3* $56 # * + (L + 5I$ * + # F+ ; S+ 5$5

2

3* 1D&$6 * + * 1?+ D2+ ; ' - ! + .5&$0+ $$IB 3@ $6 @ + 1 / ' + 2- .5 $0+ BI5 3@76 @ + S' + * ;' " ; *+ 4 9 @ + 77

© 2004 by CRC Press LLC

,7

+ 4 D *

Section 5.6

5.6

445

Applications to Timetabling

APPLICATIONS TO TIMETABLING ) ( % & " *+ ,

) #"+ - ./

$ @ ) ' 1 S $ !1 1 $ - 1 $ - 1 $ @ 1 *'

Introduction 1 ' '

/ ' , 2 + ! ! + + , 2 V , ; ) ' - ,' ' G ! !

1 ' , -

- , ' 9 ' : ! + - + - !

+ , ' 2 - ! - 9

' - - ' Automated Timetabling: Historical Perspective

1 ' - - ! ) ' - 7 " 55 - 3"5$6 ' ' 5$7 55 1 , ) ,

5$7 5&7 1 , ' 5&7+ , 2 5B7 2 ' - $7 55 + ' ( ' S 1 ' ; 1 .S;1;10 3"* 5$6 ( 55$ ; ' % ! * @ 9 2 ; 1 , + - 77 ' $7 REMARKS

: 1 ) ' - , - '

+ ,

+ - + + + , -

'

© 2004 by CRC Press LLC

446

Chapter 5

COLORINGS and RELATED TOPICS

: 9 S , 39S $&6 - , ! 5$& 1 ) ' '

- ; ' , !

:

(

- ' K 5B$ - 3 B$6 - -, ' !

+ D - 55 3 D5$6 1 '

- - ) .+ 3B !6+ 3"5$6+ 395$6+ 3"4895&6+ 3 D5B6+ 3@5560

1 + ' -

5.6.1 Specification of Timetabling Problems 1 - , + '

' ) - 3"8S5&6+ 38776+ 376 % ) - The General Problem DEFINITION

: ; , ' : / + ) '

A ,+ ) ' A A + ) ' A ' + ) ' 1 '

' 1 ' ) , Times

; , - ' + / - ) ) ' - ' DEFINITIONS

: ;

:

$

' '

/

' '

; -

FACTS

: :

1

.) - -0

( + - - ' ' - # + ' ,

, - ? +

'

' ' ,2+

: @ : - ,2+ - , ,2+ @

- % .+ 0

© 2004 by CRC Press LLC

Section 5.6

447

Applications to Timetabling

Resources E +

+ ' + . ' 0+ + , ,

DEFINITIONS

:

;

-

' '

,

' '

:

;

-

FACT

:

*

' .) - -0 @ !

EXAMPLE

:

1 ' + ,

+ + +

+ + ' ' % / ' L # + ' - +

' ' + , )

+ + L

L

Meetings DEFINITION

:

;

&

'

;

-

'

'

EXAMPLES

:

( + ,

:

+ ' .

0+

:

(

+ ,

?

,2+ , '

+

+

. ' 0

:

( L + ,

L '

-+ , ' L

+

Constraints 1 - / ' L

/ - - +

- -

'

9 -

© 2004 by CRC Press LLC

448

Chapter 5

COLORINGS and RELATED TOPICS

- - ' 7 ' + - -

DEFINITIONS

:

D 6 ' - ;

!- ' . : 6

..+0 <

:

;

7 + ) ' +

' ,

'

7

,

+

+ ..+0 < 7 ' .

:

) ; ,

6 ) +

D 6 ' - ;

6

+

+ ' ; , ' ' ) : 6

I

1 +

' , ).+0 '

:

D 6 ' - 1

' ' 3 : 6 3.+0 - ' +

6

6

I

:

1

:

1

. # 0

-

- -

,

:

1

) -

' ' / # + ! -

1 #

EXAMPLE

:

D ' - . . .

' !

) ) ) ;

'

, ' -:

3.60 <

. .60 C

+ ) .60

, , + - V

' + , +

REMARKS

:

( - + ! ,

' ' ' ' . ' - 0

:

9

+ ,

+ , ) - ' ,

© 2004 by CRC Press LLC

#

Section 5.6

Applications to Timetabling

449

+

9

.

+ + 0

? A ' + S -,

' /

:

1 - ' ) ? , - '

:

'

' : - ' A - A , /A ,2 , /A

5.6.2 Class-Teacher Timetabling ! ' , !

+

+ '

9 ) - ' +

/ " $ . ! 0+ , - @

+ - , L

- , L The Basic Class-Teacher Timetabling Problem DEFINITIONS

: 1 3 $6 ! , !

+

+

1 ! -

: ; ' ! ' ? L

.0

:

1 ' +

¼.0+ ' L ' ! ' FACTS

: 1 +

9 , '

+ , - - ,

-

: 1 ! *' #"- "# #$ 3"B+ B !6

' + + & ! , & (' + ,

© 2004 by CRC Press LLC

450

Chapter 5

COLORINGS and RELATED TOPICS

, , ;

A ! -

!

: ; - , ! ' + +

' L ' ! + - 8M K .# B0 ' ! + ,

- , + , J : 38M + 5$6 D 1

'

'0

J.0 @ + ¼

.0 < J .@ 3W556+ Ü7+

REMARKS

: ; J L ,!

3"B6 1 ) E Ü ' !

: 1 , ! ! , ) 3 $ 6+ 3@@ B76

Extensions to the Basic Class-Teacher Problem

9 - ' ! EXAMPLES

: @ - ' ' ' ' 1 , ) + ) + , S! 3 -( @&$6 ; , + , - ' , : E

, 1 ' 2 34 &56+ , + , S! : @ = > - - - ! ' 1 S! +

3;76

: *

(

! +

- + '

( + '

L L +

('

+ , - S! - , - $ Graph Models for Subproblems of the Class-Teacher Problem

( ' - ! ' - , ', 2 ('

+ ' . ' -

© 2004 by CRC Press LLC

Section 5.6

Applications to Timetabling

451

)0 (' - ' '

+ , 3 $6 MODELING EXAMPLES

: @ , , ' # , 2

1 , 2

' - ' , 1 - '

. 2 ,0+

, - ' , 1# #$ .*:

' + - , + ?

) - ,- - '

) 1 '

- ' 3 856 :

7 / , , 2 ,

- -

1# #$ .*: 1 ' ' .& $) -)0+

, & + $)

' &+ -)

' & 1

- ' ' 1 ' .- $0+ , - $ , - 1 - ' . ' 0 ; ? -

-

# + ' $) , , ? A ' -)

, , ? -

, + ' - + '

: 9 - ,

-

-

9 2 , , ' , 2

A ' ) - .+ 0 1

+

, , - - -

1# #$ .*: 1 '

' A +

' / ,

#

+ , ' , ; , '

'+ ,

K

+ ' 1 , ' -

3B !+ 856

: (' ' ' # + + - ' '

'

' 1 + , L - + %3A $ :)0 ) .*: # " 3

' + , ' '

© 2004 by CRC Press LLC

452

Chapter 5

COLORINGS and RELATED TOPICS

# ! , '

-

3

' 1 ! - ' - # ! + 2+ , ' , ;

% A

3

, , ' '

' '

REMARK

:

'

# ' ! , 2 V ,+ + ' + 3SB6

!

Ü

5.6.3 University Course Timetabling - L ' ! ! '

' +

' - '

' ,

Basic Model 1 ' ,

, ' '

D

<

'½ '

' L

,2 = + , = -, ' 9 ' . ½ ' # ! + + ' < ' ' ' # ) + 6 ' )

DEFINITIONS

:

;

' '

= ' #

+ ' ' #

:

- +

, ' '

#

6 '

' <

' ' ) +

(

, + ,

) -

: 1 V !' . 0 REMARKS

:

# , .+ !

' 0

:

( , - ' + ' '

( , , ! V 1 ' /

- ' V

; ' - ' V '

, - 2

' 1 ' / ) ,

© 2004 by CRC Press LLC

Section 5.6

453

Applications to Timetabling

A Graph Formulation % ' + ? '

+ , , - ' V

DEFINITIONS

:

'

( " +

1

+

'

+ ' V , ,

'

' , - 2

' + +

+

$ ' ' 6

<

# ' # '½ ' '. + ' - ) - ) ' ' - ) :

;

+

!, ) ' ,:

+

<

+ '

+

7+ , ,

'

+

+

<

+

,

( + , ,

'

'

,

"$

+

'

#

, '

1 ?

C

3

C

6 1

V - ! @

'

C

.

:

0

C

(

0

'

:

;

? ,

C

!

+ ' !?

.

'

'

'

? - L ;

!

L

FACTS

:

1 ) <

=

+

6 6

;& .

:

0 <

+ .

' 6 :

' !

' ' 6

;

&

-

'

' ,

.

0

0

( ! !

. V !'0

&

+ /

;& .

&

'

,

;& .

0

<

7:

, '

- +

=

0 < 7 - ! +

' L L

-+ - ! + !

- .

0

'

<

=

' ,

' '

;& .

0 < 7

1+

!

:

! ! : !

, ' ! ! + -

© 2004 by CRC Press LLC

454

Chapter 5

COLORINGS and RELATED TOPICS

: # ' + ' ' , ! . 0 A ' ' . 3"B60 " S! ! .- ! 0 Ü Ü EXAMPLE

: # $ - ' - (

-+ , -

- - 9 - L @

+ , , , - , , . ' L 0 - - 1 ,

' + , !

, , - < < ' < ' < '½

'

< '½ ' < ' ' 6 < ' ' ' 6 < ' ' 6 < ' 6½

+

6

+

+ + + +

< < < < <7 <

Figure 5.6.1 7 8-# &9 -", ' '

' '

: :

;

' '

' '

.& 0 < .!(!3'

'

: :

60

Scheduling Multi-Section Courses

@ ' & + ' ' ' + ;+ '

+ ' ,2

. ' ' + , . . . 1 ' , ' ! ' ' & + +

' + ( ( (+

. ' ' 1 ' ,

© 2004 by CRC Press LLC

Section 5.6

455

Applications to Timetabling

!# : < . , 0+ , '

< '½ ' ' + , < . + ' < &+ 3' 6 ' < .

!# : S ' $ ' ' (' J ' # 8M K .# B0+

+ + J < & .

J! + ,

# ! ' + .'0+ < &+

' ' . 0 ' %-

.'0 < . + < &

!# : - < (

( ( ' + < . , 0+ , ' < ' ' '+ , < ( ( (+ ' + < & < + 3' ( 6 ' ' ( ' !# : ; ' '

' ' '

, ! '

D (.'0

' ' FACT

) @ + ' !( .' 0 < . ' + J! ' ' ' ' ' ' @ .@ 3;FM5B60 : 3FMB6 -

REMARK

: ( ' + # ' ' ' ' - ' '+ + (.'0 <

.'0 .< . 0+ < &+ '

' ! ' ' EXAMPLE

: .@ 0 1 , # $

Figure 5.6.2 1# #$

© 2004 by CRC Press LLC

&<

..

&<

. . 0 < .$ 0 ,

* ..

. . 0 < .$ 0,

456

Chapter 5

COLORINGS and RELATED TOPICS

.@ 0 # # $ -+ J < $+ $! 2 3 # ! ' , , . 0 3' 6 $ " '½ 2 3 # ! ## ' 3 ! 2 ## ' # 2 ! ! $ ' 2

' !

1+ ' - ' , .' 0

' ' . ! 0 ' ' '+ < &:

.' 0 < 2 3 # !

.' 0 < 2 3 !

.' 0 < 2 # !

.' 0 < 2

.' 0 < ! .@ 0 1 , # $ , ) ' - + ( ( ( # + ( ' + '+ '+ ( ' '

Figure 5.6.3 1# #$

,

.@ 0 ; $! ' # $ , %- ' - ! K ' @

( ( ' ! ' ' ' ! 2 '

© 2004 by CRC Press LLC

( ( ( ( ( " 2 3 # 3 ! 2# # # 2 ! # #$ !

Section 5.6

457

Applications to Timetabling

5.6.4 University Examination Timetabling Basic Model

L ' - ' , F ,-+ - ' 9 - ' ½ -

. 0

1 ' - ( - ' - ' , ' ,2 1 / L '

DEFINITION

: - + # ' , 2 FACTS

:

( ' . 0 - -

( , + ' +

- '

Z

: ( +

- '

- %

+ ' '

@ ' - 2 1

- $ , - .½ 0

)- (

'

9

V '

) EXAMPLE

:

#

+ - . 0 )-

V ( + ! + ? , , -

9 , , .0 ! ' , - 2

' 1 , # $ , V , .-

2

0 F ,-+ + -+ , + + &+ - 1 ) ! ! , ! - ! 1 ' , ) )- . 0 Period 1 Exam-1 Exam-5

© 2004 by CRC Press LLC

Period 2 Exam-3 Exam-4

Period 3 Exam-2

Period 4 Exam-6

Period 5 Exam-7

458

Chapter 5

Exam-1

COLORINGS and RELATED TOPICS

Exam-2

1 7

3 7

Exam-3

5 Exam-4

3

Exam-5

2

Exam-7

7

Exam-6

Figure 5.6.4

7 #$ -* #9# #"-,

A More Compact Schedule

( - -

P (' , ', + , ; - # $ , Period 1 Exam-1 Exam-5

Figure 5.6.5

Period 2 Exam-3 Exam-4 Exam-6 Exam-7

Period 3 Exam-2

Period 4

Period 5

7 & $ #"- $ && 9 $ &,

REMARK

:

1 # $ . 0+ )-+

' ' F ,-+ ' , + , 2 + & , . 0 # $

%

+ - 2 # $

# $ 2 FACTS

:

1 ! ' + ' '

+

' + # +

+ - - 2 - '

: ( , ! + ? -

/ ' - V " ' ,

' ' : ,

+ , + DEFINITION

:

1

© 2004 by CRC Press LLC

Section 5.6

Applications to Timetabling

459

) ' - ' ' ' ' . - )0 The Breadth and Variation of Exam Timetabling Constraints

( 55$+ "2+ + # + 9 3" # 95$6 ! / ' ! ' $ " - 1 ' , ! ' " - . 55 0 1 2 !

; 5 , 1 - ' " - REMARK

: 1 ' - ' ! ! ' ! , ' , + V , ' ,

( ,

, ' ' K

, Heuristic Methods

- ! 3"$6+ 3 $6 ! ; + , , - 9 S , 5$& 39S $&6 1 ! - - ' ! 1 - 3 B$6 D 3 D5$6 - ! --, ' - ' ! ! ' ! FACTS

: % ' ' - !

. 0 ' , Æ . 3 B$60 1 -

:

1 ' ) !

Æ

' ' ' ! ! , HEURISTICS

+: %& : 1 2 , .

' 0 ) 1 - V ,

+: %& ;$* : 1 D

, ' , - - V

+: : F+ , ) - '

V .0 ,

-

© 2004 by CRC Press LLC

460

Chapter 5

COLORINGS and RELATED TOPICS

+ !

: : 1

) - ' - , - REMARKS

:

9 '

' - + - 2 ! ! ! '

3"$6+ 3 $6+ 39S $&6+ 39 $B6+ 3EB6+ 3EB6+ 3" 956 # ' + - 3B !6+ 3 B$6+ 3"5$6+ 395$6+ 3"4895&6+ 3 D5$6+ 3 D5B6+ 3@556

:

1 - ,

- . ' 0 2

1 ; 2 ! ' 2 ' ' 3 D 56 3 DD5$6

: , ' ) + - )

' V ; , ? -

@ 3 4 76 ' ' Two Different Random-Selection Strategies

"2+ ,+ 9 3"95B!6 ' ! 1

2 , - ! ! 3" . V ! ,0+ , 1 , / 3"95B!6 , .0 ; ' + Æ ' , . 0 .0 1 Æ . 0+ ' Hybrid Graph-Coloring/Meta-Heuristic Approaches

1 557+ ! + + 3 "+ $ "+ , - - ' - ! @) ! , ' ! ! #

' - - ' ' + 3B ! 6+ 3 B$6+ 3"5$6+ 395$6+ 3"4895&6+ 3 D5$6+ 3 D5B6+ 3@556 1 ) ! -+ 38 76 EXAMPLES

: , 1 31 5$!+ 1 5$!6

G ! ' - ! ! - ' 9 @, 1 , 2 , 1 )

© 2004 by CRC Press LLC

Section 5.6

Applications to Timetabling

461

) .$ 0 : .0 ,

+ .0 ,+ .0 ' + .0 ' L + .0 ' + .'0 , , ,+ .0 77 - - 1 '

/ ' ' : .0 / ' , - 77 ' 7+ .0 / ' ' - !

-

: "2+ , 9 55B 3"95B!6 !

.D + + @ 0 , )! E ' - ! . ' 0 , . ' !0 1 - , , , 2 , 2 ! 3"95$6

: "2 , - ? ,

3"556 1 - , ' + , , - F ,-+

' - ' -!

3 B6

: @' 3@76 + , 2

' F / 9 3FB&6+ ! 1 , ' ,

, - - V , ' +

, ' 2 # ' 2 + , - , . +

0 ! '! ! 77 REMARK

: ;

,2 , 5

' ' "2 ,

2 Æ # ! 3"556+ ! , / ' 7 ' 77 ' , L - 1

2!

Æ ( , ) ' - . C 0

© 2004 by CRC Press LLC

462

Chapter 5

COLORINGS and RELATED TOPICS

5.6.5 Sports Timetabling 1 ' - ' '

9 , , '

! ! @ , - - - / 1

.11S0

3

176 ( '

+ , ' 9

' 3"B6 ' ' 3B6 '

DEFINITIONS

:

;

' (

'

'

<

'

' , K

:

:

+ F+ '

- '

( .'

(' ,

'

" + ;+ '

<

+

+

'

! 0 , ' '

:

, ,

A

'

REMARK

:

# ' + , '

'

A Simple Graph Model

1 '

) , '

,

! +

; ,

'

.

, '

+ ?

%- '

3

6 ;

0 '

DEFINITIONS

:

D

'

'

! + ' ,

:

'

A .

#

0 ,

, '

1 ' '

© 2004 by CRC Press LLC

;

1 ! )

'

! ! , '

1 '

! '

4

Section 5.6

463

Applications to Timetabling

9 ! .+ 0+

4

1 ' ' - Ü : D # # # ' ! ' # # + < + A ' # 4

' # # - ! ' + A : ; ' . 0 ' .0 - ' .0 : ; ' + + ' !

' ' ' ; ' 4 4 ' ' !' / ' ; !' / 4

)

; ' !' + , - -!+ ' E + + ' ' / FACTS

:

; !' / ' ! ' +

# ' ' + < 1+ ' !' /+ ! '

:

; # + ! ' 4 '

!' /!

:

1 ! !' / ' ' + + L

@ + ! + !' / ' ! ' + ,

EXAMPLE

:

# $$ , ! . 0 '

< ' < $

Figure 5.6.6

© 2004 by CRC Press LLC

7 * ' ,

464

Chapter 5

COLORINGS and RELATED TOPICS

4½ 4 0+ ' 4 , , 1 !' / + . .# $&0 1 , ' 1 ' / ) ! ' ( + , # + + + 4 .0 4 .0 4 .0 4 .0 ! 4 . 0

. $0 . 0 . 0 .$ 0 . 0 . 0 . $0 . 0 . 0 .$ 0 . 0 . 0 . $0 . 0 . 0

Figure 5.6.7

" ## ## $

7 -# &$*,

%- ' )- + , # 1 2 ' , ' ! ' . 0!' / + , '

Profiles, Breaks, and Home-Away Patterns of a Schedule DEFINITIONS

: D 6 ' ' 1 " .F;S0 , 6 . 3B60+

" .6 0+ . 0 )

;% , % & & . .6 0 < F ' ' '

: # - . 0! 6 ' ' + 4 '

, ' " .6 0 1+ ) ' FK ;K , , '

: # - 6 + ) ' , ' ' + '

,

: # - 6 + . C 0 ' . .6 0 < . .6 0 (

, + ) ' , - FK , - ;K+ , ' . C 0 EXAMPLE

:

# $B , F;S , ! ' # $$ # $& 1 2 !

- )

© 2004 by CRC Press LLC

Section 5.6

465

Applications to Timetabling days

1 teams

Figure 5.6.8

1

2

3

4

5

A

H

A

H

A

A

H

H

A

2

A

H

H

3

H

A

A

4

A

H

A

H

H

5

H

A

H

A

A

6

H

A

H

A

H

$ +7 &&* )$ $ &$* ,,,

A Lower Bound on the Number of Breaks

%' ! + , '

+ , .+

' 2 /0 DEFINITION

: ; ' !? . 0 1 ' +

.0+ / ' @

' ! .0 ' .0 FACTS

3BB6 D !' / + .4½ 4 4 0 !' / ' ! ' 1 . .00 2

:

:

@ ' + . 4 0 ' 2 0! + .4 REMARK

: # - # $&

' 2 Irreducible and Compact Schedules DEFINITIONS

: ; '+ ,- ,

+ ' 2

: ; ' .+ F;S 8 0 FACTS

: ; ! !' / '

© 2004 by CRC Press LLC

466

Chapter 5

COLORINGS and RELATED TOPICS

:

( + ' , ; ) ' C +

, F ) ' C 1+ + $ % .( # $B+ , 0

:

" - ' . , 2 ' ' 0+ , ' 6

- 2 6 # ' + , ., ' 0 ,

# 5 , ' ,

-

6

! + 3

, 2 + D ' 2 3 ., 3½ 3 3 0 ( + , ) 3 < 3 < C

: .0

! 1 ' , - :

6 + , D 2 3 <

! ' .0 1 .3 3 0! , - ' < C .0 < < D ' < D

.0

EXAMPLE

:

# 6 # $& . F;S # $B0+ .3 3 3 30 < . $0 D < D < + .0 ' # 5 ) 1 , .0 )+ < A < A 1 - ' :

<

< < $ < < $ < < $ ( , ' , -' .0 ) Complementarity

;

' ' DEFINITION

: ; 6 ' '

? / / , / - ) FACTS

:

3BB6 (' 6 .' 0 2+ 6 . # &0

© 2004 by CRC Press LLC

Section 5.6

467

Applications to Timetabling

: (' 6 .'

0 , 2+ 6

: 1 , ,

- 2 EXAMPLES

: @ # $& F;S+

- # $B 1 / < $+ / < + / < ,

@

: # $5 , !' / ' < 6 ( F;S , 6

- . , 20 a

d a

b

d a

c b

d a

c b

K4

c b

F1

"

! 5 5

5 " 5 "

c F3

F2

"

d

"

5 " # # " $ 5

Figure 5.6.9 7 *" -# &$*

,

Constructing a Compact Schedule with a Minimum Number of Breaks

9

+ , < ; $ , - . + 0 - 2+ ,+ # + EXAMPLE

: # $7 ; $ ' %- ! - # $$

Figure 5.6.10 $ '*9 &$*

© 2004 by CRC Press LLC

#** "9 7$- ,,,

468

Chapter 5

Algorithm 5.6.1: 7:

7 . 0' 9 !$*

COLORINGS and RELATED TOPICS

;$ 10&

. 0! , 2

=:

@ . 0!' / ' : # < < 3 6 3 C 6 . 0 : < @ % : # < ('

3 6 . 0

3 6 . 0 # < ('

3 C 6 . C 0

3 C 6 . C 0

REMARK

:

1 ' / ) @ ' 3"B+ 6 ( ' / 3BB6

An Alternate View of the Canonical Factorization

D J J J J

+ -+ . 0 ' / ; $ . 0 ' ! ' + ) < < + < FACTS

: # < + ) ! X < J J J < J J J ( + < < J 1+ # 5+ ) , J - 2 3 < C + <

: # ' / - 2 ' EXAMPLE

: # + , - < < < J < J < J < $+ ' F;S ' # $B+ < J < - 2 + , < J < - 2 + , < J < $ - 2

© 2004 by CRC Press LLC

Section 5.6

469

Applications to Timetabling

Some Characterization Results FACTS

:

D 6½ 6 , ' + , 2 ('

- 3 3 3 ' , 2 + F;S+ " .6 0 " .6 0+ . ' ,0

:

- +

3 < 3 < + , , < .3 3 3 3 3 3 0+ , ' ' .3 3 0! ' ! ' ) # .# + ' # $& 7 < . 0 .3 3 3 30 < . $00

7

:

- 7 < . 0 , C C < + , F;S ' ,: ' ) ' , ; 2 C C C A ) ' , ; 2 # ) ' J ' ) ' EXAMPLE

:

1 F;S # $ # & ' < $

"

5

"

"

5

"

5

"

5

5

"

5

"

5

"

5

"

5

5

"

5

"

5

"

"

5

"

5

"

5

J J

!

J

Figure 5.6.11

7 +7 &#*

7

7

< . 0

" ## ## ## $ < . 0

,

REMARK

:

; 7 < . 0 ' - , C C <

- F;S ' , 2 # + F;S # $

' Feasible Sequences DEFINITION

: ; 7 < . 0 F;S ' ' , 2 FACTS

:

3BB6 (' 7 < . 0 ' ' + 7 < . 0 ' 7 7 '

© 2004 by CRC Press LLC

470

Chapter 5

COLORINGS and RELATED TOPICS

: / ' ' A ,-+ ' ' - / . 3E(,E760 : - .½ 0+ , F;S

: , J J J + , J - ! )A J - 2 C C C . < 0A ) ' , ;

# - F;S '

/ ' + , ) ./ 0 < / . < 5 / . < "

./ 0

: 3E(,E76 (' - F;S '+ '

/ ' + ./ 0 ' , / E -+ -

- + , -

./ 0

/

: 3E(,E76 ( ' 2 / + Æ

. J J 0+

/ - ' -

1+ ' 2 %. 0 - + , + '+ ' 7 < . 0A ' $+ 7 ,

' F;S CONJECTURE

3E(,E76 1 - # Æ ' 7 ' F;S REMARKS

: 1 , ' -

1 ' - '

: ; + ' 2 .'

+ - , ) 0+

,

' , ' 2

: @ ' - ' ! - 3 176 : #+ - ' /

'!

/ ' 1

' ' / ' . , -

- , - 0 @ 3B6 3B !6

© 2004 by CRC Press LLC

Section 5.6

471

Applications to Timetabling

References 3;FM5B6 ; @ ; + 1 E 4 + * FM 2- + + - S+ 55B

)

3;76 ; 9+ ; / !1 E ' @ 1 S + 7 / , .770+ I 3"5$6 ; "+ ; @

- 1 : 1 , 9-+ "2 S * . 0+ *

;* 8..G= 7 = H= < ;+ & = > 88GD< @ + .55$0+ I

3"B6 "+

+ !+ S+ 5B

3"$6 @ " + # @+ 5I5B

& .5$0+

* 2 ;* 8..9= = = = < ;+ & = > 8B:A<+ @ + 55B

3" 5B6 "2 E . 0+

* > ;* -::-= = ) = = < ;+ & = > -9B:<+ @ + 77

3"76 "2 S 2 . 0+

3" 956 8 "2+ + * # 9+ ; - 1 @ " E + , 7 & .550+ IB 3" # 95$6 8 "2+ + S F # * # 9+ ! 1 " - : @-+ "2 S * . 0+ * ;* 8..G= 7 2 = H= < ;+ & = > 88GD<+ @ + .55$0+ &$I57

* 2 ;* -:::= # = = = < ;+ & = > -:9.<+ @ + 77

3" 76 "2 9 . 0+

3"4895&6 8 "2+ 8 @ 42 + 4 F 8 + * # 9+ ; 1 : 1 @ ' ; + 7 .55&0+ $ I & 3"8S5&6 8 "2+ 4 F 8 S 9 S + ; @ # ' 1 ( + "2 E . 0+ *

;* 8..9= = = = < ;+ & = > 8B:A<+ @ + 55B

3"556 8 "2 4 S ,+ ; E ! - ; '

1 S + 777 7" .5550+ $I&

© 2004 by CRC Press LLC

472

Chapter 5

COLORINGS and RELATED TOPICS

3"95$6 8"2+ 4S,+ *#9+ ; E ; ' -! 1 + "2 S * . 0+ *

;* 8..G= 7 = H= < ;+ & = > 88GD< @ + .55$0+ I 7

3"95B!6 8 "2+ 4 S ,+ * # 9+ ; @ F @ ' 1 S + * 7 ;7@.A<+ ( @ !;+ , W 2 .55B0+ & I &5 3"95B!6 8 "2+ 4 S ,+ * # 9+ ( / @ - - 1 + 7" $ .55$0+ BI7 3"* 5$6 "2 S * . 0+ *

2 ;* 8..G= 7 = H= < ;+ & 2 = > 88GD<+ @ + 55$

3 B6 E 9 + ; ; ' S 1 S ! + 1! * AD2:6+ ' ( + - ' 1

.5B0 3 B$6 E 9 + ; @- ' S ; ' ; + / , .5B$0+ 5I7

1

3 4 76 E 9 4 + ( / !

1 + / , + .770+ BI 3 D5$6 E 9 D + * - S 1 + "2 S * . 0+ *

;* 8..G= 7 = H= < ;+ & = > 88GD<+ @ + .55$0+ I

3 D5B6 E 9 D + * - S 1 + "2 E . 0+ *

;* 8..9= = = = < ;+ & = > 8B:A<+ @ + .55B0+ I5

3 D 56 E 9 + D + 4 9 2+ ; @ @ + .550+ 75I7

3 DD5$6 E 9 + D + @ D+ 1 : ; ! @ ; + / , & .55$0+ &IB 3 &6 S T + ; , S ! @ * + E + 4 " #+ E 8 2 . 0+ , + @ + .5&0+ &I& 3 $6 ; 4 + 1 S ' 1 @ @ + & .5$0+ &I

© 2004 by CRC Press LLC

Section 5.6

473

Applications to Timetabling

3 856 1 "

4 F 8 + 1 @ ' * ( ' 1 S + $ .550+ $ I$ 3 $ 6 4 + " 2 @ + - ' 1 + 5$

*+ S + @

' !

3B6 9+ @ @ + S F . 0+ * + !F + .5B0+ BI5

3B6 9+ E/ ( @ @ 1 + .5B0+ &I$ 3B !6 9+ ; ( 1 + 7 I$

/ ,

3B !6 9+ + F ! 1 + , ; = $,< 5 .5B 0+ 7I

5 .5B 0+

/

3B !6 9+ % E ' - : ' ' @ @+ &'! .5B 0+ I$ 3BB6 9+ @ E ' ' @ @ + .5BB0+ &I$ 35&6 9+ 1 ' 1 + .55&0+ 7I

7 / ,

2 5$

376 S 2 "+ * ' - ! F S * + "2 S 2 . 0+

* > ;* -::-= = ) = = < ;+ & = > -9B:<+ @ + 77

3 176 8 + + E 12+ @ - 1- 1 ! S : ; ( S S ; + "2 S 2 . 0+ * 2

> ;* -::-= = ) = = < ;+ & = > -9B:<+ @ + 77

3 -( @&$6 @ -+ ; ( + ; @+ % ' ! V , + .5&$0+ $5I&7 34 &56 E * @ 4 + + 9F # + 5&5 38 76 # - 8 8 +

! + 8,+ 77

3 $6

+ 1 ' !1 1 + $* + .5$0+ &I&& 3W556 4 D 4 W+ 555

© 2004 by CRC Press LLC

*

+ * S+

474

3FMB6 * FM2- + ,

Chapter 5

COLORINGS and RELATED TOPICS

7 ) + E 5B

3FB&6 ; F / 9+ 1 @ 1 ' + 5 .5B&0+ I 38776 4 F 8 + E 1 S , @11D+ "2 9 . 0+ * ;+ & = > -:9.<+ @ + .770+ I 7 3EB6 8 E + 1 ; ' E @ S + .5B0+ &I$ 3EB6 8 E + ; " E @ + .5B0+ B I5B

72

3E(,E76 * E + F (,2+ 1 E + $ * ' & )!+ "2 S 2 . 0+

* > ;* -::-= = ) = = < ;+ & = > -9B:<+ @ + 77

3SB6 F S 8 @ /+ 5 + S !F+ 5B

/ %

3@556 ; @'+ ; @- ' ; 1 + .5550+ B&I&

0 ," '

3@@ B76 @ 1 @ M

+ 1 U ;

" ! + .5B70+ 7&I$ 31 5$!6 4 1 8 ,+ ' @ ; ' 1 S + / , $ .55$0+ 7 I B 31 5$!6 4 1 8 ,+

@ ' @ ; " 1 @ + "2 S * . 0+ *2

;* 8..G= 7 = H= < ;+ & = > 88GD<+ @ + .55$0+ I$

39S $&6 4 ; 9 E " S ,+ ; " '

' ; 1 S + 7 .5$&0+ B IB$ 39 $B6 9

+ ; @ ' - 1 + .5$B0+ I& 395$6 ; 9+ @+ 1 * I ; @ * P+ "2 S * . 0+ *

;* 8..G= 7 = H= < ;+ & = > 88GD<+ @ + .55$0+ $I&

© 2004 by CRC Press LLC

475

Chapter 5 Glossary

GLOSSARY FOR CHAPTER 5 $- -"

:

I '

' - !

' ,

-&

. ' 0 ! '

C 6+ ' 7

3

& . !0 I : 6 $: ' ##8- < ##8-= $-:

.

:

' '

'

0 #( . 6 0 6

2

'

-. 0'##8- $- I ' -. 0 :

Ê

"9

- ,

.0 -. 0 .0 .,

: ' -

- + - ' 0

I '

:

' !?

,

'&&- I - .0 ' : , < . / < 0 ' - - .- !0 &* # I : ' - , ' - , - ?

-

&9-#9 -& &9 .

5550:

< .0 #( . 6 0 6

-

. 0+ '

S

'" . ' 0 I : 6 1 #$: , " #$: , C !' ' - ' ,

C

L -

.

0

"'$9##$: , "* -" .0 I ' :

B

)

0 B . 0 < .0 "# #$: , - , - ' - , - $ -" I ' : - !

" .0 . ( 0'$&" #$ I ' , ' ( : Æ: ' < .0 ' - - +

' ' < (.0 ' ' < ' '$&" #$: ! ' - ! '$&" #$ I ' ' : Æ: ! ' - , < .0 '

.

0

.

$* #$:

<

<

, - '

? ! - - 0

© 2004 by CRC Press LLC

"

.+

476

Chapter 5

COLORINGS and RELATED TOPICS

'$- #$: ' - < !

. 0!

$- *8 I ' :

' ! '

A '

.0 $- -" I ' : ' - '

.0 $- #9- I ' : ' - + - ' ! ' A

. 90 $- &- I ' : ' - ! , $ A

[.0 && > : ' ' J.0+ ' ' .0 < J.0 C ) I ' : ' ½ )' #$ : ? I ' : ' ' - - ' ? $9##$ I ' < . 0: ,

- ,! +

- (

.0 ? * -8 I ' : 7! +

7.0+ , , - ' ' A + . 0 ' -

' ? -" I ' +

.0: / ' ? # -" I ' : # .0 ' # .0 ' ' .0 ? I : ' - ? ¼

¼

. : !' ) 0

?

I

:

? ' - . :

!

' ) 0

&&

I - .0 : ' - .0 -

&: '" #$: - , " -8* $9##$:

!

'" I + , : ' - . 0 ' -" I ' : - ' - ' A

.0 " ' I ' : ' " ': ,

© 2004 by CRC Press LLC

477

Chapter 5 Glossary

-#"9 #$:

, ?

2 30 2 #0 -#- I ' : . < . . 0 , < . 0 ' ,: - ) . C 0 . ' . C0

-# -' - $9##$: -! ' - A

# . ' - 0 - + + ,- .

3 #0

.

.

( #$:

, - .+ + 0

, , , -

& 2& ##9 I 7! .:

' , '

.

K -

& $&:

' ,

-

& $- -"

I '

+

, -

':

'

!

& &: - , ' #$: ! , ,! -

-# #$:

,

' - !

'

99 '*'* #$:

,

- , +

" '*-#& #"-:

) ' ,: - )

! '

*&" "9 $:

;

' " P

, ' !

$ $

$

$

"+

* '" #$: ! , * $ -" I ' : ' ' A

0 *@' '@ 7'* I ' : *'$- -" I ' : ' L ' ! ' A

.0 "

¼

.

¼

*': ' . 0 + ##: ! , , - - 8 $-: - / !

© 2004 by CRC Press LLC

478

Chapter 5

! '

:

'

COLORINGS and RELATED TOPICS

0'**': ' , - . 0 . 0+ ' ' .0 - + .( 0': ' ( . 0 - .0 . 0 ' .0 + 32 36': ' ' ' , 2 . 0 3+ ' .0+ , 2 3 2 3 + ': ' - !? , .0 + ': ' , + .

,

'

. ,

0: " ' .0+ ' .0+ , + ! -+ !- ' .0 + ' I ' " : ½ ' ' " . 0 - .0 + ': ! ' #"- 5'/ .0: ) (@1; : ; " b @1(%: " !' P / I ' : ' ' ½ ' ! ? ' ' .0 .( 0' &: - .( 0!' ' ' .!' ' - . (.0 ..0 .0 ' .0 +

I '

. 0 <

&" & $- -"

: )

I ' :

- - A

.0

8' Ê

: ' ' ' ' -

+ ' .( 0!' I ' : - < ./0 , .0 7 / #/ ( . 0 . 0 . 0 F . 0 < + ,

- , - #/ = >

= > ' ! 32 36'#$: , 2 . 0 3+ ' - - .0 #$ I : ' ! -

' . 0

*9 -" I ' : ' - ! , , -

' +-- *& I , < . 0 < . 0 ' - : ' ' , < +-- #$ " . 0: , - - , !

; ' - " . 0 ? ' F , + - -

$* &

: )

© 2004 by CRC Press LLC

479

Chapter 5 Glossary

$ I : ' , + + , ! - -

" : - ' - ' " , ' - ' "

' : " A '

: "

$--#$&- I ' .0

$9##$:

. 0

! < . " 0 , .- 0 "

$9#$- I : , F + ' ,

+ ' .0 $9#" I : , F + ' , + ' .0

*#* -": ' - !/

'

*#* & I ' ?

:

' -

!

*#* &: !? ' - *#*'& I ' : ' ' - - '

: 7! +

1.0+ , , - ' ' A + . 0

' ' -

*#*'& * -8 I '

& #$ I ' ' ' ' - :

. 0 , ! ! , ' - ' , -

' ? ' ' , ' , !

#$: ' , ' ' -

+ + ! ! , - ' - ' , - ' ? ' ' - - '

##: - , - '

,

- ' -* I ' - : ' ' - ' , ' I < . 0: ' ,

? ! 5 0 < + ' ' , , - '

.0 1 .5 5 0 .0 00 ' +

,

.0 # < + - '

' 5 ' 0

.0 # < + ' 5 0 ' - ' +

© 2004 by CRC Press LLC

480

Chapter 5

.' I < . 0:

5 0 ' 7

COLORINGS and RELATED TOPICS

' -

+

' ' , :

5 0 0 + - - .0 1 .5 ' 0+ .5 0+ .0 7 0 .0 0 ' .0 1 , ' , ' - : .5 7 0+ .5 0+ .0 ' 0 .0 0 #$ I ' : +

.0+ , - ' ' , - ' .0 ? ' ' ,

, ' "9 I ' : ' ! ' , .0 ' &: ' , ' & &&- I - ' : ' = ,> , - ' .0

- -

& $- *8 & * $- -": & $- -": & " #$: ! , !

&$:

' ' +

+

'

"$ "

' +

+ +

-#-:

, ,

' ,

)

) $- -" I ' : A

.

!0

'

-$ -" ' : / ' A

E .0 -8-- ?: ' - !/ ' -8* $9##$: , ' + . ' 0 + ' ': , 7 ! + 7 ': , ' ! @ + ":

- + ?9 " : ? + '

$"$* -#8 - '

+

I '

:

, -

- -

)$'/ '()

I :

7 +

, ' '

!V , . ' , ,

- 0 !V , . ! 0+ -

**'9 ##9:

, - '

- - ?

' :

- -

½ ' + ' ½ ½

- - 2 ,

© 2004 by CRC Press LLC

481

Chapter 5 Glossary

* '

:

I '

! ! , !

' A ' ,

# # -" - G

'

: ' ' - G

I '

+

#$ #$ $-@ $ I '

# #$ # -8

/ ' -

' ,

! ' ' '

: ' :

! '

! ' ! '

: : 7!

7

, ' - !- -

+ !- -

:

G.0:

E

8

8

- - '

'# #$: , - . ! 0 - ! " ' .! " < 0+ ." 0 < G." 0 '# #$: , - . ! 0 - ! " ' .! " < 0+ ." 0 < ." 0 #- I ' 5: ' ) 5 < 2½) 2) 2) , - ? ' A

5 #- #$: , - ' , ? ' . 0+ ' + ' ' ' ?. 0 ?.0 9.0: - ' 5 # * I ' : - ' ' +

' ' , A

< . 0

# #$ # 8&

:

8

: 2 , -

' + -

'

$'8:

S

,

$

-

-

## 8'

I '

'

*- #$ +

: D

!

!:

- ,

' -

' -+ ,

- ,

7 ,

, .

0

I '

:

1

' '

. 0

0 -"

!

!

!

< 1

' - !

, - ' A

0

.

$' #$ : © 2004 by CRC Press LLC

, - $ $ .0

482

Chapter 5

COLORINGS and RELATED TOPICS

$' "-A 1 ) : - - $+ $! P &-# 8 I ' : - , - '

&0: ! ' + , !!

& &: +

+ ' &? I ' : ' ? - A

&"9 -" I ' +

.0: / '

&** &-#8: J < , : 7 < < &$ I ' : ' , & ' I ' : - ! , +

- ' ! , - , - 7! ,

- , *L ! #$ $-@ $: ' '

' !

!/0&B; -": " &$: , , -

' !

-" #"-:

'

' '

: ' - . - 0 +

##:

, , ? - G -

#$ I ' < . 0: - + C ? '

A

/ .0 $&& I ' : ) C ,

,

?

6 #. 6 0 6 .0 , 2 - 6 .0 #. 6 0

' ' 6

.0

: '+ , ' ' -' - $9##$: - - - ?9 <8'=" #$: ? , ' + '

?9 *'" #$: ? ! , ' + '

## $- -" I ' : ' A

.!0

8 I ' : ' .0 - '

'

© 2004 by CRC Press LLC

483

Chapter 5 Glossary

'8' #$: ! ,- -

,

8': ' - . - 0 + ##: , , ? - - 8' -" I ' : / ' - - A

: .

0

*9 -8

I '

- .

8'** &"#$:

C

.

0

¼

.

0

© 2004 by CRC Press LLC

7! , , '

+

:

"

C C

< .

¼

- ? ¼

0 '

' , -

8')$* #$: ,

0 ' '

, - -

¼

+ .

< .

0

¼

-

0 ,

' '

! -

Chapter

6

ALGEBRAIC GRAPH THEORY 6.1

AUTOMORPHISMS

6.2

CAYLEY GRAPHS

6.3

6.4

ENUMERATION

! GRAPHS AND VECTOR SPACES

"!# ! $

6.5

SPECTRAL GRAPH THEORY

6.6

MATROIDAL METHODS IN GRAPH THEORY

%& &

' $( ) GLOSSARY

© 2004 by CRC Press LLC

Section 6.1

6.1

485

Automorphisms

AUTOMORPHISMS

!

" # $%

Introduction &

' ( ) ) ) $

* + , - ) ) + !( !, $ ) ) . / 0 + , $ ( ) % ( ) . % /( ( % ( ) )

1 + , ( )

6.1.1 The Automorphism Group DEFINITIONS

2

( + , + , +, +, + ,

+ ,

2 ( ( )

+ ,

( ) + ,

)

2

© 2004 by CRC Press LLC

486

Chapter 6

ALGEBRAIC GRAPH THEORY

REMARKS

2

1 & + , ( &

2 $ ) ( & ) + , + ,

2

1 . / ( ) ) & ) + , $ &

) ( + , ) EXAMPLES

2

3

+ , 4 4 ( + , 4 ( ) %& ) ( %& ) + , 4 5 - ( + , ( ) ) )

2

2 67!8 5 +,(

% )9

2

& + % , % ½ ( ) + % ,

FACTS

2

3 + ,

)

+ ,

2

7 (

2 % ( : & :

%& )

6.1.2 Graphs with Given Group DEFINITIONS

2

( & +, 4

2 ) + , + , $ - ( & - ( &

© 2004 by CRC Press LLC

Section 6.1

487

Automorphisms

2 +½ ¾, +½ ¾ , & +½, 4 ½ +¾, 4 ¾

2 7 4 ( ) 1 ½ ¾ &

; 2 ½ ¾ ) 9 2 ½ ¾

++,, 4 6;+,8+ +,, ½ ½

( 7

Figure 6.1.1

$ ( (

2 ½ ¾ ) 9 2 +½, +¾ , ½ ¾

& ½

+½ , +¾ ,

& ¾ 2 2 )

+ ,< NOTATION

7 % ( + , + , + ,

EXAMPLES

2 + ,

5 +, = )

2

7 + , +, ) +, ) ) ( ( ) 5 +,

2 ¿ ½¿ +¿, ) ) +½¿, FACTS

> : < (

+ ,( + , 4 +, +,

2 6= "8 + , +) ,

& ¾

© 2004 by CRC Press LLC

488

Chapter 6

ALGEBRAIC GRAPH THEORY

2 618 3 ) ( $ & ( 2 67"8 ! ( & % + , +) , ) ) 67?8

# (

2 65!8 $ 7* ( + , (

)

(

+ ,(

( @ )

$ ( 7 & ( ) - + , ) +$ + , )

( + , ) ) ) , ' ( . / )

.) /

2

) ) 9 & & ( +, 4 ! 2 6A)!8 $ % 0 ( ( ( + , 2

+ , 4 ?< +

, 4 B< + , 4 +5 65 !8,

REMARK

2 >&

( 7*

FURTHER READING

C 671!8

6.1.3 Groups of Graph Products $ ) ( ) D ) D ( % ) & + , + ,(

) ) + ), 9 $ ) + 6$ $C!8, & & B % E

( ( ( + D ,D D+ D ,

© 2004 by CRC Press LLC

Section 6.1

489

Automorphisms

DEFINITIONS

%

2 3 ) ) 3 ) + , ( ) + , 5 + , + , + , 6 + , 4 8 6 4 + ,8

) ( 4

<

6 + , 4 8 6 4 + ,8 6 + , + ,8

( 4

< + , + , ( 4 < + , 6 4 + ,8 ( 4 6 8

7 ( )

Figure 6.1.2

" # $% #

2 + D, & 4 D 4 D

2

+ D, ( ( &

2 + D,

2 3 ! ) " ( 1 % ! ) # " & & × ! #+ $, 4 ++ , «× +$,, + $, " GENERAL FACTS

2

6$ ?8 &

$ ( 6 8 4 6 8( ) ( ) ( ) + & ,

2

& ( ( 6 8 4 6 8

© 2004 by CRC Press LLC

490

Chapter 6

ALGEBRAIC GRAPH THEORY

FACTS ABOUT CONNECTEDNESS

2 + ( ,

) 2 3 ) ) ) ) 2 6 8 FACTS ABOUT DECOMPOSITION

2 65 B8 > :

2 7 ) )

2 6$ !8 &

) ) (

( : FACTS ABOUT AUTOMORPHISMS

2 65 B8 $ ( + , )

7 B

2 3 ) 4

+ , + , C &

7 ) ( % + , 4 + , + , 2 + , + , $ & 4 ( % +, 1 % ) &+ , + , + , ) + , &+ , % +, 4 % +,< + , + , % +, 4 % +, < + , F+ , 4 2 65 8 3 ) + 6 8, 4 + ,+ , 2 + , &+ , 4 F+ , ( +

, + , 4 F+ , 2 > D & ) & FURTHER READING

7 ( 6$ G BB8

© 2004 by CRC Press LLC

Section 6.1

491

Automorphisms

6.1.4 Transitivity DEFINITIONS

2 $ ( + , ) 4 2 +, 4

) C & + , + , ) ) + ,

2 + ,( & + , +, 4 + ,

2 + , + , 9 ( & + , +, 4 +, 4

2

& ) )

2 ) &

)

2 ( +', ) ' &( ) ) ( % % % % ) + , 4 6+',) 8 $ + , ( B( $ + , 4 B( $ 6 +* ,)' 8 )(

$ )& )

( FACTS

2

$ & ( + , 2

+ , + , 9 ) + ,< + , 4 + , 2 +, 4 < $ % ( + , 4 + , + ,

2

$ & (

&

2

$ ) & ( ) $ ( + , & ) + ,( ( )

2 2 2

67 !8

B $ ( ) & 6 8

! > %

7 & (

© 2004 by CRC Press LLC

492

Chapter 6

ALGEBRAIC GRAPH THEORY

2 6A !B8 7 ( & ? + H , 2 6= "8 ! 2 61"?8 > % )&

2 6"8 3

) ( & ( % : 2

<

+ , + , % + ) , ) - + ,) % )% ) C + ,) ) %

I % * 6A!8

( 2

2 $ ( % ( &

(

( ( B + 6 +,)' 8 + ( )

2 6 "8 & + , - % ) % ( ) % )

2 65?8 7 %

EXAMPLES

2 1 , 4 ( ) & 2 - ) $ 4 ( - & ) ) ( 7 +- , 4 = ( - )< +- , 4 " 2 & ) J ' + * , 2 3

4 ) ) (

4

'

2 ( ( & + % ,

% &

2 % ) ) - % : 7 - * + 7 , )

+ H , + 61"?8, REMARK

2 7 & &

( 6$"!8

© 2004 by CRC Press LLC

Section 6.1

6.1.5

493

Automorphisms

-Regularity and -Transitivity

$ & ) %

) ) DEFINITIONS

2 $ (

4

2 1 )

2 7 B( -

2 ) + , *+ +

2 ) + ,

2

, ,

FACTS

2

$ (

( : +, 4 $ % ( 4

2

6 8 3 ) & <

. + , H ( . + , <

( + H ,

2

61!8 3 ) % + H / , ( / ( ( / $ + , ) ( ! 4 + C %

5 65?8, EXAMPLES

2 2 2 2 2

B

) ) ) 4 : < :

© 2004 by CRC Press LLC

494

Chapter 6

2 2

ALGEBRAIC GRAPH THEORY

$ :

: ! ( % 3 +! , ) 7 * 4 B ( & * ) 9 * ( * H ( * H

6.1.6 Graphical Regular Representations DEFINITIONS

2 ( + , + , ( (

2 +% % , 2

) )

&

0

&

0

4

0

& < 4

,

,

0

<

<

" :

1

C

FACTS

2

$ I

( + , 4 $ )

)

2

6I ( 5 ( $ !B8 )

) 4

2 2

6J "( 1!8 J C

) C

2

(

4 61!8<

+, 61!8,<

1

4 61!8<

) ! & 6J 1!)8< ) " &

6J 1!( 1!8

( (

2

6=! 8 > % ) ) ( C ( ) &

2 2 2

61!8 5 +,

+,

6 "8 > % )

6A) "8 3 ) ) 2 3 + , ) )

© 2004 by CRC Press LLC

Section 6.1

495

Automorphisms

+ , * & 2

( ( 3+ ,

1 ) ) % *

$ ( 68 4 *

2

2 *

61! 8 3 ( 2 * 68 )

9 + , ( + , 4 ¾

$ ¾ + , % (

%

6.1.7 Primitivity DEFINITIONS

2 3 ) ) 4 +

, ( +4 , 4 4 +4 , 4 4 I ( )

$

) -<

) - ) -(

< )

) -(

2 4

$

4 ) -( (

2 )

& EXAMPLES

2

+ ,

$ ( ) - ) ) &

2

( )

< )

+ ,

2 2

>& (

) ) - C ( &

& FACTS

2

& & ) - + , =

2

$

( : E + ,(

&

© 2004 by CRC Press LLC

496

Chapter 6

ALGEBRAIC GRAPH THEORY

2

61""8 3 ) %

( ( ( 1

%

2

6K1"?8 3 ) & % ( % (

4 4

2

6K1!!)8 3 ) & + , 4 Æ )

& ) )

( ( )

2 61B8 3 ) % ( %

+ , 4 ( ( & ) & ) ) # ( ) ) / /

2 6 $ 511 "?8 3 ) % ( ( &

6.1.8 More Automorphisms of Infinite Graphs . / % ) ) 0 ./ ( )

% ( % ) = 6= 8 $ ) ( ) %

( + , DEFINITIONS

2 3 & & + , 1 & & & & & & + , ) +& &, +& &,

%

$ : + ,

2

:

+ , ( % 5+ , 4

+ ,

2 $ ) & + ,( & ) ) & $ & & + ,( & &

2

2 + , + ,

+ ,(

+ , +, +, + ,

$ ) 9 ( > % ) 9

© 2004 by CRC Press LLC

Section 6.1

497

Automorphisms

2

%& % ) + ,( ( + , 4 %

2 ) + , + , % ) 1 + ,

2 % ) % FACTS

&' ()"*+ *,

2 6GL 82 3 ) + ) % , : % ( ( 9 ) 5 ( & 9 &

2

$ ( % $ 5+ , 4 B

2

6= 8 3

& &

+ ,

% % )

: 2

& & J % ) & & % 9 +% , &&

2

1 % ( 5+ , : )

% % ) + , I + , + ,

2

6= !8 > %& % )

2

6= !8 $ % %& % ) (

2

% ( ) % + , ) % + ) ) ,

2

6= !8 > % %& )

2

6= !8 $ % ( + , ( 5+ , 4 ( A ) 7 = K 6K"8( )

% %

2 2

$

( % ( 5+ , 4 ¼

5 $ ( 5+ , 4 + . / ) , $

( 5+ , 4 $ & ( 5+ , 4 ¼

© 2004 by CRC Press LLC

498

Chapter 6

ALGEBRAIC GRAPH THEORY

2 65?!8 $ : ( + , ) + , % ) 2 65?!8 ) :

2 6A)1"B8 5

% $ + , ) " + ,( 5+ , 4 $ " + , ( + , + H , 6+ , 4

2 65?8 3

) ( % ( & + ,

) % ) & + ,

2 6= "8 >

9 &

( ( ) %

9 ( %

9 $ (

9 )

& EXAMPLES

2 ) ¼ ( C (

) E ( % 5 % (

%

2 3 ) ) ( ) $ % (

& = ( % (

& $ (

+ ) , 2 3 + , 4 ( + , +, , +, H , +, B, +, H B, , ( ) 7

: 5+ , 4

2 % + % ,

¼

2

> + , & : &( +

, : &( +

, & & : &

2 ) +(

&, & ( ) & Strips

© 2004 by CRC Press LLC

Section 6.1

499

Automorphisms

DEFINITIONS

2

%+ ,,

+

2

& )

+ , %+ ,

+ ,

%

( ½ + ,(

+ ,

%

REMARK

2

$

(

) % %

½

(

$

9 +5 6K1!!8,

FACTS

2

7 % ( : 2

<

% + , % )

6K1"8<

2

% ( 5+ , 4 ( + , 6$ 5""8

6$ 5""8 3 ) (

% ( &

2

6K1"8 5 + , )

) ! + ,

½

! 4

+ , 4

(

( 5+ , 4 (

7

% ) 7

2

61?8 $ (

2

61?8 3 ) '

+, 4 +

+ , + H ' H ', 2 + ,8 + ,

,)8 8 4

9 +

,8

& + ,8

& ) )

2

6K?8 3 )

)

+ ,

% )

+ , %&

$ (

) + ,( % %

Some Results Involving Distance DEFINITIONS

+ ,( + +,, + 9

© 2004 by CRC Press LLC

+ ,

& 9 ( B

2

500

Chapter 6

ALGEBRAIC GRAPH THEORY

2 ) ) 9 )

2

)

3 ) ) :+, %

:+, 4

+ , + ,

+, + , ) 9

2 2

) & ) :+&, ( B

$ + ,( & +&, &

( & FACTS

2

6K1"8 ) ) + , & GL * .' -

/( - % 2

2

6 1?8 7 & + , ( & )

2 B :+, ) $ ( :+, 4 ( )

2 6 1?8 $ ; + , %& ) ( ) %& ) C ;

2

6 1?8 5 ; + , %& ) $ :+ , + ( ; %& ) :+ , ( :+ , $ :+ , 4 (

2

6KJ ?8 $ ; + , ( :+&, 2 & ; 4

EXAMPLE

2

3 + , 4 & + , 9 + , + ,( % ) , 3 ; + , 4 + H H , + , + , 3 ) ) + , + H , + H , + H H , :+, 4 ( J & + , 2 ; ( %& ) ;

References 6A)!8 3 A) ( E

( +?!,( !M!B

© 2004 by CRC Press LLC

Section 6.1

501

Automorphisms

6A) "8 3 A) I N ( E I ( +?",( ?M 6A)1"B8 3 A) # > 1- ( I % ( +?"B,( ?BM? 6A!8 = A(

% ( +?!,( BM

6A !B8 $ O A ( P& ) (

+?!B,( M! 6I 8( IQ I ( E 5) ( ?M?

+?

67 !8 K 7 - (

(

+?

,(

!,( M

67!8 7( N * G L ( +? R!,( !M 67"8 7( =

) )- ( ! +?",( ?MB 67?8 7( ) ( +??,( M!"

671!8 7( K > ( # > 1- ( C ( " +?!,( M" 6 "8 I N ( * ) ( 2 " ! # + I 5C ?!",( 3 3 SC P 5S ( ,( I : 5 KS A ( J = ( ( ?"( M? 6 $ 511 "?8 I N ( 1 $ ( J 5 ( # > 1- ( 1 1 ( E ) % ( # ! +?"?,( M " 61""8 K > # > 1- ( C %

( +?"",( ?M B 6 "8 $ -( N % ( $% &' +?",( ?MBB L 6= 8 = ( ') 1 (

+?

,( M!

6= "8 = ( N #& C C 1 ( ( +? ",( ?M! 6= !8 = ( % % ( $ " +?!,( M"

© 2004 by CRC Press LLC

502

Chapter 6

ALGEBRAIC GRAPH THEORY

6= "8 7 = > # ( E ( # 5 = +? ",( M ? L 6=! 8 N =C ( ') L N TL ) ( ! $ !+ ( ?!

) *

6= "8 N 7 = ( ) ( +?",( BMB L 6$ ?8 1 $ ( ') & - - ( ! +? ?,( "M

#

6$ !B8 1 $ ( ) ( 2 &! ! +I : # 5 KS A I : A U ( = ? ?,( >U ( ( P 5S ( J = ( ( ?!B( M 6$ !8 1 $ ( C - ( ( / &&( +?!,( BM?

+ ,! - .!!

6$ $C!8 1 $ = $C) - ( ( +?!,( !!M" 6$ G BB8 1 $ 5 G VC( 1 D 5 ( $( J Q -( BBB

!

# !* % " ( K

6$ 5""8 1 $ J 5 ( ( +?""R"?,( M! 6$"!8 P $ ( E ) & ( +?"!,( !M" 6K"8 = K( % ( "M"" 6K?8 = K( E % %& % ( M

)!

! )!

+?",(

)! +??,(

6KJ ?8 = K J ( N % ( ( +??,( "MB 6K1!!8 = K # > 1- ( E % % ( ! +?!!,( M 6K1!!)8 = K # > 1- ( E % & ( )! ( M 6K1"8 = K # > 1- ( 7 % ( ! +?",( ?M 6K1"?8 = K # > 1- ( %

( +?"?,( M !

© 2004 by CRC Press LLC

Section 6.1

503

Automorphisms

6GL 8 N GL ( ( 3 C ( ?

# ( -

P

6J "8 3 J C( E & C ( +? ",( ?M 6J 1!8 3 J C # > 1- ( ) ( $( +?!,( ??MBB" 6J 1!)8 3 J C # > 1- ( ) ( $$( +?!,( BB?MB" 6 1?8 J # > 1- ( E ) ( " +??,( M

65!8 5) ( ( +?!,( M 65 B8 5) ( (

0! +?

65 8 5) ( & ( !M!" 65 8 5) ( P& ( 65 !8 5) ( W (

B,( M!

) - +?

! +?

% ( J

,(

,( M"

( # ? !

65?8 J 5 (

( +??,( M

65?!8 J 5 P $ % ( : ( +??!,( BMB 61"?8 I # > 1- ( $% & ( ( ( +?"?,( "M 6"8 P $ % (

( $% - +?",( J ( BM! 6 8 1 (

# !( ' ( ( ?

61!8 # > 1- ( E ) (

+?!,( ?MB

61!8 # > 1- ( E 1( 2 # &! !( +Q ( N 3 -( 1 ( , 5 P ( A ( ?!( BM 61!8 # > 1- ( (

(

( 1 +?!,( BMB 61! 8 # > 1- ( (

+?! ,( !M

© 2004 by CRC Press LLC

504

61?8 # > 1- ( > ( BM!

Chapter 6

ALGEBRAIC GRAPH THEORY

+??,(

61B8 # > 1- K > ( C %

( ( L 61!8 # 1 ( ') L ( ?M

+?!,(

618 = 1 ( I ( +?,( BM "

© 2004 by CRC Press LLC

Section 6.2

6.2

505

Cayley Graphs

CAYLEY GRAPHS I $

5) 7 C 7

Introduction :

: E

I )9

6.2.1 Construction and Recognition 1 % ( % ( ) ) % 1 I % I % ( I % I % ( DEFINITIONS

2 3 ) % 3 ) ) 4 ½ ( ( ½

( I+ < ,( %2 I+ < , < 9 2 < I+ < , < 4 2

1 : $ ) ) ( ) ( 4 < 4 2 H

I

I+ ,

2 I 1 I +< ,

2 2 2 2

) 2 +<, 4 2<

2 ) 9 & + , ! " ! +, +,"

© 2004 by CRC Press LLC

506

Chapter 6

+ ,

2

)

2

3

+ ,

+ ,

) % X

ALGEBRAIC GRAPH THEORY

: ( )

$

%

2

%& X <

4X<

=½ =

X(

: 2

= 2 4 =

EXAMPLES

2

) 1

) C I )

(

C ) 1

) C

I ( ) C 9

2

) I

( 1

)

2

( , (

C) ,( )

C

2

% % 7 +>,( >

: 7 +>,(

2

#

+ ,(

4

!

Figure 6.2.1

)

-

- "- .' $ *"" "

FACTS

2

> I &

2

I I+

2

65"8

<

,

/$"

2 I + ,

)

REMARKS

2

5) * ) )

- C

) I $ )

2

I

$

© 2004 by CRC Press LLC

-

Section 6.2

507

Cayley Graphs

6.2.2 Prevalence

I &

I

:

% &

& I (

& I &

Figure 6.2.2 0 #1%*% 23# 2 " EXAMPLE

2

$

(

)

) -

) & 9 I ( &

)

(

( Æ %

FACTS % ) C )

:

/

2

2

( ( ) ) :

2

3

/ > ) / + >

/>

2

/ > < ¾ > 4 / ( > 4 < > 4 H / / 4 > 4 / 4 H < / 4 !( > 4 <

© 2004 by CRC Press LLC

<

508

Chapter 6

2 3 / > ) 2 /

>

/

# + ,

/

4 !(

/

# > # + ,(

/

# > # + ,(

/

4 !(

>

/>

<

>

/ + >

ALGEBRAIC GRAPH THEORY

>

# + ,<

4 <

/

/

( /

>

>

4 <

<

4 ?

2 3 / > ) / + > + />

/>( / > ) ( />( / > ) ) 2 ·½ /> 4 + H ,+ H ,( $<

/>

/>

4 + H ,+ ,( <

4 +

4 + H ,)<

/>

/>

4

H ,)

/>

4 + ,)

4 +> H ,)<

/

4 +> ,)(

/

<

4 H

( 4 / > < / > /

> <

>

4 +/ H ,)

>

4 + ,)

/

4 / H (

4 +' H ,)+' H ,(

'

/

/

4

'

' H (

/

/ ( >

/ ( >

4

4 +' ,)+' ,(

) +'

4

H (

) <

/

/

H <

4 + H ,)<

4 +' ,)+' ,(

+'

H ,)+' H ,( ' + ,) /

4

>

>

>

>

4

+'

/ ( >

,)+' ,( 4 +' ,)+' ,(

) <

,)+' ,(

4 +' ,)+' ,(

'

<

/

/

4 ( > 4 + H ,)( 4 ( < 4 + H ,)( > 4 ( 4 H ( 4 <

/

4 (

>

4

4 ?<

/

4 !(

>

4 !

4 !

RESEARCH PROBLEM

42 $ ) ' ( B ' Y - C )

6.2.3 Isomorphism 5 - I :

I

© 2004 by CRC Press LLC

Section 6.2

509

Cayley Graphs

DEFINITIONS AND NOTATIONS

2

I I+ < , " I+ < , 4 I+ < ,( & %+ , 4 +,

2 2

" I

I$

3

2

3 4 /½½ /¾ / ) C 7 , B , + ( + , 4 +, , , ,

,

: $

,

# ,+ / , B ,

+ /

( 4 $

3 & +, 4 ,+, 2 B , + 7 ) & $( & +, & +, , 4 B " 1 ( & & 2 3 4 /½ /¾ / ) C $ 4 + , $

+ , J

4 B( 4 $

3 + , )

( & $ $ ( &( 4 ( ( +&, 4 +&,) $ ( +&, 4 +&,( +&,

&

EXAMPLES

2 I +!< , I +!< , - 2 2 N% ) ( )

2

7

4 ( 6 4

4

? ? B 8

? B ?

I +< , I +< , ) E (

4 % ( I$ FACTS

2 6A!!8 3 ) I % I$ ) + , 9 + ,

© 2004 by CRC Press LLC

510

Chapter 6

2 63 B8 $

I$ (

ALGEBRAIC GRAPH THEORY

)

2 6#?!8 I$ 4 ,( :

B (

,

" ? "

2 3 / ) / I$

E (

/ 4

/ H

( I$ :

E I$ I I$ ) S * 6A 8 &

2 6 !8 $ / ( ) & /

;+, /

+/ ,) ; >

2 6 # B8 $

4 / / / ( / 4 ( )

/ / /

;+,

½ ¾

;+,;+ , ;+,

& Ê

- ) ( - ) & $ / 4 ( & 4

& - ) EXAMPLE

2 1 7 4 B ) +(,( +(,( +(,( +(,( +(, +(, ( & ( +(, 5 / 4 ( & 4 ( ) & 4 +, 4 7 & 4 ( +&, 4 & 4 - ) 7 & 4 ( +&, 4 & 4 "( ) ) 7 ( & 4 ( +&, 4 & 4 " ) 1 ) % B RESEARCH PROBLEM

42 7 /(

I$

© 2004 by CRC Press LLC

Section 6.2

511

Cayley Graphs

6.2.4 Subgraphs : ) I 5

& (

$ 9 &

& )

DEFINITIONS

2

½

( (

2

)

(

( 4 ( ! "( ( ! " ! "

%&

)

! "

4

(

=

(

$

- +

2

0 4

)

4(

0

=

FACTS

2

3

$

2

)

(

$

& $

(

&

& (

2

6#!( 1!B8 $

&

2

6 ?8 $

I+

2

2

<

,

&

!

:

6IZ "8 3

2

( I

,(

&

)

) I % ) $

)

7

,

& ( & 5 ( B( & & 5 H

7 (

(

(

=

) $

© 2004 by CRC Press LLC

)

=

61 "8 > I

( =

/ ( /

512

Chapter 6

ALGEBRAIC GRAPH THEORY

6.2.5 Factorization DEFINITIONS

2 2

( ) ½ )

2 =

( =

2

) ) FACTS

2

65"8 > I 2 4 '<

) <

C

2

65"8 I

I C 2 <

) ) & < / / / (

2 $ ) I ) ) & ( C

2

63 ? ( 3 B8 $ 4 I+ , I )

I ( =

2

67 ?B8 $ " ( ) 1

C ) " 7 ( C " )

RESEARCH PROBLEM

42 3 ) ( I ( &

$ ( + ,(

C ) Y

© 2004 by CRC Press LLC

Section 6.2

513

Cayley Graphs

6.2.6 Further Reading REMARKS

2 & ) ) I

) ) 5 I ! ) - 6"!8( 6 !8( 61B8 & 6 K1B8

2 I I ) 7 6OB8 I : - ) - 6[B8 6-G"?8 I - & & 6 ?8 63?8

2 I 63 B8 2 ) & I 6A?8 ) 6$B8

References 6-G"?8 5 - A G (

-( & ! +?"?,( M 6 ?8 A ( I ( +??,( !M?

& !

6 ?8 J ( )(

- 2 ! 3 &&( 3 ( # L 3 3 S C( #$ J = ( ??( !?M!" 6 # B8 A # # ( > I 2 $( )! +BB,( !M? 6A!!8 3 A) ( $ )

( " ? +?!!,( ?M

6A?8 3 A) ( ( ( (

- 2

! 3 &&( 3 ( # L 3 3 SC( #$ J = ( ??( !MB 6A 8 J A 9( S * ( I ( 1 ( J Q -( ?

© 2004 by CRC Press LLC

!(

514

Chapter 6

ALGEBRAIC GRAPH THEORY

6IZ"8 I I I J Z ( E

) ( I ) # P$$$( ( ! ! ( 5 P ( ?"( M 67 ?B8 K 7 -( E ) ( +??B,( BM 6"!8 K 3 1 -( ?"!

#

" # ( 1 ( J Q -(

6$B8 # $ I > ( E I & ( +BB,( M? 63 ??8 I = 3 ( 7 I$ ) ( ?M

63 B8 I = 3 ( E % I \ ( +BB,( BM

+???,(

)!

63 ? 8 K 3 ( =

I ) ( ( +?? ,( !M" 63 B8 K 3 ( =

I ) ( ( 63?8 3) C-( I 2 ( & -( ( +??,( M"? 6#!8 1 #( > > ( +?!,( M S * 9 ( 6#?!8 # #C-( E S +??!,( ?!MB

!

)!

V 6 > B8 A ( K 5 S V( K9( - # 1- ( I ( 6 !8 (

( 5 P ( J Q -( ?!

65"8 5) ( E %& ( +?",( "BBM"B 65"8 5 ( E C) I ( +?",( ?"MB!

6 !8 K (

) ( +? !,( M 61!B8 # 1- ( I ( M? 61B8 1 ( # ! 2 # ! = ( ( BB

© 2004 by CRC Press LLC

2!(

+?!B,(

# 5 ( J

Section 6.2

515

Cayley Graphs

61 "8 N 1 ( I

(

6[B8 K [(

+?",( B!M

" !! 2 & (4 -!

( G (

N ( BB

6OB8 [ O( I )2 ( B

© 2004 by CRC Press LLC

)!

+BB,( !M

516

6.3

Chapter 6

ALGEBRAIC GRAPH THEORY

ENUMERATION

! I I I I I

5 # N I

I 5

Introduction $ - I ) C :

(

- ) C

$( ( : & ) # : ) ?! - S + 6S "!8 > , 7= 6=8 & (

( (

) I 6I!( I"?8( % ./ #

) A =C 6A =( A =)8

) S 6S "!8( E 6E"8( = 6=?8 & ( ) )

( ) 6=!8 ( - % :( : - % 65 ?8

6.3.1 Counting Simple Graphs and Multigraphs DEFINITIONS

2 ) ( ½ ¾ ( ) )

)

2 4

© 2004 by CRC Press LLC

Section 6.3

517

Enumeration

2

. ) .

% ) 2 ' . +, . +, .

2

. .

)

FACTS

2

) ) , )

5 ) Æ ,

2

+ , 7 , ( ¾ ( ) ) , ) ) ,

2

) )

Table 6.3.1 ,

B

5 )

5$*" * - 2 " "

B B B B B B

(B

B

! " ? B

+¾ ,

"

,

!

"

B ( (BB (BB ( ( (BB (BB ( B

B (B (?" B(? ( ("B B(?B ?(?B (! (! ?(?B B(?B ("B

" !" (! B(! ?"("B ! (!B ("(BB (B"(B (?B (?BB ((B (!("B B((! !(( B B( ( BB

(! "

(B?!(

2 6 8 ) )

4

+¾ ,

4

+¾ ,

"((

(

)

5 )

2 S * ( ( +* , 4 * )

© 2004 by CRC Press LLC

518

Chapter 6

Table 6.3.2

ALGEBRAIC GRAPH THEORY

1 " *$*" * - 2

"

!"

! (!B

("

(

" ("(?

) - C ' +* ,( ' 4 7 & ( +* , !( * 4 ( * 4 ( * 4 * 4 * 4 * 4 * 4 B

2

5 + ,

( ( + , 4

]

] ' * ]

+ ,+

¾ ¾·½ ,

¾

= - +* , ( + , + ,

( >& + , 2

4 4 4 H ]

H

]

]

]

4 H ? H " H

4 H B H B H 4 H

H B H B H

H B H B H "B H B

H H B H B

2

) )

+¾ ,

2 +, 4

) ,

2

6=( S "!8 2 +, & ) ) ) & + , ) ) H 5 )

2

) )

& + , ) ) ) 5 )

© 2004 by CRC Press LLC

Section 6.3

519

Enumeration

Table 6.3.3 /* -

, B

! " ? B

2 "

, "

? ?

!

"

B ?! " " ?!

B ?"B ( (! (

(B

(

+¾ ,

2 ( ) % 2 )

, +, 4

)

)

, 2 6=( S "!8 , +, &

) ) ) & + , ) ) % H H H H 5 )

9

½ 9

Table 6.3.4 5

* * -

, B ! " ? B

© 2004 by CRC Press LLC

2 " , "

! " B

! ! ! ? ? ? ?!

" " ! B

" ! ( " (!

520

Chapter 6

ALGEBRAIC GRAPH THEORY

EXAMPLES

2 7 0 0 ) % ) B 0 )

Figure 6.3.1

/* - 2 " "

2 7

7 & 0

Figure 6.3.2

0"" * *

* * - 2 " "

6.3.2 Counting Digraphs and Tournaments DEFINITIONS

2 ) ( ½ ( ) ) )

2 + , ( ( ( & ) )

2 + , ( ( &

2

. ) . 4 % ) . 2 + , ' +. +, . +,

2

)

. . FACTS

2 ) ) , )

Æ 5 )

2

7 , ( ( ) ) , ) ) + , ,

© 2004 by CRC Press LLC

Section 6.3

521

Enumeration

2 " ,

Table 6.3.5 5$*" *

* " -

,

B

B ?B (B (" (B "(! B !!(B (?!B !(? B "(! (B"(!

B

B ? !? ? !? ? B

! " ? B

(B?

2 ) ) )

5

+¾ ,

2 ) )

)

(

2 5 + ,

( ( + , 4

]

] ' * ]

+¾ ,

= - +* , ( + , + ,

(

>& + , 2

4 4 H

]

]

]

H " H H

]

]

4 H H 4 H

4 H B H B H H B H B H

4 H H B H

H ?B H B

H H H ?B H B H B

© 2004 by CRC Press LLC

522

Chapter 6

ALGEBRAIC GRAPH THEORY

2 +, ) )

+, 4

) ,

2

6=( S "!8 +, &

) ) ) & + , ) ) H 5 )

2

) )

& + , ) ) ) 5 )

Table 6.3.6

- 2 " ,

,

B

! " " " !

!? !B! ( (?B ( !B

"

?( B"

! " ? B

2 2

( ) %

´ ½µ

6N8 ) " ) ] " 4 ] ' * ] +* , C ) -(

+* , 4

+ ,* *

*

5 ) !

2

6# "8 3 " +, 4 H H H H H H ) ( 7 +, 4 H H H H H )

" +, +, 4 H " +, 5 ) ! J &

© 2004 by CRC Press LLC

Section 6.3

523

Enumeration

Table 6.3.7

" - 2

5

(""B ?( ?(!(B ?B(!(" (B"(( "

B

! " ? B

(BB" !"( ?((?? ""( (?B (B(?(!

EXAMPLES

2 7 0 ) % ) B 0 )

Figure 6.3.3

*

* " - 2 "

2 7 E

Figure 6.3.4

- 2

6.3.3 Counting Generic Trees DEFINITIONS

2 ) ( ½ ¾ ( ) ) ) )

2 &( ( %

© 2004 by CRC Press LLC

524

Chapter 6

ALGEBRAIC GRAPH THEORY

2 + , FACTS

2 1%*% * 6I"?82 )

5 ) "

2

) ) 5 ) "

Table 6.3.8

5$*" "

" *$*" - 2

3)

3)

? !(!! !( ? (B?!( (B (! (BBB(BBB(BBB (?!(( B !(BB"(!B( "" (?"(B"((" !?(!(!!(( ?(?(? (B(?B( ((?(B( B (" (?"B

! " ? B

2

)

(? ("B! ( (!"(? ? BB(BBB(BBB (!(?!( ? (?!( ( (!?( B(?(B! ( ?(?(!(? (? (?(B "(?(! !(B!(?(B!(?!(?

) &

+, 4

½ &

4 H H H H ? H B H

2 6I!8 Æ & +, )

+, 4

½

+ ,

½

% & S + 6S "!8,

+, 4 & 5 ) ?

© 2004 by CRC Press LLC

+ ,

Section 6.3

525

Enumeration

2 ) "

$+, 4

½

" 4 H H H H H H

2 6 * 6E"82 Æ " $+, )

+, 7 ! )

$+, 4 +, +, +,

5 ) ?

2 I : &

+, 4

½

7 4 H H H H H H B H

2 Æ 7 & )

+, 4

½

+ , H

% &

+ , 4

H

&

½

+ ,

2 ) !

<+, 4

½

! 4 H H H H H H H

2 6=?8 Æ ! <+, )

& +, 7 )

<+, 4 + H , +,

H +, H

+ ,

5 ) ? ) J & EXAMPLES

2 7 B 0

) % 0 ) 0 )

Figure 6.3.5

© 2004 by CRC Press LLC

- 2

526

Chapter 6

Table 6.3.9

ALGEBRAIC GRAPH THEORY

" 7 7 " "" - 2

? B " " !? (" (! (" (?! "!(" (" ("! (!(? ( ""( ! (" (" ((" ?!(B(" "("(" !(!(?" (B !(!( (!?( (B (B"(!(? (BB!(B ( ? (" ((B" ( ("!(?! ??!(!((??" ("B?(?((!BB !(??("?(!"( (B?(( !( " ((!B("(B !?( (?B(B( B?(""(B?("B( B (!(B("("(B? (((?(?(B (!B(!"B(B!?( (

B B !" ? " ( (?"" ( ! B(?" (B? (! "(" B(! (!? ?(? ( (B!B (( ? (B?(" B(!( " B((!? ( B( "(B("" ! !?(? B("(! !(?!(B! ( (B!( (BB( "("

! " ? B ! " ? B ! " ? B ! " ? B

! B (B (? !(! ?(B "( ? (" ! !(? "(B ((B ( (! (""(B! ?(??("?! B( ("?B !?(!?(B !(B ( B (B((B ( ?( (" ("B("!("B B(B("?(BB B?(?!(B( BB( "(" ("B "(!!?( (! ( ( ((! ( (B (B!(!" !( ?( !!(?B(! !( ((( (?B((!!?( (??B(!(!"(

2

0 % 7 ( 0 ( 0 ?

2

7

© 2004 by CRC Press LLC

Section 6.3

527

Enumeration

6.3.4 Counting Trees in Chemistry DEFINITIONS

2 2

#$ & # #$ &

REMARKS

2

) + -,

I = ) H

2 ) )

I = E= ) ( H ( E= FACTS

2

) 0 H + ,

+, 4

½

0 4 H H H H

H " H ! H

2

6A =8 Æ 0 +, )

+, 4 H +, H +, +, H +, 5 ) B

2

I + , : % ( & ) ½

2+, 4

4 H H H H ? H " H

2

Æ 2+, )

+, 7 ) 2+, 4 +,

H

+, + , H " +, +, H +, H + ,

2

) 4 + , H

+, 4 © 2004 by CRC Press LLC

½

4

4 H H H H H H H

528

Chapter 6

ALGEBRAIC GRAPH THEORY

2

6A =)8 Æ 4 +, )

2+, +,

+, 4 2+ , H +,

½¾

+,¾

+¾ ,

5 ) B ) EXAMPLES

2

7 0

Figure 6.3.6

# - 2 "

2

% 7 ) & 0 ( 0 ( " 0

6.3.5 Counting Trees in Computer Science DEFINITIONS

2

% & :

B ) & 9 ) ) $½ $¾ $ ,

2 & ) ) > ) ) % ) )

2 & ) )

2

) & B

REMARKS

2 2

$ (

E

( ) ) ( )

© 2004 by CRC Press LLC

Section 6.3

529

Enumeration

Table 6.3.10 #

" # " # -

! " ? B ! " ? B ! " ? B ! " ? B

2 "

+ ,

+ -,

" ! ? "? B! (" (B! !( ? ?( "(" (?B (?" "B(? ( (BB ( (B? (!(" "( ?( B("(?! ?(BB(" !( ( ! ("?(??( (B(!B("" ((!("B (" (B(" ? ?(??( ("" !((" (B ?B(?"((B ("!("((? (BB(B( B!( (" (B(B!( ? (B("(B(!" ?"("B(!!?(!B(" ("((BBB(!! !B("B!(?! ("(! (?(BB(BB( ("?

? " ! ? "B ("" (! B(? ("? B( "(" (? ?B(! (!"( " (!("B (?B( (!?!("" ?("?( B(("B !(B( (?B(B!( ((" (! B( B(B!(!? !(!((! ? !((B""( B ""( ( (? ?(!"(?(?B (?(?!(""(" (B(?B(!"B( "(? (!!(!(? ( !(!"(?(? ? ("("B(!(

2 A E )

© 2004 by CRC Press LLC

530

Chapter 6

ALGEBRAIC GRAPH THEORY

2 3 : & ( ) ) H ( ( ( (

FACTS

2 % )

4

4

4

H H

H

+,] 4 H + H ,] ]

B

5 )

1* 8$

Table 6.3.11

I J )

I J )

? (B (" (!? "(!" B"(B !(?BB ( !(B ?( ?(" (!( !B

! " ? B ! " ? B

?( (!?B !!( "(!BB (! !( (?B ( (B(B ( ( !(BB ?("( ( B (B?( ( B ("?(?B(!( (" (? (B( "( !((B!( ?((B(? (BB (!!(?(!B( B (BB(( ( ( " ("(?" (B(B?(B (( (B?( (?B? ((B ("!!(B"(?"

! " ? B

2 ) I ) 5 )

2 ) ) 5 ) 2 ) H 5 ) EXAMPLES

2 7 ! ( 7 " ) ( 7 ? !

© 2004 by CRC Press LLC

Section 6.3

531

Enumeration

Figure 6.3.7

Figure 6.3.8

Figure 6.3.9

"" - 2

$% - 2

*# - 2

References 6A =8 I # A = =C( ) ( +?,( BMB 6A =)8 I # A = =C( ) )

( +?,( B!!MB" 6I!8 I ( E ( +"!,( ?MB 6I"?8 I ( (

6 +""?,( !

6N8 3 N ( 5

(

MB

! "

M!"

!

+?,(

6=8 7 =( ) ( ( ( ( ! !" +?,( M 6= ?8 7 =(

# ( 1 ( ?

6=!8 7 = > # ( #

© 2004 by CRC Press LLC

?

(

( ?!

532

Chapter 6

ALGEBRAIC GRAPH THEORY

6=?8 7 = ( ) ) ( ( B +??,( M 6 8 > J )( > ) ( 6# "8 K 1 # (

! !( = ( 1 ( ?

6E"8 E( ) (

"

2 ? +?",( "M??

6S "!8 S I ( !( 5 P ( ?"! 6 "8 K (

" +? ,( BM

2 # !* # !*

& !!( 1 ( ?"

65 ?8 J K 5 5 0(

( ??

© 2004 by CRC Press LLC

2 &" 1 !(

Section 6.4

6.4

533

Graphs and Vector Spaces

GRAPHS AND VECTOR SPACES "!# ! $ A I N% I 5) ' I 5) ' ) I I 5) I I 5 N I RIA ! C I I 5

Introduction > )

) ) 2 G 0* ( G 0* E * > ) ( ( ) ) G 0* : ) C ( G 0* : ) C ( % ) ) ( $ ( ( 9

5

) % ) ) ) & ) T # )

C

6.4.1 Basic Concepts and Definitions

( ( )

) I 7 - ( ) ) % 7 ( 6Q??8 65?8

' %(

(

4 ½ ¾ (

© 2004 by CRC Press LLC

,

(

4 + , + , 4 ½ ¾

534

Chapter 6

ALGEBRAIC GRAPH THEORY

$ + , ( ) ( ) + , DEFINITION

2 &

) ) ^ REMARK

2

$ + ( ,

EXAMPLE

2

>& ? 7

v1 e1

e3

v2

v3 e2

e4

e5

e7 e6

v5

e8

v4

Figure 6.4.1 Subgraphs and Complements DEFINITIONS

2

4 + , 4 + ,

2

> ) % : ) 4 + , 4 + ,( ) % J )

2

> ) % : ) 4 + , 4 + ,( ) & J & )

2

) 4 + , 4 + ,( ) , + ,

© 2004 by CRC Press LLC

4 +

Section 6.4

535

Graphs and Vector Spaces

EXAMPLES

2 7

4 ½ ( )

7 7 +, 7 4 ( & ) 7 +),

v1 e1

e3

v2

v3

v5

v4

e8

(a) An edge- induced subgraph of the graph G

v1 e1 v2

v4 (b) A vertex-induced subgraph of the graph G

Figure 6.4.2 0 "#"" $ " 23#"" $

2 )

7 +),

7 +,

v1

v1 e3

v2

e1 v3

v2

v3

e2 e4

e5

e7 e6

v5

v4

v5

(a) Subgraph G'

Figure 6.4.3 0 $

© 2004 by CRC Press LLC

e8

v4

(b) Complement of G' in the graph G

" *

7

536

Chapter 6

ALGEBRAIC GRAPH THEORY

Components, Spanning Trees, and Cospanning Trees DEFINITIONS

2

-

( N% B

2 &

5 $

(

% N% !

2 2

)

& ) & )

2 ) $

"

2 / (

/

2 "

"

2 3 ) & , / 8+ , + , ) 8+ , 4 / + , 4 , H /

EXAMPLE

2

" 7 v1 v1

7

e1

e3

v2

v3

v2

v3

e2 e4

v5

v4 (a) A spanning tree T of G

Figure 6.4.4

e5

e7

v5

e6

e8

v4

(b) The cospanning tree with respect to T

0 " "

FACTS

2

& )

© 2004 by CRC Press LLC

Section 6.4

537

Graphs and Vector Spaces

2 & ) , H / / ) , H / REMARK

2

' (

Cuts and Cutsets DEFINITIONS

2 I 4 + , 3 ½ ) 9 ) 4 + ( , & ! " & (

2 $ (

( >: (

EXAMPLE

2

7 7 ( ! "( 4 4 ( ( 7 +, 5 ( ! " (

7 +),

v1 v3

e1 e2

v2 e4 e7

v5

v4

e8 (a) A cut of the graph G

v1 v2 v3

e5 e4 e6

v5 v4

e7 (b) A cutset of the graph G

Figure 6.4.5

© 2004 by CRC Press LLC

0 "

538

Chapter 6

ALGEBRAIC GRAPH THEORY

The Vector Space of a Graph under Ring Sum of Its Edge Subsets DEFINITIONS

2

5 4 ½ ) ) )

7 & ( ) + B B B B, ) 7

2

+ ' , ( ( ) ) )

2 , 4 + , 4 + , 4 +? ? ? ? ? , ( ? 4 ) ) ( + + ( ) B 4 < B ) 4 < B ) B 4 B< ) 4 B, FACT

2 , ) , ) , ) + , , 7+,( % (

) + ) , ) _+ , REMARKS

2 2

)

$ ) ) , + , ) ) , ) E) + ^, B _+ ,

2 2

$ ( 6 8

) & 65 8( 6I!)8( 6N!8( 65?8( 65"8

6.4.2 The Circuit Subspace in an Undirected Graph DEFINITIONS

2 & I (

2

) ` + , $ ( ` + ,

9 ^,

© 2004 by CRC Press LLC

+

Section 6.4

539

Graphs and Vector Spaces

FACTS

2 2

)

9 ( ) ) )

2

) ( ` + ,

2 ` + , ) _+ ,

EXAMPLE

2 7 ( ( 7 v1 e1

v1 e3

v2

e1

e3

v3 v 2

v3

v2

e2

v3

e2 e4

e7

e7

e4

e6

e6 v5

v4 v 5

e8 (a) Circ G1

Figure 6.4.6

e8

v4

(b) Circ G2

v5 (c) G1

G2

- "

REMARKS

2 2

7 ) P) 6P8

( + Ü,

( ( &

Fundamental Circuits and the Dimension of the Circuit Subspace DEFINITION

2 " : (

$

" (

FACTS

2

" ( , H ( "

© 2004 by CRC Press LLC

540

Chapter 6

ALGEBRAIC GRAPH THEORY

2 " ( (

"

2

+, H , ` + , )

2 $ ( ) &

2

) ) ` + ,( ( ` + ,

: , H ( + ,

2

) ` + , : + , 4 , H /

/

EXAMPLE

2

7

"

4

½

I I I I

4

4

4

4

$ ) % ( ( ( ( ( 7

6.4.3 The Cutset Subspace in an Undirected Graph % 5 ) ) & DEFINITION

2 9

) 6+ , ^ ) 6+ , FACTS

2

> 9 ( 6+ ,

2 2

6+ , ) _+ ,

< ( 6+ ,

© 2004 by CRC Press LLC

Section 6.4

541

Graphs and Vector Spaces

EXAMPLE

2

I 7 ½ 4 !½ " 4 ! " 7 ( 4 ( 4 ( 4 4 4 ( 4 ( 4 # ( ) 4 !0 4 "( 0 4 4 ( 4

4 4 (

4 4 (

4 4

$ ( ) 7 Fundamental Cutsets and the Dimension of the Cutset Subspace DEFINITIONS

2

3 " ) ( ) ) " $ & " ( ! "

) "

$ ) )

2 &

"

(

FACTS

2

( ) "

"

(

2

) " )( ( ) ) "

2

& ) 6+ ,

2 $ ) ( ) & 5

2

) ) 6+ , ( 6+ ,

: ( - 8+ ,

2 ) 6+ , : 8+ , 4 /

/

2 & ) ) 6+ ,

© 2004 by CRC Press LLC

542

Chapter 6

ALGEBRAIC GRAPH THEORY

EXAMPLE

2 7 7 ( " 4 ½ A A A

4 (

4 (

4

,(

A 4 $ ) % 4 ) ( ( ( ( ( ( 7 B

6.4.4 Relationship between Circuit and Cutset Subspaces A ) ( 7 7 5 - ) & & Orthogonality of Circuit and Cutset Subspaces DEFINITIONS

2 ) , < ) < ,

,

2

)

+ , C J C ) ) FACTS

2

)

=( )

2 7+, C

2 ) ) ) )

) ) >: (

2

) ) ) )

) ) >: (

© 2004 by CRC Press LLC

Section 6.4

2

543

Graphs and Vector Spaces

)

Circ/Cut-Based Decomposition of Graphs and Subgraphs DEFINITION

2

)

) & J C

FACTS

2

$ ) (

2

6I!8 )

)

2

$ ) ( ) + , ) &

2

6I!)( 1 #!8 > ) $ ) : '( EXAMPLES

2

I 7 ! $ ) % ) ) 5 )

) _+ , E ) ½ ( ( ( 2

4 + B B B

B,

4 +B B B

B,

4 +B B B

B ,

4 +B B B

B B ,

4 + B

B B,

4 + B B

B B,

4 +B B

B B ,

$ ) ) & ( 7

© 2004 by CRC Press LLC

544

Chapter 6

ALGEBRAIC GRAPH THEORY

7 ( +B B B ,( ) ) ( ) & 2 +B B 4

+

B ,

B B B

4

B B, + B ,

+ B B B B B ,

( + B ,

e4

e7 e3

e2

e5

e6

e1 Figure 6.4.7 9

**

2 I

7 " $ ( ( ( = ) ) ) & = ( 7 ( ) % 2

+ ,

4

+ B B

B, +B B

+ B B B , ( ( ( (

B

( +B B B ,

v1 e1 e6 v2 Figure 6.4.8 9

© 2004 by CRC Press LLC

e4

e2 v4 e5 e3

,

v3

**

Section 6.4

545

Graphs and Vector Spaces

6.4.5 The Circuit and Cutset Spaces in a Directed Graph $ ( A(

&( 0

:

$ (

)

&

( (

)

( ( ( (

Circuit and Cut Vectors and Matrices DEFINITIONS

2

) ( -

- + ,

2

3 ) 4 + ,

(

% )

2

+ , ) (

2

( 4 + ,

% )

2

3 ) 4

½ ¾ (

)

,

+½ ¾ ,(

4

2

B

3 ) 4

½ ¾ (

)

,

+½ ¾ ,(

4

2

B

3 ) 4

½ ¾

3 ½ ¾

½ ¾ ) ( (

© 2004 by CRC Press LLC

$

,

&

546

Chapter 6

ALGEBRAIC GRAPH THEORY

,

&

The Fundamental Circuit, Fundamental Cutset, and Incidence Matrices

J&( % & REMARK

2

% % )( ( ( & + ( ,( & ( & ) +½ ,(

4 B

+ ,

+ ,

DEFINITIONS

2

) " ( ) ( ( +, H , ) & & 5

( & " ( ) 1 ( + , ) & & 3

"

2 ( 0 ( ) & & ) & & ) 0

2 & )

: ) & & : ( B EXAMPLES

2 I 7 ?+, ) 7 ?+), +,( + B B B , + B B B B ,(

2 I " 7 ?+, ( ( ( " &

& 2 7 I # &2 I I I

© 2004 by CRC Press LLC

B

B

B

B B B B B B B B

Section 6.4

547

Graphs and Vector Spaces

v5 e4

e7 v1 e3

v3

v2

e1

v1

v4 e2

e2

e2

v3

e6

e5

v4

v2 e5

e5

v5

e6 v3 v1

e3

v4

v2

e1 (a) A directed graph.

(b) A circuit with orientation

(c) A cut with orientation

Figure 6.4.9 0 "" 7 7 " - 7

I # &2 A A A A

$ # &2 J J J J J

½

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B B

B

Orthogonality and the Matrix Tree Theorem

)

)

FACTS

2

)

'

$

( )

'(

( C

( + ,

3

)

) +,

,

)

% ) ` + +),

% )

6+ ,

© 2004 by CRC Press LLC

)

548

Chapter 6

2

$

2

ALGEBRAIC GRAPH THEORY

% $ ( ) &

) Æ

)

) &

5

(

)

Æ )

2

` + , 6+ , $ (

(

2

` + , 6+ ,

%

) ) ( : ,

H /(

( :

/(

- ( / )

2

)

2

2

I "

½

)

)

4

& 4

& C ,

5

4

H 4

C

1

1

4 6@

) & 1

2

1

@

4 64

( 8( @

) & 4

) & 1

8(

" 5

(

@

&

" # ( 1

4

4

&

) 5

( &

2

+

( ! !

& 0 0

, 7 (

: )

EXAMPLE

2

>&

7

REMARKS

2

A

)

% (

2 +,

- H ,(

+ N% ,

© 2004 by CRC Press LLC

& +

% -

,

Section 6.4

Graphs and Vector Spaces

549

2

& 0 0 ( $ ) % + , & : & 0 + * , : ) & & + , 7 ) 65?8 &

65"8 Minty’s Painting Theorem

+ , ) DEFINITIONS

2

2

2

&( ( 4 1 C ( ) ( ( ) ( & ) - FACTS

2

+ ! , 6# 8 3 ) 7 ( & 2 & - ) ) ( - & - ) ( -

2

> ( ) ) )

REMARK

2

# * + - , )

+ 65?8, 7 7 E - ) # ) % 6# B( # 8 5

) ) 6PI"B( I! ( 1 !B8

6.4.6 Two Circ/Cut-Based Tripartitions of a Graph $ Ü A ) 9

© 2004 by CRC Press LLC

550

Chapter 6

ALGEBRAIC GRAPH THEORY

1 ) ) Bicycle-Based Tripartition DEFINITION

2 ) ) ( ) EXAMPLE

2

½ ( ( ( 7 " )

FACT

2

6 !"8

2

)

)

)

% ) 7

REMARK

2

6 !"8 7 ) 6?8

A Tripartition Based on Maximally Distant Spanning Trees DEFINITIONS

2 ( +" ",( ) % +" ", 4 +" , +" , 4 +" , +" ,

"

2 " " +" ", " "

+

"

" "

,

& ) )

2 & " " 5

" " ) 2

3 A ) " % ) 3 A ) " % ) A

© 2004 by CRC Press LLC

Section 6.4

551

Graphs and Vector Spaces

) ( ) : A½ ( A A 4 A ) A + & -

2

) N%

)

2

+

& " " , ')

) ')

) FACTS

2

6G G ?8 3 " " & " " % )

" "

) " " % )

) " "

2

6G G ?8 I 4 + , 3 ) ( ( 4 + , + , ( ( + , + , &

EXAMPLE

2 $ ) % " 4 " 4

& 7 B 2 4 ( 4 ( 4

e1

v2

v3

e5 v1

e8 e4

e2

e6 e7

v4

e3

v5

Figure 6.4.10 REMARKS

2 $ ) + - , : , H ) ) ) )

, H

) : ) :

© 2004 by CRC Press LLC

552

Chapter 6

ALGEBRAIC GRAPH THEORY

) ) ) C : ) ) , H ( - E- ( $ C- ( 1) 6E$1!B8 ) ( & ) ( (

) ) : ) ) )

- ) G G9 6G G ?8 ) E- ( $ C- ( 1) 6E$1!B8 ) - 5 65"8 & )

2 3 63 ! 8 A 1 ) 6A1!8 &

6.4.7 Realization of Circuit and Cutset Spaces $ ) ( & -

&

& ) ) 6?8 I) 6I"8 6 "8 ) ) 1 Æ C) & & 5 65 8

) ( & Æ * C) DEFINITIONS

2

7

& - +

v5

v6

v4

v1 v2

v3 (a)

Figure 6.4.11

2

v5

v1

v4 v2

v3 (b)

- & - +

& 7 7 4 67 @ 8( @ &( )

© 2004 by CRC Press LLC

Section 6.4

553

Graphs and Vector Spaces

7

2 &

) Æ ( ( B 2

9

(

( B(

& ) + Æ ( ( B, C + , ) C

2

)

& ) * * -

FACTS

2

7 (

&

2 2

0

"

&

7

& ( ( B ( * ) * - ) + - & ,

7

2 & & (

B 4 B B

B

7

B

2

7

2

7

& C) & & G - 7

7

& C) &

& G - 7

7

REMARKS

2

# 6#!B8 * C) ( * ( !

2 > - % 6 B( 8 A 6A!B8 % & ) & & ) &

& C ) A * 65"8

2

& C )

- $ & I)

6I"( I?8 C ) $ ( A 6A!B8 A Q 6A Q 8 C

© 2004 by CRC Press LLC

554

Chapter 6

ALGEBRAIC GRAPH THEORY

& & N A * ) ) 65"8 Whitney and Kuratowski

1 ) - ) 1 618 G - 6GB8 1 ) ) ( 1 ) = % ) + 65?8, - ) N% : DEFINITION

2 ¾ ½ ) ¾ ¾ ½ ½ FACTS

2

$ ) ¾

½( ½

2 2

¾

618

$ -( G - 6GB8 ) G - REMARK

2

5 61B8 G - * $ : G - * C) & C )

References 6A!B8 P P A ( 5 2 (4 - &! !( N ( N > > ( $ $ ( #( $ ( ?!B 6A Q 8 7 A N I Q ( 5 -( ! +? ,( MB

&

6A1!8 K A 3 1 )(

( " &! ! +!,( !M 6I"8 $ I) ( # )

( B( ! +",( M 6I?8 $ I) ( & ) - ( +??,( !M!

© 2004 by CRC Press LLC

&% !

Section 6.4

555

Graphs and Vector Spaces

6I!8 1 G I( E ( B +?!,( M? 6I!)8 1 G I(

# ( J =

&

( ( ?!

6I! 8 3 E I N # (

-( & ! ! !! +?! ,( ?M 6N!8 J N ( # = ( ?!

4 ! " " (

6> 8 3 > ( 5 ) 5 ( " +! ,( "MB 6 "8 3 ( ( 6Q??8 K 3 K Q ( ???

! " +?",( ?M

# &! !( II (

6G G ?8 G Q G9 ( #& ( & ! +? ?,( MB 6GB8 I G - ( 5 ) I ) ( 7 +?B,( !M" 63 ! 8 # 3 ( ( & ! !! +?! ,( M" 6#!B8 1 #( * C) ( ! +?!B,( B M 6# B8 K # ( # -(

&

& !

% * !+?

B,( ?M

6# 8 K # ( 5 - a * ( &% ! " +? ,( ??MB 6# 8 K # ( E & ( - -

( +? ,( "MB 6E$1!B8 E- ( Q $ C- ( = 1)(

& -( & ! ! +?!B,( ?M?? 6?8 G ( ! ( J N ( $ ( ??

# ( #=

6 8 # A ( 2 ) ( I" +? ,( !M

) I

& !

6 !"8 I ( E ( ! 2 )! ! +?!",( ?M

© 2004 by CRC Press LLC

556

Chapter 6

65 8 5 5 # A ( 1 ( ?

ALGEBRAIC GRAPH THEORY

# ! (4 -!(

65"8 # J 5 5 G ( 1 +$ ,( ?"

# !* (4 -! " !(

65?8 G # J 5 5 ( 1 +$ ,( ?? 6?8 1 ( # ( !M

# !8 " !(

! 2

?B +??,(

6 B8 1 (

) ( +? B,( ?BM?! 6 8 1 ( 7 (

+? ,( B"M!

6PI"B8 K P 3 E I( ) ( & ! ! !! ! +?"B,( " M" 6P8 E P) (

!! !( # 5 ( ?

61B8 N A 1(

& # ( =

618 = 1 ( (

( BB

7 +?,( !M"

61 #!8 1 1 3 # #& (

) ( & B +?!,( "M"? 61 !B8 N = 1 ( -( & ! ! !! ! +?!B,( M!

© 2004 by CRC Press LLC

Section 6.5

6.5

557

Spectral Graph Theory

SPECTRAL GRAPH THEORY

%& & A # & 1 - 5

3 ( 5 ( > A N 5 I C 3

Introduction 5 + , + ) ,

A

& ) B 9 & )9 ) 3 I C ' 5 C 6I 5 !8 ?! 5 )9 ( )

6.5.1 Basic Matrix Properties # & - & ) 6# # 8 6 B8 DEFINITIONS

2 ) + , : & 0 + 0 , 0 4

! B

9

2

9 &

2 2

+C

0,

6 )

2 7 (

© 2004 by CRC Press LLC

558

Chapter 6

ALGEBRAIC GRAPH THEORY

2 6 0 4 6 FACTS

2 ) 9 &

( . ( ( : & 9 + =

& ,

2 2 2

) :

9 & 0 ) C( ( :

& @ + , @ 0@ 4 @ 0@ ½ &

2 9 & ( ( ( B( 9 & B

2 ( 9 ) - 9 &

2 $ ( 6 6 +5 9 & ) ( - 7 ) M 6# # 8,

2 $ ( $ + : 7 ) , 5 9 &

( )

: ) )

2 3 ) )

2

1 ) )

) +

7 ?,

2 6

)

B( (

6

REMARK

2

A 7 (

) ) (

EXAMPLES

& + (

- ,

2 2

+ ,½ ½ * ½ ) 2 , B

© 2004 by CRC Press LLC

2

*,

Section 6.5

2 2

559

Spectral Graph Theory

2

+'#) H ,½ ' 4

2

-

2 +'#),½ ' 4 J (

(

2 I+, 5 2 + ,½ B ½

2 2

)

2

1

+ ',+, ' 4 B

+ H , 9 * & 5 2 +'#),½ ' 4 H

2

2

+ + 4 I+,,2 B * * 2 B * *

2 )

1

2

6.5.2 Walks and the Spectrum Walks and the Coefficients of the Characteristic Polynomial DEFINITIONS

2 )

2

& 9 %

FACTS

2

2

$ '

$ 0 9 & ( ' &

2 2 2 2

) -

9 & ( & 0

0

) -

0

) -

0

0

0

'

)

& )

65 8 $ H H H H

( 4 4 B ( ) ) + &

+C 0,,

© 2004 by CRC Press LLC

560

Chapter 6

2

ALGEBRAIC GRAPH THEORY

65 8 $ H ½ ½ H H ½ H

( 4 +, ( ) % ' ( +4 , ) 4 ( +4 , ) 4 REMARK

2

7 " Æ! $

6IN 5?8 $ & 7 ! Æ

Æ )

) & ' %& + , 7 ( + , + , (

Æ ) % ' 7 ( ) %( )

+ - ( - , )

Walks and the Minimal Polynomial DEFINITIONS

2

>+, ( >+0 , 4 B

2 * + ) + , H FACTS

2

,+, 4 -

+ 6 ,

2 $ 9 & 0( 0 4 B ' + $( 0 4 B

2

$ , ( , H ) ) $ 7 B

2 2

+ , &

2

) +

7 ( ) \ - \ : ) , EXAMPLES

2

J & ( ( ( I+,( - ( ( 1 )

2

& )

© 2004 by CRC Press LLC

Section 6.5

561

Spectral Graph Theory

OPEN PROBLEM

4

I C ) : E (

) Regular Graphs DEFINITION

2 ' (

+ 6 , <+, 4 + 6 ,

) - :

3 D : & : FACTS

2 + , + 7 ) ,

2

B + ) 0 ,

2

) +> ) ,

2 2

D

2

0

D

9 & : 0 C = 6 ( 6 0 :

6= 8 7 9 &

<+, <+0, 4 D

0

= 0

6.5.3 Line Graphs, Root Systems, and Eigenvalue Bounds ) ) ) E ) ) & " E

6=?8 DEFINITIONS

2

: & ) )

2 ( A+ ,( A+ , 9 ( (

© 2004 by CRC Press LLC

562

Chapter 6

2

ALGEBRAIC GRAPH THEORY

&

( B $ ) + , ( ( )

2

½ (

A+ < ½ ,

A+ , -

I+ ½ , I+ ,

& A+ , 9

2 %( - 9

$ (

( 9

I+ , I+ ,

NOTATIONS

6+ ,

b+ ,

FACTS ABOUT INTERLACING

2

3 0 )

6

2

& 6½

) &

6

6

$ ! ) (

! ( 6

6

6

6

6

6 (

6

(

6

4 ,

FACTS ABOUT THE SMALLEST EIGENVALUE 6+ ,

2

$ ! ) ( 6+! ,

2

$ : C

6+ ,

2

J

) B

2

*

2

2

%

)

2

+

*

:

*

)

>& ,

Line Graphs and Generalized Line Graphs FACTS ABOUT THE LINE GRAPH A+ ,

2

$ & & (

2

$ & & ( &

% (

© 2004 by CRC Press LLC

4 C H0+A+ ,,

Section 6.5

563

Spectral Graph Theory

2 6= !8 7 ( 6+A+ ,, 2 6N !B8 % 6+A+ ,, & ( )

(

*

2 % 6+ , + FACTS ABOUT THE GENERALIZED LINE GRAPH 6+A+ < ½ ,

C

7

& ) & & 7 ( C 7 B ' )

2 6IN5"8 7 6+A+ < ½ , 2 6IN5"8

% ) 6+A+ < ½ , ( 4 A+" < B B, " 4 A+! , ! ! & ( )

2 6= !8 $

4 6½ ( 6 6 6 A+ , 6 H ( 4 + , + ,)

EXAMPLES

2 7 +>& ,

+

A

,

2 ) 7 +>& ,( ¾ A+ , Root Systems

7 &

%

3 ) + 6I"?8( & , ) FACTS ABOUT ROOT SYSTEMS

) 0 ( ( ( ( + H ,)( + ,( ( ( B ( 0 E )

- 0 ( ( ( ( 0 4 C H 2 6I55! 8 ! 4 A+ , ) )

2 6I55! 8 6+! , &

0

2 6I55! 8 ! 4 A+ < , )

© 2004 by CRC Press LLC

564

Chapter 6

ALGEBRAIC GRAPH THEORY

2 6I55! 8 1 % ) & ( 6+! , !

C & + % ) , ( ( ) & ) 7 " ' : +) , - ) &

2 $ 6+! , ( ( ! ) ) ( ! B & ( B (

C 6+ , ( ) 61 J?8

*

) 0 C

EXAMPLE

2 3 ) ) Ê I 2 4 H H $ (

' % ( ) ( & : 0 4 C H

7

Figure 6.5.1

Eigenvalue Bounds FACTS ABOUT THE LARGEST EIGENVALUE b+ ,

# ) ) ) )

2 $ ! ) ( b+! , b+ , $ ! ) ( b+! , + b+ , 2 $ c & ( c b+ , >: 2 6IN "8 $ b+ , + ( - ( " + ,( " + ,( " + ,( " + , " + * ', ) - ( ( E E ! ( ( E $ & ( (

Figure 6.5.2

© 2004 by CRC Press LLC

" + * ',

Section 6.5

2

565

Spectral Graph Theory

6IN "8 $ b+ , 4 ( ½

( " + ,( " + ,( " + ,

* ,

5

) b+ , + + H

+ 6IN "8,

FACTS ABOUT THE SECOND LARGEST EIGENVALUE A ( 6 + ,( ) )

2 2

+

65 !B8 6 + ,

B

63 ?8 6 + ,

*

,

6 + ,

*

6 + , +

*

+ H

* ,

,)

)

) 6I5 ?8( 63 ?8(

65 ?8

FURTHER READING 0 ) )

6B8

6.5.4 Distance-Regular Graphs >

DEFINITIONS

2

/ B

* '

+ * ', ) '( ) % * /

2

> ) / $

! + ,(

+ ,

9 ( (

)

) ) ( ( 0

2

$

,

D + ,(

)

9 ( )(

) ' ( )(

'

EXAMPLES

2

! + , ) 1

$

2

D + , A+ , $

© 2004 by CRC Press LLC

566

Chapter 6

ALGEBRAIC GRAPH THEORY

Distance-Regular Graphs and the Hoffman Polynomial

+ / , ( = 0

DEFINITIONS

2

F

) % 0 4 C "

0 4 0 9 &( 0

&(

& + *,

F B

2

F

/

& -

+ *,

FACTS ABOUT THE MATRIX 0"

2

( 2

0

7 ( 0 0

0

4 D (

/ 0

&

) )

4

0

0 0

H + 0

,

0

) ' 4 B

2

) 9 & 0

H

2

= 0

2

)

FACTS ABOUT THE PARAMETERS /

2

- -

4

/ - (

) )

)

2

0 0

- -

0 - ( 0 -

2

-

2

(

Strongly Regular Graphs DEFINITION

2

+ 6 , ) ( ( 6 4 / 4

/

(

EXAMPLES

2

2 A+ , ++

2

3 2 A+

© 2004 by CRC Press LLC

,) ,

, +

,

Section 6.5

567

Spectral Graph Theory

2

2 - +/ , % % 7+/ ,

9 ( % ( 0 : +

)

/

+/ +/

,) +/

,) +/

) , ,),

FACTS

- &

) & 0

&(

) ( 9 &(

2 +

6

7 + 6 ,( +

, 4

6,

¾

2

5 0( 0 ( C (

+ : ( 9( 9 , * C = 0

$

( + 6 ,(

¾

0

2

H +

6,0 H +

,C 4 D

+ 6 ,

2

¾

H +

6, H +

,

$ ( ( ( 0 B

2

7 + 6 ,(

6½ 4 ( 6¾ 4 +6

,) H F

½¾

6¿ 4 +6

,)

F

½¾(

¾

¾

F 4

6 H 6

6 ( ,¾ ( ,¿ ,¾ H ,¿ 4

,¾ 6¾ H ,¿ 6¿ 4

2

$ F 7 ! :( ,¾ 4 ,¿ 5

2

E

5 6¾ 4 6¿ ( ,¾ ,¿ :

+ 6 ,

)

,¾ ,¿

EXAMPLE

2

¾

7 (

½¾ + /

¾,(

H +

,¾ 4 ,¿ 4 +/

6, H + , 4 ¾ H +/ ,)

H ,)(

6¾ 6¿ 4

FURTHER READING 7 ) &

6 B8

6AI J"?8

© 2004 by CRC Press LLC

568

Chapter 6

ALGEBRAIC GRAPH THEORY

6.5.5 Spectral Characterization E : - 2 C ) Y 7 )

EXAMPLES

2 7 (

½ B & ) : + - ) , )

5

( : )

)

Figure 6.5.3

** *

2 7

) + ,+ H ,+ H ,

Figure 6.5.4

** " *

2 7 ) )

+ , B

+ H ,+ ,

Figure 6.5.5 1 * - $ 2

© 2004 by CRC Press LLC

Section 6.5

569

Spectral Graph Theory

2

7 ½ 5 ( ( % $ ( 9 A+ , + , A ( + - " , ) a

b

c

d

a

e

e

f

f

g

g

a

b

c

d

a

0 *% * *

Figure 6.5.6

Eigenvalues and Graph Operations

E ) DEFINITIONS

2 ( ) ( & ) + , & & 9 ( ( 9

2

+ 9 , ( ( ) ( ( ) & 9

FACTS

2

&

2 6

) ) H 6( 6 6

2

3 ) 7 $ 4 B ,( % ! ) - I ) ) ) ) ) $ , $ !

2

65!8 3

+ , )

-#

+ , 4 - ½ +,- ¾ ¾ +, H - ½ ½ +,- ¾ +, - ½

-½ ¾

2

!

½ -¾

¾

65!8 ) ( ))

© 2004 by CRC Press LLC

570

Chapter 6

ALGEBRAIC GRAPH THEORY

EXAMPLES % & ) K = 0

+ K = 0 6# !8, 3 ½ )

2

N% !

) - $ ,

( $ 4 B , !

$ -

( (

2

=

! + ,

) )

( )

REMARK

2

) C J>5 ) N I- S

6IN 5?8

6.5.6 The Laplacian 3 9 & ) 9

$ ( (

- )

3 (

- )

DEFINITIONS

2

-

: &

&< 0

$ ( A 4

0

9 B 0(

&

9 &

$ ( 4 C ( 0 A ) )

9 &

3

2

67 !8

6

6

6

3 A

% ) 6

FACTS ) 3 ) 3 ;

4 ; + , ) )

) ) * A

3 A

) &

(

9 A )

9 A

+A ,

2

; + , 4 +

2

9+A, 4 ; D

2

B A )

,

2

6

4 ; + ,

© 2004 by CRC Press LLC

Section 6.5

571

Spectral Graph Theory

2 5 6¾ 4 6¿ 4 4 6 4 ( ; + , 4 ¾ 2 ) 2

+

6¾ ½

) ¾, 4

6¾+ ½ , 6¾+ ¾ ,

2 6JBB8 $ (

½ + ,

6¾

EXAMPLES

2

6¾ -

2

6¾

2 2

6¾

2

6¾

+

, 4 + +#),,

+

, 4 + +#),,

6¾ 1

+

, 4

+

, 4

+ , 4

,

FURTHER READING

3 ) A # 6# ?8 & ) # J 6JBB8

References 6AI J"?8 > A ( # I ( J ( )! 5 P ( ?"? 6I"?8 1 I(

%" # !(

# ! 2 ( K 1 D 5 ( ?"?

6I 5 !8 3 I C ' 5 C( 5 - ( $ " +?!,( !M

6IN "8 N I- S(* # N )( $ ( E 1 5 N J >& + H ,½¾( ! +?",( M? 6IN5"8 N I- S( # N )( 5 5

S( C ( +?",( "M?? 6IN 5?8 N I- S( # N )( = 5( A( ??

#

2 # !( K )

6I5 ?8 N I- S 5 5

S( E * & + ,)( )! +??,( M! 6N?8 > N ( ( " +??,( ?M

" !

6N=B8 > N 1 = = ( 1

) Y( " &! (

© 2004 by CRC Press LLC

572

Chapter 6

ALGEBRAIC GRAPH THEORY

6N !B8 # N )( ( 2 *

* ( +?!B,( M 67 !8 # 7 ( ) ( ?"MB 6 B8 7 (

'! - +?!,(

2 !* P $( $$( I ( ?

B

6I55! 8 K # ( I ( K 5 ( > 5 ( 3 ( ( " +?! ,( BM! 6 ?8 I ( "

!( I =

6 B8 I ( "

( ??

" ( 5 P ( BB

6=?8 1 = ( $ ( " ! # +??,( ?M 6= 8 K = 0 ( E

( BM

+?

6= !8 K = 0 ( > ( M ! &&( N 7 - ( >( # +?!B,

,(

# *

63 ?8 K 3 ( " ! 2 " !( N ( ' # )( ?? 6# # 8 = # # #( 2 5 ( 1) D 5

( ? 6# ?8 A # ( 3 \ ( !M"

5 &1 !(

)! +??,(

6# !8 # C(

( +?!,( !!M? 6JBB8 # J ( # )( BBB

! 2 " !* # ( '

6B8 # S O 9 S( 5 ( ' G9( BB

! " !( 7

65 8 = 5( AC C G -

( ) +? ,( ?M 65!8 K 5-( !

! ! ( J N +>2 7 =,(

( ?!( !MB! 65 ?8 * 5 5

S( 5 + ,)( " +??,( ?M!

© 2004 by CRC Press LLC

Section 6.5

573

Spectral Graph Theory

65 !B8 K = 5

( 5 ( BMB ! !( >2 ( = = ( J 5( K 5L ( A +?!B, 61 J?8 E M" +??,( !!M?

© 2004 by CRC Press LLC

*

(

" !

574

6.6

Chapter 6

ALGEBRAIC GRAPH THEORY

MATROIDAL METHODS IN GRAPH THEORY ' $( ) # 2 A N% >& & 5

N # ' $ I : 7 E ! I # " ' # ? >& # I C B 1 ( 1 ( 5

) I # ' # I

Introduction > (

: ( E ( C

- ) 1 +?!MBB, 6!?82 .$ ) ) & )) & % )

/ )

6.6.1 Matroids: Basic Definitions and Examples

&

= 1 +?B!M?"?,

% ? 618 DEFINITIONS

2 9 % +9 , + 9 , +9 , ) ) ( ½ ¾ ) +9 , ½ ¾( ) ¿ +9 , ¿ +½ ¾ ,

7: ( +9 , +9 , ))

© 2004 by CRC Press LLC

Section 6.6

575

Matroidal Methods in Graph Theory

2 ) ) (

2 2

+ , &

9½

G +9½ , +9¾ ,

G+ , 9¾

( 9½ 4 9¾(

9½

9¾

) -+9 ,(

9

) ,+9 ,

REMARK

2

$ % ) EXAMPLES

2

0 & )

$* 3* " #E$N 9

+ ,( %* "

E'JN 5> +9 ,

9

I$I'$5 +9 ,

$JN>>JN>J 5>5( ,+9 ,

A5>5 -+9 ,

C + , 2 C

7 2

) 0

C 2 C )

&

2 4 , H

C 2 C ,

4 2 4 4 ,

+ ,(

2 " 6 8(

9 0

& 0 % 7

) "( @

+B , ,

2

3 9 ) +9 , 4 +9 , 4 9 4 9 + ½ , 4 9 + ¾,( ½ ¾ 7 -+9 , 4 ( 9 4 9 608( 0 &

B B B

© 2004 by CRC Press LLC

B B

B B

B B

B B

576

Chapter 6

ALGEBRAIC GRAPH THEORY

1 1 2

3

4

2 G

3

1

5

4 G 2

6 5

6

Figure 6.6.1 9 ½ " %*" " 9

2 $ ) ( % 9 % % $* / * " I

I

9 4 9 + ,

)

9 4 9 608 & 0 %

)

) 7 +,( %

) 7 +,

) %

FACTS

$ ( 9

2 : @ 2 $ 9 ( 9 4 9 + , 2 +1 * $ 618,

) ) ) : 2 + , & < +

, + ,< +

, ) ) 9 ) ( ) ( (

:

2 :

) 2 $ 9 ( 9 2 9 9 ) ) ) ( & )

B

© 2004 by CRC Press LLC

Section 6.6

577

Matroidal Methods in Graph Theory

6.6.2 Alternative Axiom Systems # ) C ) 0 & & E ) ( & ( 6E&?8 (

% $ )

"" 3 ) , $

:; ,< :; ) ) , , +, ,< :; , (

,

< 3 ) - $ ) :<; - < :<; ½ - (

4

4

4

4

4

4 +4 , -

DEFINITIONS

$ ( 9

2 $ 0 ( & ) 0 ( 0( +0, + % +0,,

2 + , - + ,

2 ) - + , 2 & 2 + , 2 + , 4 + , 2 . + , + , 4 2 2 $ ( 2 + , 9

9

9

9

9

9

2

2

FACT

2

$ 6 8 ) ) ( % + , 4 + 6 8, '+ 6 8, '+ 6 8, )

6 8 EXAMPLE

2

7 7 ( +9 + ,, 4 + , 4 +9 + ,, 4 + , 4 $ ( + , 4 T -

© 2004 by CRC Press LLC

578

Chapter 6

ALGEBRAIC GRAPH THEORY

6.6.3 The Greedy Algorithm # -

C )

G- * %

)- Æ - ) )

Algorithm 6.6.1:

9"% 0*

3 ) %

,

, E,

) )

( (

+

(

E+ , )

+ , 5 4

* 4 B

+

, $

& (

4

&

$

3 E )

, ( (

E+,

+

,<

4 4 + ,

+

, * +

, + , 5

EXAMPLE

2

3 ) +, N% E+, 4

+,

9 G- * ( 4

FACT

2

,

) %

(

E (

4 ) & )

,

&

6.6.4 Duality # & )

DEFINITIONS

2

9

9 +9 , - +9 , 4 +9 , 4 2 4 -+9 ,

) + 7 ? ) ,

2

( ( ( 9 ( 9

( )(

9

2

9 4 9 + ,

7 ( +9 + ,,

+ , 9

© 2004 by CRC Press LLC

)

Section 6.6

579

Matroidal Methods in Graph Theory

2 ) EXAMPLES

2

) %

$* * $ 3* # N

9 + , @ 9 + , @

9 6C 8 & 6C 8 9 6 C 8(

C

) 6 8

2

9

½ 7 ½ 7 E) + ½ , + ½ , + ½,

( ( ( ( (

9

2

4 1

6

3 G* 1

Figure 6.6.2

5

9 ½ * "* ½

FACTS

92 2 - +9 , 4 +9 , 4 2 4 -+9 , ) +9 , 2 +9 , 4 9 2 - 9 + , 4 +9 , H + , 2 9

) 9 2 9

4 9 2 7 ( 9 + ,

$ ( 9 + , 7

2

+ ,

2

C

C

> & 6 8 6

© 2004 by CRC Press LLC

8

580

Chapter 6

ALGEBRAIC GRAPH THEORY

2 2 (

) %& % ( (

$* /* " * )

)

9 9

"

9

9

9

9

9

9

REMARKS

2

) )

)

2

# $ ( & ) 9 + , 9 + , ( &( &

6.6.5 Matroid Union and Its Consequences ( ) J1 6J 8( :

DEFINITION

2 9½ 9 9

) C C C C ,+9 , +5 7 ", $ ) 9 . 9 . . 9 FACTS

2 3 9 9 9 )

) C C C C ,+9 , ( %

2

$

9

- ( -

9

. 9 . . 9

+ , H 2

Covering and Packing Results

- ) - ( 7 ) ) 7 B ( ( J 7 B

© 2004 by CRC Press LLC

Section 6.6

581

Matroidal Methods in Graph Theory

FACTS

2

9 ' 9 ) ( ) '+ , H +9 , '+9 , 2 6> 8 9 ' +9 ,

( ) +9 ,( '+ , +9 ,(

6> 8

2

'

6 ( J 8 9

( + ,( ) 9 0 + , )

#

' #

2

#

6 8 ( ) + ,(

#

) ' 9

'

+ 6 8, + ,

2

6> 8 3 ) 4

) )

) 4 4

4 +)

) , : 2 + ,

% 4

)

9 +

, 7 # + ,( ) 9 +# , +

,

6.6.6 Fundamental Operations N ) ) ( ( % ) ) ( ( C > C DEFINITIONS

9"

9)"

9 + " , 9+ " , 9 " 2 B 9 B ) ) 9 ) :

B B 4 9 2 ! ! ) ) ) :

2

( (

7 :

9

© 2004 by CRC Press LLC

9)

9 ( 9 9)

582

Chapter 6

$* #

+9 , "

"

"

C +9 , " C , +9 ,

+9 , "

"

9½ 9¾ (

+9½ , +9¾ ,

2

+9½ , +9¾ ,

4

-+9 " ,

C½ C¾

" 9½

2

C 4 , +9 ,

)

9

C +9 , "

+9 ,

2

9

2

+9 ,

9)" (

,

+9 , "

"*

$ "

5

9 " (

ALGEBRAIC GRAPH THEORY

2

C , +9 ,

9¾ (

+9½ , +9¾ ,

4

EXAMPLES

2 9 + , 4 9 + ,( ) ) 2 9 + ,) 4 9 + ),( ) ) )

(

( )

2 @ 4 @

½

2 @ ) 4 @ ½

½

,4

,4B

@

@ ½

½

@ ) 4 @

½

4

2 9 608 & ) )

&

0

2

$

0(

& ) ) )

2

$

½

¾

9 608)

& 9 (

9 + ½, 9 + ¾,

½ ¾ # ( ½ & ½ ¾ & ¾ ( 9 + ½ , 9 + ¾ , ) ) ½ ¾ ( ) ½ ¾

) ) - 9

FACTS $

2

(

9)" ,

+

9 ( 9½ (

4

9 " <

9¾

+

9 " ,

4

9)"

+9½ , +9¾ , 4

+N

,

2

$ 9 ) +9 ,( 9 4 9 + , 9 < 9)) 4 9)+ , 4 9) ) < 9 ) 4 9)

© 2004 by CRC Press LLC

4

Section 6.6

2 2 2 9¾

$

9½

583

Matroidal Methods in Graph Theory

+9 , " ( % + , 4 % + " , % +" ,

9¾ 4 9¾ 9½

3 B½ B¾ ) -C 4 +9½ B¾ , . +B½ 9¾ ,

+

9½

,

+

9¾

,

9½

6.6.7 2- and 3-Connectedness for Graphs and Matroids ( DEFINITION

2 9 ( +9 ,(

7 ( . / . / ) . ) / EXAMPLE

2

$ 7 ( ½ ¾ ( ( 9 + ½ , 4 + , ( ( 9 + , 9 ¾

FACTS

2

3 ) + , ( (

2 9

2 2 2

9

)

6 8 $

9

+9 ,(

9

9)

63 8 $ 9 ( 9 :

) %& +9 ,

A ) 7 B ( ) 2

2 3 ) + , ( ( ) Bounds on the Number of Elements

7 9 ( +9 ,

C ( (

© 2004 by CRC Press LLC

9

+9 ,

)

584

Chapter 6

ALGEBRAIC GRAPH THEORY

$ +9 , ( +9 , +9 , ) C ( ( 9 9 < +9 , +9 , ) C 9 9 2 63E&B8 3 9 ) + , $ 9 ( +9 , + +9 , ,+ +9 , , H +

, +9 , ¾½ +9 , +9 , 2 +3 : 63!?8, 3 9 ) ( +9 ,

+9 , + +9 , ,+ +9 , , H &

: 7

2 3 ) + , & C ) 2 61?!8 3 ) ) C )

+ ,

1 61BB8 : ) ( 1* ) ) ) ( 0

2 +>U , 6>?8 3 )

+ , + + , ,

) : 7 +

, : )

C 7 ?

2 6J '??8 > ) + ,

2 6#BY8 > ) C

) P + 6E&B8,( OPEN PROBLEM

4! 3 9 ) N 9

+9 , Y

© 2004 by CRC Press LLC

Section 6.6

585

Matroidal Methods in Graph Theory

2-sums and 3-sums

>& >( C

C (

C ) DEFINITIONS

2 9½ 9¾ 9 ( ( /½ +9½ , /¾ +9¾ ,

+ +9½ , /½, + +9¾ , /¾,( 9½ /½ ( 9¾ /¾ (

+½ /½, +¾ /¾, 9 / $ ) 9½ ¾ 9¾

2 )

2 3 9½ 9¾ ) ) 5 +9½ , +9¾ , 4 " " 9½ 9¾ " 9½ 9¾ 9½ 9¾ + +9½ , +9¾ ,, " T 7 " 7 +9 , T 9 EXAMPLES

2

3 ½ ¾ ) / ) 3 ) ) ) ½ ¾ ) /½ /¾ % ( ( ½ ¾ 9 + ½ , ¾ 9 + ¾, 4 9 + ,

2

3 ½ ¾ ) 7 ( + ½, + ¾, 4 ) ) - ½ ¾ ½ ¾ 9 + ,

9 + ½, 9 + ¾, 1

1 3 2

3 2

G 1

Figure 6.6.3

G 2

+ ,

9

G

#

+

9 ½

,

"

+

9 ¾

,

FACTS

2 2 2

+9½ ¾ 9¾ , 4 9½ 9¾ 9

9

3 ) + , + ,

9

© 2004 by CRC Press LLC

586

Chapter 6

ALGEBRAIC GRAPH THEORY

REMARK

2

6 8 % ' ' ) $ ) ) ' ' 5 6I"( $1"( E&"8 ' ( C ' )

6.6.8 Graphs and Totally Unimodular Matrices E % 5 * ) )

( )

C +( & ( 65" 8, ) & EXAMPLES

2 3 ) ) ) &

& & 9 + , %

2

6A !!8 I & 0 7 +, & 3 &½ ) ) 0 & 4 & # ( &( & 4 9 +, &) 4 9 +, FACT

2

65"B8 ) ) ( ( ( ( &

6.6.9 Excluded-Minor Characterizations G - 1 6GB( 1!8

) &

DEFINITIONS

2

)

2

B (

B

© 2004 by CRC Press LLC

Section 6.6

587

Matroidal Methods in Graph Theory

FACTS

2

2 (

( ( ) %& % ( (

2 6 5BY8 7

( &

%

2 63"8 7 % B ( ( =( ( 1( ) % &

EXAMPLES

2 %

+) ,(

( - 7 ( ) > ! E 7 ( 7 $ 0 7 ( 7 ( ) (

J

1 4

(a)

1 5

7

2

(b)

3

4

5

7

2

3

6

Figure 6.6.4 :;

6

# "7 7 :$; "7 7

2 ) % &

) 6"( "( ?8 C ) A &) 6A !?8 5 65!?8 $* 3*""# > * " I

>&

@

@

)

@

@ 7 7

I

>&

@ @ 7 7

@ 7 7 9

+ , 9 + ,

+

,

+

@ 7 7 9 9

,

2

)

CONJECTURES

9 )

© 2004 by CRC Press LLC

588

Chapter 6

ALGEBRAIC GRAPH THEORY

1! + * I 9 6 !8, 7 % % ( % &

) 1! 7 % % ( /

) ( % &

/ REMARKS

2

7 " )) (

Æ 7 ? 7 " & 9 ) 7 " ) &)

2

7 ) ( > ( &

7 +>, ) % ( ( G 6GBB8 7 +,( ) & &

* I 9 > & 5 I 9 I ) ) 1 61B8 ) ( ( 1 61B8

6.6.10 Wheels, Whirls, and the Splitter Theorem 6 8 % ) ) ) - 5): ( C 6 8 5 ( C ( ) 5 65"B8 ( ( ) J

6J"8 DEFINITIONS

2 7 ( 0 ) & 9 ) + , &

2

7 ( 0 0 0 & ( -

2 3 9

B

B

)

9

B

9

2

/ - ½

/ ( ' ( ' - ' H

EXAMPLES

2

7 0 ( ( 9 +0 , 0 9 +0 , 0

© 2004 by CRC Press LLC

Section 6.6

589

Matroidal Methods in Graph Theory 1 5

1

6

1

(a)

(b)

6

5

(c)

6

5

2

3

2

4

Figure 6.6.5 :;

3

2

3

4

0

4

" :$; 9 +0 , " :; 0

2 A 7 7 & - 2 : - ) - & 6C C 8 +, C & ) ) C

& C (

7

FACTS

2 6 8 3 : 2

)

+ , 7 ( +

,

) )

-

2 6 8 3 9 ) : 2

+ , 7 +

,

9

9 9)

9 +0 , 0

9

(

2 + 5 65"B8, 3 9 B )

B

9 ( +B , ( 9 5 B 4 9 +0 ,( 4 0 ( 9 0

< B 9 9 +0 ,

: 9 9 9

9 4 9 < 9 4 B < ( ( 9

9

) 5 ( I 6I "8 + 6I E&?8,( 5 * 5 ) $ - 7 ) ( % 5

2 61 B8 3

7

!

)

) 4 !( ! #L

Figure 6.6.6

© 2004 by CRC Press LLC

# .' $ *""7

!

590

Chapter 6

2 6=8 3 ) 4

ALGEBRAIC GRAPH THEORY

2 65"B8 3 9 ) ) 9 9 4 7

2

3 4 0

2

)

6E&"?8 3

9

7

0

)

0

+ ,

7 !< +

, ' ( ) ) 9 & )

Figure 6.6.7

- #-*

) 6?8 ( ( (

2

6E E&?8 7

B

2

( B

0

6N E E&P?!8 7 ( B B

@

@

+

9

, 9 + , 9 +0 , 0

-

6.6.11 Removable Circuits # ' ) '

C ' 4 = ( #* DEFINITIONS

2 ' & )

) '

'

2 7 ' (

© 2004 by CRC Press LLC

' 9 &

9

Section 6.6

Matroidal Methods in Graph Theory

591

FACT

2 +#, 6#!8 $ '

' H ( ) #* + , ½ +' H , + , &

) )

& )

2 63E&??8 3 9 ) ) 9 $ +9 , +9 , H +9 ,( 9 9 9 +9 , 4 +9 , $ ( + , 4 +9 , +9 , +9 , H ( 9 ) 2 63E&??8 3 9 ) ) 9 $

+9 ,

"

+9 , H +9 , 4 +9 , H ( +9 , H +9 ,

9 9 9 +9 , 4 +9 , & ) )

2 3 ) ) $ + ,

+ , + ,( )

$ (

+ , + ,( ) 2 3 ) ) 5

+ ,

"

+ ,

( + , + ,

2 6 #?!8 3 )

$

( 9 )

2 6#BY8 3 ) $

( ) ) 7 ( #* +7 B, ) ) )

2 65 ?"8 3 )

%

) >& >? ) C & C ) & )

2 6 K??8 3 9 ) ) % $ 9

) 7 7 ( 9 ) +9 , 4 +9 ,

© 2004 by CRC Press LLC

592

Chapter 6

ALGEBRAIC GRAPH THEORY

REMARKS

2 7 ( )

)

2 7 ) ) ( (

( ) ) ( ( (

2 = )) ) ) - >

) EXAMPLES

2 63E&??8 I 2 )

& ! " ? B< ) ! " ? B( & E& 9 ) ) C 9 + , C A 9 + , ) ) ) C 7 B

) ' 4

2 K- 6K"B8 ( ( ) + 6K"B8, = ))* : +?, ) % 7 "+, 2 6 #?!8 7 ) ( 7 "+),

) ) ) 9( # 7

2 7 ( @ ( H ( )

(a) G 1

(b) G 2

Figure 6.6.8 8

+

9

,

9

+ ,

2$*

PROBLEMS

4:;! 6 K??8 $ $ ) $ ) Y 4:;! 6 K??8 $ 9 ) (

© 2004 by CRC Press LLC

9

) Y

Section 6.6

593

Matroidal Methods in Graph Theory

6.6.12 Minimally '-Connected Graphs and Matroids 7 ' ( ' ' ' 7 ' ( DEFINITIONS

2 7 ' ( ' '

'

2 7 ' ( ' 9 '

2 3

9

9

)

9

9

'

EXAMPLES

2

$ , ' (

' 7 ( - 0

0

2

7 7 )

FACTS FOR ARBITRARY CONNECTIVITY

2 6#!8 7 & '

'

(

'

2

6#!?8 7 ' ( )

'

'

'

'

+' , + , H ' '

2

6E&")8 7 ' ( ) + , + , H ' REMARKS

!

7 !B ' 4 ) N 6N !8

6 "8( ( ' 4 ) = 6= ?8 7 ! )

!

) 7 !( ) 7 ? )

( : ) 7 !B

!

6#? 8 ) - ) 9 &

)

& 7 ? ) ) '

© 2004 by CRC Press LLC

594

Chapter 6

ALGEBRAIC GRAPH THEORY

FACTS FOR SMALL CONNECTIVITY

2 $ 9 9 4 9 + ,

( ) +9 , 9 & 2 6N !( "( = ?8 7 ' (

'

'

2 6E&"( E&")8 7 ' ( + , +

,

9

2 61?"8 3

9

9

2 6= ?8 3

+9 , H + '

)

)

+ , +

,

)

'

' < , ' '

9

9

<

9

+9 , H

)

# ( + ,

"

+ , + ,

+ , + , < + , ? + , !

: ) 0

¿

,

2 6E&"8 3 9 )

+9 ,

"

+9 , +9 ,

+9 , <

+9 ,

) : ) 9 +¿ , ,

9

+0 ,

E ) 7! 7!+

, ( 2

2 6E&")8 3

)

# + , H ! ¿

# + , +

¿

+ , ? + , + , < ? + , + , + , ?

, H

7!+

, 7!+

, 2

2 3 ) + , $ ( ) + ,

© 2004 by CRC Press LLC

Section 6.6

595

Matroidal Methods in Graph Theory

+

, 3 )

) ) + , H

)

6.6.13 Conclusion # ) ) ) )

( ( T +5 6AE&?8 61??8, $

( & ) &

( C ( ) )

References 6A !!8 > A &)( G - * 1* (

+?!!,( M

6A !?8 > A &)( E * C (

+?!?,( !MB

6AE&?8 A - K E& (

( M + J 1 ,( I ) ' ( ??

6I "8 I I ( .# # 9 A # /( N ( J ' ( ?" 6I E&?8 I I K E& ( >& * 1 1 ( +??,( BMB 6I"8 1 = I ( E ( +?!?,( ?M??

B

6N E E&P?!8 N ( A E - ( K E& ( N P ( ' )

( ! +??!,( M?? 6> 8 K > ( 3 * J 1 ( %! ( ! ?A +? ,( !M! 6>?8 >U ( E & ( " B +??,( !M

61B8 K 1 ( A * 9( " +BB,( MB

© 2004 by CRC Press LLC

596

Chapter 6

ALGEBRAIC GRAPH THEORY

6GBB8 K 7 ( # = ( G ( &

7 +,M ) (

!? +BBB,( !M?? 61B8 K ( # = ( 1 ( A : ( " +BB,( "!

6 K??8 3 A K- ( ) ) ( +???,( ?M

6 #?!8 3 ( K = ( 5 # ( )

( ! +??!,( BM 6?8 A 5 )(

( )! ?M! 6=8 N 1 = (

- ( M!

B +??,(

? +?,(

6= ?8 = ( ' L)

( " +? ?,( !M""

C

L

6= ?8 = ( O C

L ( $ " +? ?,( M 6$1"8 $- 3 1 )( 1 ( )! ! +?",( M

& "

6K"B8 A K- ( )

( 9 +?"B,( "M? 6GB8 G G - ( 5 ) d ) ( 7 +?B,( !M" 63"8 3C ( ) ( +?",( M 63 8 3 ( 5 ( +? ,( "!M! 63!?8 3 ( E : ( BM!

& !

" " !

+?!?,(

63E&B8 # 3 K E& ( ) C ( ! +BB,( B?MB 6#!8 1 #( >-

( +?!,( ?M

C

L

6#!8 1 #( GC 1 ( " +?!,( "!MB

$

6#!?8 1 #( I % ( M? ! ! + A A )S,( I ) ' ( ?!?

© 2004 by CRC Press LLC

Section 6.6

597

Matroidal Methods in Graph Theory

6#? 8 1 #( E

( !* :

! ! " 3 9 + N # - S( P 5S ( 5CU ,( KS A # 5 ( A ( ?? ( M? 6#BY8 5 # ( I &

( )

6#BY8 5 # ( I ) ) ( )

6J 8 I 5K J1 ( > 9 % ( +? ,( MB

6J 8 I 5K J1 ( (

2 # ! +$ 5 ( ,( N ( ( ? ( M 6J"8 5 J

( C ( +?",( ?M! 6J '??8 P J 3( > I ( K ' ( ) ( )! ?!R?" +???,( M 6E E&?8 A E - ( K E& ( ( ) ( ! +??,( ?M! 6E&"8 K E& ( E C ( ?B +?",( B!M 6E&"8 K E& ( E ( ?MB"

6 ,52 9 +?",(

6E&")8 K E& ( E ( +?",( !M"

!

6E&"?8 K E& (

( +?"?,( ?MB

( E& ' ( ??

6E&?8 K E& (

6E&B8 K E& ( E ) ( ! !* 9;;< + K 1 = ,( 3 # 5 3 J ""( I ) ' ( BB( ??M? 6 !8 I ( I ) ( ( +J ( 5 ?!B,( P ( ( ?!( ?M

& "

6 5BY8 J ) N 5 (

[[ 1* 9( ( 65" 8 5 9(

2 &" " "( 1 ( ?"

65!?8 N 5 ( # +?!?,( ?M!

© 2004 by CRC Press LLC

7 +,(

598

Chapter 6

ALGEBRAIC GRAPH THEORY

65"B8 N 5 ( N ( +?"B,( BM?

"

65 ?"8 5 ( .5 5- > I /( N ( ' 3 ( ??" 6"8 1 ( $( ! +?",( M B 6"8 1 ( $$( "" +?",( M! 6?8 1 ( # ( !M

""

!

!

+??,(

6 8 1 ( E ) ( +? ,( MB 6 8 1 ( 3 ( +? ,( M!

%! ( !

6 8 1 ( I (

" +?

L 61!8 G 1( ') > 5C G - ( +?!,( "BM" 61 B8 G 1( A - C = P ( M 61??8 N 1 (

( BM" 618 = 1 ( (

?A

,( BM

)

+? B,(

% ! " !

+???,(

+?,( M

618 = 1 ( E ) ( ! +?,( B?M

61?"8 = 1( E & ) (

! +??",( "M?!

61?!8 3 1( ) ) (

+??!,( B!M 61BB8 3 1( >& ) ( )! +BBB,( ??MB"

© 2004 by CRC Press LLC

599

Chapter 6 Glossary

GLOSSARY FOR CHAPTER 6 "?% 3 M

2 B & 0

( )

9 (

F # "2

%

9 &(

0"

0

4

C

&(

F

* 2 M % 2

% )

4

0

B

*$ 2% 3 6

* 2 2

0

& + * ,

6

6

2

%$+ ,

#2

-

# 2 2

+ ,

% 2

2

M

(

(

M (

%$+ ,

2

$ "" M % 2

) ) &

$ @2 $

& 9 %

M

$%*

92

&

9

M 2 ) )

$% 2 M ) 2 B $* + M

)9 2 )

1

(

# +4 ,

2*2

4

(

)9 # 1 (

(

4

) -

9

4

( ( )9

$ "

M

2

)

$

M &

$

M 2

2

"2

& )

1* $ 2 : ) % ) 4 4 H H H 1%*% " M

% 2 % ( %

( +% ,

© 2004 by CRC Press LLC

&

&

I

600

Chapter 6

1%*% ½2

ALGEBRAIC GRAPH THEORY

I

1%*% 2

I

1%*% 2

)

1%*%

2 ) ) (

*% * M

2

+C

0, 9

&

"

M 2

1#2 1# 2

% 2 , +% 2 , 4 +% 2 ,( %+%, 4 + ,

I +

&

%

%

I

I$

M 2 9

(

M 2 )

""

M 2

(

2$*

M ' 9 2 '

(

9

9 '

3

M + ,2 &

(

M 2

)

$

` + , % ( )

M 2

<

` ) + ,

2

M 2 ) ,

2

M 2 , <

* 2

I

* " " "*% *

M

/

/

2

)

/

M 9 2 & ) +9 ,

-

* 2

M + 9 , (

) <

(

M 9 2

+* %

I+,2

9

+

* $

M 2

<

4 + + , + ,+ ,,

M 2 & )

" 2

© 2004 by CRC Press LLC

)

601

Chapter 6 Glossary

2

M " 9 2 9)" +9 ,

M + ,2

) :

2

"

+9 ,

"

M " 2

3

2

M " 2

" <

"

M 2

"

&

)

, <

2 M !½ ¾"

2 ) ,

M 4 + ,2 & ½

¾ 4 M ½

½

M 2

)

¾

M 2 &

(

""

M 2

$ ½

$ ¾

%* " M

M 2

) 7 +, M 2

) 2

"

M &2 ) &

"* M

" 9 2 9

9 +9 ,

""

"

+9 ,

"

) (

½ ¾ (

)

)

"

M 9 2 9

"7 *$*"2

"

)

M 9½ 9¾ 9 2 +9½ ,

+9¾ ,

9½ 9¾

" #*

M /

(

B

* ' 2

'( )

"2 M H2 % * /

0 4( & 0H4 (

2

½

" $*% 2 2

© 2004 by CRC Press LLC

602

Chapter 6

ALGEBRAIC GRAPH THEORY

"*½

+ ,2 ) &

( ) & ( + Ü! ,( 9

"*¾

+ ,2 N% Ü M 9 2 +9 , ) ) 9 "# 2

"* 9

"# 2

"# 2 ) 9

9 9

"# 2 2

2* M 2

2* #" :* -; $ " M )

+ , H

*% @2 )

¾

2

% 2 " M % 2 : )

) ) % )

" M 2 2 2 & +5

, 3*"" M

2 )

2 A M

9

2 & )

+

9

, %& -

"* 3 M

2 +, H , ) & & ( ) ) 4 < ( %

"* M

: )

"

"

2

"* 3 M 2

+ , ) & & < ( % "* M 2 : + ½ ¾ (( ½ ¾

*>" "%* 2 ) ) ) ( )

) ( ) % & 9 ) %

© 2004 by CRC Press LLC

603

Chapter 6 Glossary

" "

2 )

2 ( ( ) 9

"½ " H

M 2

M 2

7 *$*"

2

) (

(

) )

* *

-

2

&

' ) 8( +', ) ' %& & ( )

M %

)

9 *# 2 B* " 2

2

6 + ,

2 & ( ) M 2 =

+ , =

+ ,

B* # "

2

=

B* #*$*

(

2 ) 0 4 0 4 ( =

B ! + ,2 + , < 9 ( ( ) + ) ) (

(

0,

"% **

)

B C *% * '

'

M 2

% %

M

(

2 #

)

2

( ) -

*

%

:

!2

<+,

4

2 #

+, # * ! 2 2 # 9 ( +, + , 9 ! %* M 9 2 & ) +9 , ) 9

" 3

M 2 &

< &

( *

*

* (

&

B

" 3

M 2

© 2004 by CRC Press LLC

) & &

604

Chapter 6

ALGEBRAIC GRAPH THEORY

" M &2 & " 2 M 2

& ( ) &

" 2 M 2 )

&

9 2 9 "" $ " $ + ,2 ) & "" $ 23 $ + ,2 & @ 2% M + ,2 "" M

%

*" 232 & C

2 ) )

> M

9½ 9¾ G

+9½ , +9¾ , 9½ G+ , 9¾

" 2

2

( ) 9 )9

! 2 ) 9 2 # 2 # +, #

*

* ! 2 ) 9 2 #

+, +, 9 ! 9

D D + ,2 ) <

9 ( )( + ) ' ( )( ',

& - + 2 7

( C

5* 32 : &

( &< 0 9 B

( ) A+ ,2 & ( A+ , 9 ( (

*

( *>" A+ < ½ , M

½ 2 ) - 9 A+ , - I+ ½ , I+ ,< & A+ , 9 ( 9 I+ , I+ , *

M

© 2004 by CRC Press LLC

9 2 +9 ,

605

Chapter 6 Glossary

"

9 2 % +9 ,( 9 (

) ) +9 ,

+9 , +½ ¾,

¾ )

+9 , (

¿

$%2

+9 ,

9 ½

½

¾ (

) ¿

&

% 7 +, (

$ "

M 2

) (

2

)

( # "2

( (

) ( # "2 ) (

2

(

*2

(

*2

(

$* M %

)

& (

*2

(

@

2

2 B

, (

+" " ,( "

% :" $* %;2 %2

+, H , )

3**% " 2

%

)

+& ,

"½

"¾

+"½ "¾ ,

"

%

+

**% '# " 2

'

'

**%

' # "

"2

'

(

'

'

*% *

>+0

2

>+,

(

, 4 B

M 2 ) ) ) :

( (

M 9 2

) ) 9 ) :

(

M 9 2

9 :

#* " M

9

2

)

M 9 2

9

** 2 **% ,

M ( , / 2

H/

+ ,

© 2004 by CRC Press LLC

:

606

Chapter 6

ALGEBRAIC GRAPH THEORY

!½ " M 2 (

(

M 2

* * M 2 )

C

* $ 2 )

) ) : C

M 2

4

&

+) ,( Q

+,(

+ ,(

4*% 2 I % % >

# + ,( :

*** * 9

9

37 F # "2 &

/

+ * ,

+ ,( + ,

7 >

7 >

2

-

7 " $*% 22

)9

7 22 ) -

7 *2 )

7 *2 %

( : 2

1 %&

<

4 <

( :

#

1

+

#

, 4

2 1 + )9,(

1

( *2 &) C ( 22 )9

( )9 M 2 2 2

&

2 2 ) - * $

M

2 N%

" M ! 2 H &

# (

& ) 9 !

H!

+ M

/ 2 ) < - : / ) 8+ ,<

© 2004 by CRC Press LLC

607

Chapter 6 Glossary

+ M 0 2 & ) 0 - 9 & ) +9 , % M % 2 %

( # * M % 2 ) ) ( " $* M % 2 % "" " 3 M 2 ) & & + , * M 2 * 32 N% ? B

#* 2

*2*% 2

2$* %* M ' 2 ' - ½ 2 ) ( ) ) < ) - 2 + , + ,2 4 +? ? ? ? ? ,( ? 4 ) )

& + ) B 4 ( B ) 4 ( B ) B 4 B( ) 4 B,

% 2 + (

, &

M / 2 / ( M 9 2 ) +9 , ) +9 , M 2 2 < ( $*> & 2 ) %$ %& &

) 2 )

+ ,) + ,(

2

) !

%! ! +! , % +! %! , ! *% * M + 6 ,2 & 9 9 6 ( 9 9

# " 2 9 9 9 ( / ) 9 < / / + +9 , / , + +9 , /, 9 /( 9 / (

+ /, + /, 9 /

© 2004 by CRC Press LLC

608

Chapter 6

# " 2

)

+9½ , +9¾ ,

9½

9¾

ALGEBRAIC GRAPH THEORY

"

9½

9¾

< + +9½ ,

+9¾ ,, "

T

% "C M )

½

¾(

½

7 "

¾2 ½

) )

7 +9 ,

T

9

¾<

)

½ ¾

% 2 % 2%2 &

% "2

&

%

$ 2

7 +,

) % (

%

**% "* 32

&

: ) &

2

)

B ( ) )

(

&

(

2 ( *% "2 ( ( (

&

2 M 2 # 2 2

* 2

%& % )

&

*$*"

) (

½ ¾

)

( )

)

)

M 2 ) ( #2 & ( #

" #2 & ( $%2 & ) ) >

) ) %

)

)

**% ""2 ( *#2 ) & B ( ""2 & : $½ $¾ $ , B ) (

& 9

)

)

""2 (

"2 &( ( (

%

© 2004 by CRC Press LLC

609

Chapter 6 Glossary

2* 2

&

"* 32

&

& :

M

(

9½ 9 9

9

2 " 32

*

2

C C

) &

)

<

<

(

23#" " 3 23# 2 2

C

)

C

2 2

)(

: )

( B

M 2

(

&

&

( H ( . /(

(

. -/

-*

-*

2

2

& +

-*7

02

0

(

&,

9

) ) 9 &

+

,(

( 0

&

.)/)

-

- " M ! " ( 2

" # % # + $, 4 ++ , +$,,( ! $ "

© 2004 by CRC Press LLC

Chapter

7

TOPOLOGICAL GRAPH THEORY 7.1

GRAPHS ON SURFACES

7.2

MINIMUM AND MAXIMUM IMBEDDINGS

7.3

GENUS DISTRIBUTIONS

! " 7.4

VOLTAGE GRAPHS

! " 7.5

THE GENUS OF A GROUP

#! $ 7.6

MAPS

%& ' (% 7.7

REPRESENTATIVITY

) % ' 7.8

TRIANGULATIONS

*$ + * , 7.9

GRAPHS AND FINITE GEOMETRIES

! # # $ GLOSSARY

© 2004 by CRC Press LLC

7.1

GRAPHS ON SURFACES

Introduction ! "#$ % ! & ! ' ! #!! ! !

! ( )

7.1.1 Surfaces 2-Manifolds and 2-Pseudomanifolds DEFINITIONS

* ! + ! ! ! &

" $ ¾ , ¾ + " $ ¾ , ¾ " $ - ¾ , ¾

! . + ! #! ! ! .

* / + ! 0 !+ & + ! 1+ ! 1+ ! !0 1 / 1

* / 1

! & & + ! # 2

Figure 7.1.1

* / #!! ! ! ! !

! ! ! ! 1 !0 1

* ! 0 ! ! ! !& ! !

! ! 1

© 2004 by CRC Press LLC

612

* * *

Chapter 7

TOPOLOGICAL GRAPH THEORY

/ 0 + 1 / #!

# ! % 0 # ! ! !

! 0

! 1 !0 1

! 1 + ! !

* / 0 " 1 $ 0

& % & FACTS

* ! & ! ! !

! + !

* .&

0 & 0 % EXAMPLES

* * * *

! . 0 ! 1 #! ! !0 1 0 #!

! ! #

+ ! 2

* / "

! $ # ! #

! #! ! ! + ! # ! ! 2

Figure 7.1.2

!

Some Standard Surfaces DEFINITIONS

"* / " ¼$ ! ! ! " $ ¾ , ¾ , ¾ 3 * / " $ #!! ! ! !

" $ ¾ , ¾ 3 © 2004 by CRC Press LLC

Section 7.1

613

Graphs on Surfaces

¼

Figure 7.1.3

*

/

" ½ $ ! !

! ! 1

! & ! ! 1

*

/

$ ! ! ! " $ #! ! ! " $ " $

"

! 4 ! & !

Figure 7.1.4

# $%& & ' & ! ( &

" ½ $ ! ! ! " $ " ¾ ¾ $ , 3 3 - ! 0 *

/

! . ! 0

*

!

"

! ! !

Figure 7.1.5

"

¾ $ ! ! ¾ , ¾ 3 " $

$ " $ !

#

½ ) & ¾

FACTS

*

/ ! 1 !

*

/ 5

! 1 6 &

*

! !

! # ! "

© 2004 by CRC Press LLC

$

¾

,

¾ 3

"

$ " $

614

Chapter 7

TOPOLOGICAL GRAPH THEORY

EXAMPLES

*

! ! + ! + ! 7

*

/ 1 #! + #!

#! #

*

/ 5

#!

Surface Operations and Classification DEFINITIONS

*

!

8 ¼

#

¼

! 1 ! ! ! &

*

8½

*

Figure 7.1.6 * *

*

!

# ¾ & ¿ !

! 6 &

/ 0

+

!

! !

!

"

½

+

8 ½

! 0

! ! !

! 5

'! #

*

!

" $ ! !

!

!

*

!

! !

*

*

!

/

9 " $

0

" $ % ! * "

$ 3

" $ 3

& ! 6 1

*

*

!

/ &

# '! #

/ &

© 2004 by CRC Press LLC

1

! !

Section 7.1

615

Graphs on Surfaces

*

/ &

*

/ &

!0

!

5

FACTS

*

/ 4& # ! & !

# 6 1 ! ! ! ! +

& !

*

! # % " ! ! $

&

&

*

/ 0 ! ! # 0

. 0

*

/ 0 ! !

. 0

" *

! * +!

*

* .! ! !

! # *

*

! *, +!

*

-

* .! #!

! #! !

.!

% + & !

+ ! !

* *

8

¼

3

·

¼+

8 ¼ 3 · ¼ + 8

3 ·¾

! & & *

"$ "$ 0 "$ 0 & "$ 0 &

EXAMPLES

" *

! ! ! : ! ( 0 *

! 7 ! 6 & 0 +

# ! ! ( 0

*

! 5

0 #! + (

0

*

;! ! < & !

! & & ! !

! ! + ! !5

)

!

! ! & & ! 1

*

! ! "=!>$ 0

&

© 2004 by CRC Press LLC

616

Chapter 7

TOPOLOGICAL GRAPH THEORY

7.1.2 Polygonal Complexes DEFINITIONS

"

*

/

! #

& " +

1# 1# $ !

*

#

! !

! ! ! !

*

/

! ! ! ! & !

"#!!

! % $ ;! +

?

!

! ! !& !

?

! ! !& !

!

*

!

! ! ! &

*

.!

& & +

#!

& ! !

*

½

/

#1 ! 0 1

#! !

! #1 + & # ! "

! +

! !

# ! + ! # !

! ! &

* *

/

!

,,$

&

! 4

! ! ! %

*

/

"

! !

"

* *

'

& #! #1

!

!

#1 ! 0 1 ! & ! !

"

*

!

! #1 4 ! ! @ & $

! !

! #1 !

*

!

!

#1 !

*

!

! % #!

0 1 ! & ! #1 &

© 2004 by CRC Press LLC

Section 7.1

617

Graphs on Surfaces

* / #! # %

! ! !

! ( !

* /

#! 0 #1

* ! 0 % ½½ ½ ½½ ½ ¾¾ ¾ ½¾ ½ ½ ½

* !

% ½ ½ ¾¾ FACTS

*

/ ! #1

* *

/ (

/ ( 0 !

! ! ? ( 0 + + & & ! ! !

! ! ! 1 "! & ! 1 $

EXAMPLES

*

/ # " $ !

! + 4 + ! ! ! ! ! !

1

Figure 7.1.7

* *

'& , - !

! 0

1 #!

%

½ ½ ½ ¾ ¾ ¾ ¿ ¿ ¿

! ½ ½ ½ ½ ½ ½ & ! ! % ½ ½ ½ # 1 !

*

! 5 %

0 1 ¿¿

½ ½ +

#!! !

*

# ! ½ ½ ½ ! 5

% . + ! ( 6 &

*

! !

"*

½ ½ ½

( ! ! #!

! ! ! ¼ ! !

© 2004 by CRC Press LLC

½ ½

½

618

Chapter 7

TOPOLOGICAL GRAPH THEORY

7.1.3 Imbeddings DEFINITIONS

*

/

" %

!

& ! & ! ! ! !

*

/

!

0 0 ? ! + ! ! ! ! !

! ! 0

! !

"

*

/

! 0 0

*

/

(

*

/

/

! ! !

*

!

! ! ! ! 1? ! + ! ! ! 1

*

/

! ! ! !

1? ! + # ! %

*

!

! "

+ $ ! 0

!

"$

! !

"$

!

* / ! "

+ $

*

!

! !

"$

*

/

!

! !

! !

!

* !

! "

+

$ ! ! ! ! !

"

*

/

"$

9

$

9 "

*

!

! ! !

! !

"$

! !

9

*

/

! 0

© 2004 by CRC Press LLC

Section 7.1

619

Graphs on Surfaces

FACTS

* .& % ! ! ( ! .0 0

* / ! ! * AB CD .& ! ! ! "* A EF D ! + ! !

!

* ACCD G ¼ ¼¼ ! 0 !

¼¼

¼

!

0

* G ¼

! 0 ! ¼ !

0 ¼¼

¼¼

* A;!D .! 0 !

4 ! ! ! ! + & * A !FHD ! ! ! I 0! * A HHD 2 & ! % ! 0 ! ! ! % ! ! ! %

! ! !

! 1 # ! & ! ! !

4 (

* . * .! ! #! & 0 + % !

, 3 " $

* A/CD 2 ! + J"$ "$ , + ! # & + !

* A/CFD / ! 0

1 * .& ! ! ! & & ! 6 6 & #! EXAMPLES

* 2 F ! # # ! ! ! + ! !

Figure 7.1.8 ( ,& !

© 2004 by CRC Press LLC

620

Chapter 7

TOPOLOGICAL GRAPH THEORY

*

! & 0& % "$"KC F$"CK$" C$"F $"KF$ ! ! ¿ ! # * 3

4

8

7

5

6

2

1

Figure 7.1.9

# ,& ! &

* # 4& ! ! # & 0& %

¿

¿

! & !

" CK$"KF $"F C$"CKF$ "F C$" FK$"CKF$"KC $ ! # 2 -

5

1 2 3

7 4

8

a

5

6

1

c 8

b 5

Figure 7.1.10

b

2

4

3

3

5

6 7

a 3

c

( - ,& !

¿

7.1.4 Combinatorial Descriptions of Maps DEFINITIONS

* / % & #! + # * ! !

( # + ! ! ! !0 #! #! ! ! ! ! ! ! & ! !0 ! & #!! !

* / ! !0 ! &

© 2004 by CRC Press LLC

Section 7.1

Graphs on Surfaces

621

* ! ! !

! !0 ! ! + ! 1 #! !

* / $ % !

! & ! ! ! !0 #! ! ! &

* ! $ % ! ! &

* ! ! ! ! ! #! ! " ! ! # &

! ! & 4 $

* / ! ! #1

!

"*

! ! 3 " $ L + #! ! ! #! & #! !

* / " "! L$

ALGORITHM

; ! ! & ! !0 ! & ; # 1 !

+ # ! ½ # !0 ! & + + ! # M ! ! ! !+ # & ( & #!! ! !0 #! ! ! ! Algorithm 7.1.1:

' # ,

- ,. !0 + & + ! /,. 0 ! - 1 !0 % ;! !0 !

" $ !0 # # ="> *3 ; *3 !" " $$ " !0 $ N 3 # ! =$> #! #! 0

! ! ( ! + & &

& + + ! AO F D

© 2004 by CRC Press LLC

622

Chapter 7

TOPOLOGICAL GRAPH THEORY

EXAMPLES

* / & # ! 2 0 / ! !

! !0 ! & + ! ! ! 2 + ! ! ! ! !

!

3 " $"

$"

$"

$

3 " $" $" $" $" $" $

! ! ! 6 #! 0 + #!! ! #1 ! N !

!0 & ! # ! &

*

! #

!

! ! 0 F0

* / ! 2 ! " ! ! / ! ! ! # ! ! ! + a

c

b

a

c

b

Figure 7.1.11 # ,& !

"

" !+ ! % ! #1

3 " $

3 " $

3 " $

+ & + ! % !

3 # * "$

* " $

* / ! 2 ! " ! / ! ! ! # ! ! +

© 2004 by CRC Press LLC

Section 7.1

623

Graphs on Surfaces

a c

b

b

c a

Figure 7.1.12 ,& !

"¿

/ #1 + ! ! %

3 " ½ ½ ½$

! % !

3 # * "$

* " ½ ½ ½ $

* / ! 2 ! "¿ ! 7 / ! ! ! # ! ! +

a

c

b

b

a c

Figure 7.1.13 ) & ,& !

"¿

/ #1 + ! ! %

3 " ½ ½ ½$

( ! % !

3 # * "$

* " ½ ½ ½ $ L 3

FACTS

"*

! ! 6 & #! !

+ ! !

* O (

*

! ! % ( ! #!

© 2004 by CRC Press LLC

624

Chapter 7

TOPOLOGICAL GRAPH THEORY

References A/HCD /! + ! ! ? & + & ! ! " 2 + /+ HHK$ "HHC$+ KPK

+ ! ! + ! N& + HCF

A/CFD O / +

A:;! HD I G : + / ;! + GGI + N&

+ H H

+

ACCD / 1 + ! + + : !+ ! " "HCC$+ F PF AO--D < G O

+ O! ! + # $ " + 7 Q +

+ ---+ K PK H

AO F D < G O

; 1 + % % + & + -- "2 . + ; 0 + HF $ A HHD : !+ / ! ! &" ! $ " "HHH$+ CPC A EF D

+ O + 1+ / O E + ' ! + " ' "HF $+ HPH A;!-D / ;! + FF+ . & + -- A;!D Q ;! + 0

! ! +

+ I !0Q ! 0 & ! " "H$+

PF

AB CD < ; B + ! !+ ! "HC$+ -PK

© 2004 by CRC Press LLC

" "

Section 7.2

7.2

625

Minimum and Maximum Imbeddings

MINIMUM AND MAXIMUM IMBEDDINGS

2

N : * N 0 G # :

7 # 10 !

K / !

Introduction ! ! ! #

! ! ! # ! % ! # ! ! ! ! ! ! !

& ! EG + ! + ! ! !

!& 6 06 / !

7.2.1 Fundamentals DEFINITIONS

* / ! 0 0 ! * ! ! !

* ! ! * ! !"$ ! + !"$ ! =! !> =! >

* *

.! !"$ ! !"$

! ! ! ! ! 0

! 0 1 "'

# ! $

* !

! *

! !

* !

! * !

! !

!

* ! "$ " ! "$$ ! ! ! ! ! !

© 2004 by CRC Press LLC

626

Chapter 7

TOPOLOGICAL GRAPH THEORY

* !

9 "$ " !

9" $$ ! ! ! ! ! !

2 !+ 9 "$ 3 -

* ! "$ ! ! ! ! ! " $ !

"

* !

"$ ! ! ! ! ! " $ ! 0

* ! , ! " ! & $ ! ! + $ "$ ! ! 0 & EXAMPLES

*

2 + ! ! - G1 # + ! !

-

! 0

!

* 2 ! 4 % + !

-

& + & * 2 ! " + !

-

" $& + &

*

! ! ! 4

-+ !

FACTS

* A:D / ! ! ! ! ! ! + 0 ! !

!

! !

*

! ! # - $ "$& + #! $ "$ ! 1 ! ! !

# - $ "$

* / . " + AO

F D$* / ! * ! #! & + + ' + % ! , ' 3

* A:Q7BCD "/& O $ G

0 ! ! " $3

"

%

% %

% !

$

* AI ;! D "/& O $ G % % % ! 0 0 ! ! " $3

© 2004 by CRC Press LLC

"

%

$

Section 7.2

627

Minimum and Maximum Imbeddings

Whitney Synthesis of 2-Edge-Connected Graphs

/ 2 K+ + # ! 0 ! ! 0 0 ! ;! R

! # ! 0 0 ! & ! & ! ! ! DEFINITIONS

* / " 3 A(½ (¾ ( D ! ! (½+ (¾+ + ( ! ! (½ (+ + ! #! #! (½ , , ( ½ .! ! ( "

# ! + ! ! ! 1 ! ! $

* ' ! * ! ! # !"$ 1 ! ! ,

* / ! # ! ! !"$ ! #

! ! ! + ! # ! # # M ! , 2 "$

*

! # ! # M ½ !"$+ ! ! ½ ¾ ! ! + ! # ! # ! " + ! # $ ! , 2 "$ ¾

+ ! # * % # ! ! # ½ ¾ & # ! !

! # ! ! ! # #! # ! ! # ! ! + & ! # # ! # !

(a) Figure 7.2.1

(b)

0 ( ,&

* ( ! * ! ! + ! # ! # # %

* ! # # M ½ ¾ !"¼ $+ ! !"¼ $ #! ! !

© 2004 by CRC Press LLC

628

Chapter 7

TOPOLOGICAL GRAPH THEORY

* ! # + + ! !"¼ $+ ! ! & ! " $ !

! ¼ + #!! ! ; ! ¼ ! !" $ # ! ! ! ! & " $+ ! ! # ! #! 1 FACTS

* *

A;!D / ! ! 0 0

! & ! + & !

7.2.2 Upper Bounds: Planarity and Upper-Imbeddability ! ! & ! ! ! -

! $ "$&

4 1+ ! DEFINITIONS

* "*

/ ! -

/ !

" ! ! ! & 2 $

" $

$ &

* / ! ! ) ! 4

0

*

/ " 3 A(½ ( ( D ! ) * + , ( , ( ! ! ! ! + & ! ( , (

+ *+ ! ! 0 ! (

! 0 ! ! ! 0 ! / 0 0 ! & 0 0 ! FACTS

*

A7-D "7 # 1 ! $ / !

! ! ! !

* A; D / ! ! ! "* A!H-D .& 0 ! ! 0 "N

+ # & # 01 # 0 ! $

*

A;!D !

!

# 0 !

*

A C-D .& 0 ! ! ! #!! & + ! + &

© 2004 by CRC Press LLC

Section 7.2

629

Minimum and Maximum Imbeddings

* A2FD .& ! ! ! #!! & !

* +

AI ;! D / ! 0 !

! ! ! 0 + !

* *

A7 D .& 0 0 ! # 0 6

.& 0 0 ! 0 " ! # ! # A7 D$

*

A.CKD

!

!

$ "$ " AO F D$

REMARKS

1*

: 2 C+ ! 0 ! !

! 0

1*

A!/OHCD 0 ! ! 0 EXAMPLE

*

! ! ¿ 0 " #! $ Deficiency

. ! 0 # & ! 0 !

! !

M & ! ( ! ! !& & & ! ( ! DEFINITIONS

* ! + ! ! " ! 0 + $

* / ) ! 0 " $ !

* ! , " + $ ! 0 +

& " $

+

% !

% !

* / + * % , " + $ 4 ! % , "$ ! ! # ! !

© 2004 by CRC Press LLC

+ )

* AS HD ! , "$

, " + $ & + ! !

*

+

+

630

Chapter 7

TOPOLOGICAL GRAPH THEORY

* AI FD 2 - ! + % " -$ " -$ ! - #! 1 #! & 1+ & + . " -$ 3 " -$ , " -$ - ! + , . "$ % ! . " -$ & - ! ! FACTS

*

# 6 ½ # !

!"$ !

*

A!7HHD / ! ! ! ! ! 0 "$ 6

"* *

+

AS HD ! "$ ! 4 "$ "$ , "$$&

AI FD . "$ 3 , "$ "$ "$ . "$$&

! "$ ! !

4

*

A!7HHD

! ( 0 ! !

*

/ ! 0 " -$ , " -$ -+ - + #! "-$ "-$ ! - #! 1 #! & 1+ &

* * *

AI FD G ! 0 A S HD 0 ! 0

A1I FHD 0 & 0 & ! ! 0

* *

! !

A1 &HD G

! 0

AQG--D " , $0 ! "$0 ! 0 EXAMPLES

* *

! !

0

! !

0

/

7.2.3 Lower Bounds ! 0 % 01 , ' 3 ! ! #! ! + ! #! !

© 2004 by CRC Press LLC

Section 7.2

631

Minimum and Maximum Imbeddings

Lower Bounds for Minimum Genus DEFINITIONS

"

* ! " + 0" $ ! 0 #1 " 0 + ! 0" $ 3 ? ! + ! ! #1 ! # # $

* ! !

* ! - *" $

! ! !

%

#! . ! ! " $3

)

,

H

FACTS

*

0%$2( 01* 2 ! 3

! !

"* * *

¾

3 #!

'

+

" $

0

# ! ! (

! ! ! ! ! ! 2 !

3 #!

'

+

'

+

" $ 0" $

*1

*

0%$2( - 1* 2 !

3 #!

'

+

*1"$' ! # ! . 02 .4 2

*

2 !

"$

*

G

G

3 +

" *1"$ $ *1"$

!

"$

*

C

,

"$

© 2004 by CRC Press LLC

" *1"$ $

*1"$

!

,

!

9 "$

9 "$

,

,

!

,

9 "$

,

632

Chapter 7

TOPOLOGICAL GRAPH THEORY

* !

3 ! + ! !

" $3

C

,

* ! 3 ! 0 + ! !

9

" $3

,

* !

3 ! 4 0 + ! !

" $3

,

"* !

3 ! 4 0 + ! !

9

" $3

,

REMARK

1* /

! ! " 0 $

! ! !+ !+ 4 ! ! ! &

& # 01 # !

E ! !

4

FACTS

" 1 !

$ ! # AO F D

* AB CFD 2 ! ! & + #! + "

$3

" $" $

9

"

$3

" $" $ C

! "A2D$ 9 " $ 3

* AB CFD "2 ! Q #

6 $ ! ! & ! 7 4 Q #

! 2 * G ! ! + ! " $

"

, $

* ACKD 2 ! ! #! / + "

© 2004 by CRC Press LLC

$3

" $" $

Section 7.2

633

Minimum and Maximum Imbeddings

*

AKKD 2 ! !

*

A< FD 2 ! !

& + #! + " $ " $ 3 ,

& + #! +

9

"

!

*

$ 3 " $

3 K+ 9 "$ 3 " $ ,

A;! -D G & + !

3

+ #!

½

"

)

"

*

" $3 ,

&

"

)

$

$

A F-D G ) 0 + *0 ! ! ! + ! )

2

& + !

9

*

*

" $3 ,

* 2

$3 ,

"* $

! + ! 9

"

)

$3 ,

"* $

Lower Bounds on Maximum Genus

: ! & ! + # % !

! 0 &

! ! ! ! 0 ! $ "$& !

# ! ! 0 0 DEFINITION

* AO7HD / "* 0$ ! "* , 0$0 * 6 ! ! 0

! ! 0 & !

Figure 7.2.2

2!3 " -$ 4 1 3 " $

© 2004 by CRC Press LLC

634

Chapter 7

TOPOLOGICAL GRAPH THEORY

FACTS

"

*

! 1 1 "* 0$ * , 0 ,

*

AO7HD

*

A!OHKD

! & 1 ! 1

! #

! 1

*

A!7OHCD G !

"$

$"$&

!

! !+ ! ! %

! #! $"$&

*

G 0 !

"$

$"$&

" A/,-D$

!

2 K ! !

& 0 0 !

*

A!7OHCD

! %

0 0 !

#! 4

*

$"$&

A/,-D G 0 !

! "$

$"$&

!

!+ 2 K

*

A!/OHCD 2 C 2 KPKK 0 ! 0

#! ! 0& !

*

2+ # ! # ! !&

& ! &+ + !+ ! 0 AQG--+ GG--D

7.2.4 Kuratowski-Type Theorems ! 7 # 1 ! 4& !

! ! ! ! 2 ! & #+ 7 # 1 ! & ! ( ! % + #!! & !

! +

! ! !

Complete Forbidden Sets for Minimum Genus .T

75

A7 CD ! 4 #! ! ! 7 # 10 ! 0 !

! ! "

0 $ %

DEFINITIONS

*

/

!

*

G

*

G

! !

+

!

/ !

!

! /

+ ! !

© 2004 by CRC Press LLC

&

!

& ! !

Section 7.2

Minimum and Maximum Imbeddings

635

FACTS

*

! ! 1 !

! " ! & & ! 7 # 1 ! $

* AOQ FD ! % !

! ! ! 6 & / AOQ; HD A/FD

"*

A/QFHD 2 & 0 + ! % !

! !

*

A FFD "2 1 # ; R 6 $ /

! ! % / "A FK+ FF+ H-+ H-+ HKD$ !

*

*

2 & -+ !

! #!

A H-D 2 & -+ ! % !

!

*

A HD ! ( ! REMARKS

1*

!"$ ! = !1> ! ! # % !

! ! & & +

! + !

! N + ! # ! ! ! & ! "$

1*

/ &

2 C # & ! A HHD ! ! ! M !

A !H D ' ! ! !+ ! ! + & & ! 3 + &

!

! ! ( ! ! ! 2 C Complete Forbidden Sets for Maximum Genus

! & 2 + ! 4 % # 0

" + ! ! #! & # 0

$ ! ! " " + ! ! A# D #

0

# + & $ Q # & + & ! "% $ 3 #! ""$ 3 - ! + 7 # 10 ! & 2 C DEFINITIONS

* G ! 0 & #! # ! # 3 "# 3 ! & $ ; ! ! ¼ ! & ¼ & ! & ! # ! & # 3

© 2004 by CRC Press LLC

636

Chapter 7

TOPOLOGICAL GRAPH THEORY

* # ! ½ !

!

! 0 & ! # ! ! ! !& ! ! FACTS

* *

/ 0 0 ! ! -

AI ;! D / ! ! - & # M ! ! !

*

A!OHD / 0 ! !

! ! ! 1 ! ! 2

Figure 7.2.3

5 ! ,- ,,

*

/ ! ! 0 ! & + ! 0 ! ! ! 1 ! ! 2

7.2.5 Algorithmic Issues 2 # G A2 GFFD ! ! ! !

&

R ! ! + ! ! # ! ! !

+ ! ! ! ! ! !

Minimum Genus Testing DEFINITION

* / ! & ! ! I ! #! ! & ! FACTS

*

AQ D ! 0 ! ! ! 0 & ! ! ! !

"*

AQ ; D ! ! ! !

! !

* A2 HD ! ! #! ! ! !

#! 4" $

© 2004 by CRC Press LLC

Section 7.2

Minimum and Maximum Imbeddings

637

* A HKD G ) % ! ! ! ! & ! #! ! )

* 2 !

+ ! ! ! ! ! !

* A HHD 2 ! % + ! 0 ! !+ &

! + !

! " ! & ! % & ! 2 ! 0 ! $

* A !FHD ! # I 0 * & ! + #! ! "$ * A !H D ! #! ! ! !

& I 0

* A HFD ! #! ! ! !

& I 0 Maximum Genus Testing FACTS

* A2OFFD ! ! !

& ! " ! S I 1T ! ( 4 ! 0 ! ! + #!!

& AOFKD$

* A!HD 2 % + ! ! !

#! ! & ! ! +

+ ! ! ! !

"* A!HD 2 % + !

! ! ! * AOHD !+ ! 4

0! 0 ! & ! & 0 ! ! ! + ! ! ! = >

* AOHD ! = > ! ! + ! !+ & 0

REMARK

1* 2 F F ! ! M !

© 2004 by CRC Press LLC

638

Chapter 7

TOPOLOGICAL GRAPH THEORY

References A/Q CD 7 / ; Q1 + .& 0 + " F "H C$+ P

&

A/FD /! + / 7 # 1 ! ! 6 & + ! % K "HF$+ PC A/,-D /! + < ! + B Q+ 7!+ G+ B G+ I + U 1 & + + &+ I 1TR ! + $ " + A/QFHD /! Q 1 + / 7 # 1 ! 0 + ! % C "HFH$+ P A:Q7BCD < : + 2 Q+ B 7 + < ; B B + /& !

!+ & " CF "HC$+ KCKPKCF A:D Q :!+ # 0 + & - "H$+ P A!H-D < ! + % ! + ! + N& + "HH-$

"

+

!

A!HD < ! + / !

! ! & + &" ! $ " "HH$+ CPC A!/OHCD < ! + /! + < G O

+ &+ $ " H "HHC$+ HPH A!OHD < ! < G O

+ 7 # 10 ! & + % K "HH$+ --P

!

A!OHKD < ! < G O

+ I # & + ", % ,, + B /& / !# 1+ ; 0 "HHK$+ FPH A!7HHD < ! 7!+ O! ! + &" ! $ " "$ "HHH$+ HP A!7OHCD < ! + 7!+ < G O

+ / ! # ! 0 !+ $ " KC "HHC$+ FP- A2FD 2+ ' ! ! + & "HF$+ HP

" ( ) *

A2 GFFD 2 # G + I &

& 0 + ! & " K"$ "HFF$+ P H A2 HD G 2 + O + < + ' ! ! 4" $ + ++ & & " % "H H$+ P

© 2004 by CRC Press LLC

Section 7.2

639

Minimum and Maximum Imbeddings

A2OFFD 2 + < G O

+ G / O !+ 2 0

! + ! & " K"$ "HFF$+ KPK AOFKD Q I O # + .Æ ! ! + , H "HFK$+ -P- AOQ FD Q O & + < Q 1 + ! ! ! 6 & % + $ " "H F$+ PKC AOQ; HD Q O & + < Q 1 + 0 ;+ - ! ! ! 6 & + ! % "H H$+ P - AO7HD < G O

+ . ; 7 + O + ' ! & !+ H "HH$+ KPC AOHD < G O

O + G 0 % ! + % K "HH$+ KHP AO F D < G O

; 1 + . + ; 0 + HF

% % + & + -- 2

AQ #H-D < Q #

+ 0 ! + AQ D < Q " "H $+ KHPKCF

!

- ! " "FH-$+ PF

6+ .Æ +

! &

AQ ; D < Q < ; + G !

! ! + . & & " % "H $+ PF AQG--D B Q B G+ !

! #! ! & & + & " - "---$+ KPKK AQG--D B Q B G+ + !+ & " + "---$+ PF A< FD < + / ! ( 0 ! + " "H F$+ -P-C

% &

! ! ! O! +

& /

A7 D 7+ : ! 6 + "H $+ HHP-

! %

A7 CD 75 +

"HC$

A7-D 7 7 # 1+ V ! + 0 K "H-$+ PF AGG--D G B G+ + ! &+ ! "---$+ CKPCK

"

A HFD : !+ ' ! ! #! + $ " F "HHF$+ KPK

© 2004 by CRC Press LLC

640

Chapter 7

TOPOLOGICAL GRAPH THEORY

A HHD : !+ / ! ! + &" ! $ " "HHH$+ CPC AI FD G I 1T+ / # ! ( ! !+ ' " ! "HF$+ C-PC

)

AI FD G I 1T+ .& + ! + ! % K "HF$+ -KP- AI ;! D . I ! + : #+ / ;! + ' ! !+ ! % "H $+ KFPC A S HD I Q S + N & ! + $ " "H H$+ PF- AB CFD O < ; B + ! Q #

0 0 + & 1 & C- "HCF$+ FPK A FKD I

+ O! W & + KP ' +234+ /

+ N&

+ HFK A FFD I

+ O! SS ; R 6 + + "HFF$ A H-D I

+ O! E 0#! # 0 4 0 + ! % F "HH-$+ PK A H-D I

+ O! E / 7 # 1 ! + ! % F "HH-$+ KKPFF A HKD I

+ O! S + ! % C "HHK$+ CKP-

! 6 !

A HD + / ! + ! + HH U A1HD 1 & + ! ! # + F "HH$+ KPF-

$ "

U A1I FHD 1 & I + ! & 0 & ! + $ " F "HFH$+ HPFC A !FHD !

+ "HFH$+ KCFPK C

! ! I 0 +

A !H D !

+ ! ! + CH "HH $+ KPKF

! & "-$

! %

A !H D !

+ /

! ! !! 0 + ! % - "HH $+ -CP A C-D ; + & ! + , " PF

© 2004 by CRC Press LLC

- "HC-$+

Section 7.2

641

Minimum and Maximum Imbeddings

5 A; D 7 ; + N . ! 7 + " & "H $+ K -PKH- A;!D Q ;! + I 0 ! + "H$+ HPC

% & "

& ! "

KK

A;!D Q ;! + / & ! + "H$+ PK

AS HD I Q S + Q # ! !+ % C "H H$+ PK AS HD I Q S + N 0 ! + ! C "H H$+ CP

© 2004 by CRC Press LLC

!

%

642

7.3

Chapter 7

TOPOLOGICAL GRAPH THEORY

GENUS DISTRIBUTIONS ! "

I

O 2

'!

K ! N C /& O

%

Introduction ! ! ! 00 0 & ! % !+ #!! # O

2 AO2F D ! & ! / 1 +

! 0 ; ! #

=! > ! !& 4& & + ! 1

! ! #

7.3.1 Ranges and Distributions of Imbeddings DEFINITIONS

* ! ! & ! ! ! ! ! ! "$

*

! ! ! ! ! ! ! ! "$

* *

! + ! & A "$ "$D

! + "$+ ! 4&

! + 4& " $+ ! ! !

*

! . ! ! 4 #! "$+ #! "

$ 4 ( + # ! 4 ! + ! % 4 (

© 2004 by CRC Press LLC

Section 7.3

*

643

Genus Distributions

!

5 "

½

$ 3

"$

* !

+

1 # ! ! ! ! ! ! ! ! "$ * !

+

1 # ! ! ! ! ! ! ! ! "$ " *

! *

+ !

!

"$ "$D

& A

+ "$+ ! !

4&

. ! ! 4 #! "$+ #! "

$ 4 ( + # ! *

!

4 !

+ ! % 4 (

*

!

5 " $ 3

*

/

½

"$

# 6 ! ) # & # & ) I * )

#

Figure 7.3.1

*

/

# &',, !

# 6 ! ) # & ) I * )

0

&

Figure 7.3.2

*

!

# 6-',, !

! 4 ! 4

© 2004 by CRC Press LLC

#!

644

Chapter 7

TOPOLOGICAL GRAPH THEORY

FACTS

* ACCD 2 &

#! ! !

+ + #! &

"$ "$+ ! "$ &

* G

! 4

! ! 4&

½

"$ 3

¾

A "$ DX

! ! ! ! ! ! & & + ! & 5 "$ & !

* A FD 2 &

"$

"

#! !

! + + #! & $+ ! "$ &

* G ! ! ! 4&

" $ 4

½

"$ ,

½

"$

3

¾

A "$ DX

! ! ! ! ! ! & ! ! ! !

!

&

* AO2F D ! 0 # ! 0

! & ! & + 4 ! ! ! & ! !

* ! & 0 ) # ! #

4 ! ! 0 ) ! +

" ) $ 3 " ) $

* ! & 0 ) # ! ! 4 ! ! 0 ) ! + " ) $ " ) $ " ) $ , * .& ! 4 & 0 )

# ! ! ! 0 ) EXAMPLES

/ ! # ! " $ !

*

! !

! ! # 4 * - -

© 2004 by CRC Press LLC

Section 7.3

*

645

Genus Distributions

! 4

%¾

! ! # 4 * - -

*

!

"

! ! # 4 * - -

*

! !

! ! # 4 *

- - - -

*

! 0

! ! # 4 *

"- F- C-F C - - $

*

! 0

! ! 4

" - - - $ ! & 0

! ! 4

" - - $

RESEARCH PROBLEM

1 /&,

* ! ! ! & # ! REMARK

1*

! & !+ & + ! & + &

! C

7.3.2 Counting Noncellular Imbeddings ! ! # ! 0 ! !

&

1 ! DEFINITIONS

* / + #!! !& !

* / !

© 2004 by CRC Press LLC

#!

646

Chapter 7

*

TOPOLOGICAL GRAPH THEORY

/ & ! !

! !

! !

*

/ & !

& #!! !

.4& + ! !

"

*

O& !

+ !

! 0

0 & ! ! #! 1

*

O& !

+

"

0

& ! ! ! ! ! ! !

* O& ! + !

! 0 0 & ! ! #! 1

FACTS

*

.& ! ! !

4 ! !

"

*

! ! 6 &

#! ! !

*

.& ! ! ( &

*

.& ! 0

! 4 ! !

*

! ! 6 & 0

#! ! ! ! ! &

*

/

!

! +

! ! ! ! + ! ! ! ! & !

!

EXAMPLES

*

! !

! & + !

C 3 F - -

Figure 7.3.3

© 2004 by CRC Press LLC

! + !

! 4

Section 7.3

647

Genus Distributions

*

.! ! # ¾ ! %& 2& %& # " + ! 0 $ ! + !

%& M

! + ! + F + . + F

$

* .! ! ! ! %& #

! ! N !

3 + ! 4 . 3 K + # ! ! " $ 4 K , F 3 C / ! + # FF

7.3.3 Genus Distribution Formulas for Special Classes .& ! ! & + # ! & !4 # M ! &

! DEFINITIONS

* ! 6 ! ! ! ( ! ! ! !+ 2

Figure 7.3.4

' ' 6

* ! 7 ! ! & ! 0& ! ( + 2 K

Figure 7.3.5

© 2004 by CRC Press LLC

&& 7

648

Chapter 7

TOPOLOGICAL GRAPH THEORY

* ! . % ! ! #! & 0

+ 2 C

+, &.

Figure 7.3.6 FACTS

* A2OFHD ! 0 !& ! # ! *

"6 $ 3

-

½

,

! #

! # ! # !

!

6 6 6 F - C 6 C F 6 FF K C F -

C C KC -

*

A2OFHD ! ! !& ! # ! *

"7 $ 3

,

-

! # ! # !

! ! 7 C 7 C C 7 C F C 7 KC - C HC

*

AO FHD ! 4 !& ! # ! 0 *

"% $ 3 " $X #! !

"$ 3 8

8Y

"$

8 ! + " & $ $ A8 3 ! Æ $ D

#! ! % !

© 2004 by CRC Press LLC

Section 7.3

649

Genus Distributions

& ! <1

A<F D ! #

! &

"% $

*

"% $ 3 - - "%¼ $ 3 "%½ $ 3 - 2 3 "%¾ $ 3 3 -

2 * " , $ "% $ 3 " $" $" $¾" $ "% , " $" $ "% ½$

¾$

! &

"% $ 3 3 K

X 3 X 3 C FKX 3 C C--F X 3 K-CF CF- FK HX 3 CFF-

REMARKS

1* G 01 ! !

! Q #

0 " A D$ O ! AOF D ! = > =5

> A --D ! 0 = > 1* ! !

# 4 ( ! AHD = > 1* AH-D ! ! ! !

= >+ #!! ! #! # & 0

/ # + <1

+ E A/<EHD

1 0! !

1* ! AHD 0 ! 1* 7#1+ 7+ G A7#7G HCD

1 !

! & 1* 1 AHKD

1 !

! 1* ! AH D ! (

1* / ! 4 ! ! !

& ! &

& "

& $ 4 " AO D AO F D$

© 2004 by CRC Press LLC

650

Chapter 7

TOPOLOGICAL GRAPH THEORY

7.3.4 Other Imbedding Distribution Calculations + 0 4

! + !

#

! ! & B ! & !

4& # ! =

1 1 > #! &

& DEFINITIONS

*

! ! ! &

55 " $ 3 5 " $ , 5 "$ 3

½

"$

,

½

"$

* O& ! ! + + ! ! 3 A/ D & ! + 3 9#* "!$ / 3 3 # - ! #

! 9#*"!$ ! ! ! ! ! ! + , ,

* ! : * : * ! ! ! ; * ! ! * ! ! ! 2 & ; # : :

Figure 7.3.7

*,, 6 ! ,& :

:

FACTS

*

;! 0 !

+ ! + ! ! $ "$ ! # #

55 " $

3

¾

A "$ DX

* A FHD G !+ + + !

!

* " $ 3

© 2004 by CRC Press LLC

"!$ 3 " $ "!$ 3 " 0 $

Section 7.3

651

Genus Distributions

! + ! ! #! 0

"* A!OHD ! 0 4 4"$ ! #! + ! 1 ! & 4"¾$ Q # & + ! 1 !

&

!

* A!OHD 0 !& ! #

55 " $ 3

½

½

5

*<#

, *<#

, 5 " $

* A!OHD ! !& ! #

55 " $ 3

½

½

5

*<#

, *<#

, 5 " $

REMARKS

1"* A;!FFD

0 ! 1

# & ! -#1"$ ! ! ! 1* A7#G HD

0 ' ! 0# ! & " $ ! & ! ! ! ! ! 1* A7#!-D & ! 4 EXAMPLES

"* ! & 2 F+ ! ! + K+ C ! + ! # + + ! 1 ! ! 0 + ! " 2 H$

© 2004 by CRC Press LLC

652

Chapter 7

TOPOLOGICAL GRAPH THEORY

Figure 7.3.8 +, ! 6 , -

+

* & 2 + A!OHD ! ! ! #! ! + 2 H ! = > & + #!! & 1

67

7 6 , - & 2 + A!OHD ! + ! !

Figure 7.3.9 2

*

#! ! + 2 -

! = > &

7

Figure 7.3.10 *&&

* 1 + 2 ! # ! # ! C M ! !

Figure 7.3.11 / ,& !

: R G

.! ! ! ! ! & ! 0

© 2004 by CRC Press LLC

Section 7.3

653

Genus Distributions

+

& ! " ! ! $0 4&

+ 2

Figure 7.3.12

! , ,

7.3.5 The Unimodality Problem DEFINITIONS

*

/ 4

"

! ! ! ½ / ·½ / & #!

* / 4

4 4 FACTS

*

/ 4 % ! + 2

Figure 7.3.13

*

# , . ! ,- ,

A7 O D / 4& !

© 2004 by CRC Press LLC

¾ ·½

½

/

654

Chapter 7

*

A2OFHD

TOPOLOGICAL GRAPH THEORY

! & 0 !

*

A2OFHD

! & ! 0

*

AO

FHD

! & 4

REMARKS

1

*

+

; & ! ! 4 % ! +

! , ! !

! +

& +

! ! ! ! 4 %

! ! 4

,

#!! 4

1

*

AHD

! !

& !

+

1

*

AH-D

! ! 4 %

! 4

!

! !

RESEARCH PROBLEM

1 /&,

* #! ! ! & !

7.3.6 Average Genus DEFINITIONS

*

!

! +

"$+ ! & & !

! + 1 &

* ! ! + $"$+ ! , ? !

! !

*

AO7HD /

"* 0$ * , 00

* 6 ! 0

! ! 0 & !

*

G ! ; ! #

!

# # # & # ! ! # ! ! # # & !

*

;

!

! # # & 3 ! & 3 ! !

! ! 0

3

© 2004 by CRC Press LLC

Section 7.3

655

Genus Distributions

* ; ! * 0 ! ! ! + ! # ! #

GENERAL FACTS ABOUT AVERAGE GENUS

*

AO7HD ! & ! #! & !

*

AO7HD ! & ! ! &

!

"*

A!OHKD 2 0 !

*

+

"$ "$

A!OHKD 2 0 !

! ! +

"$ C $ "$

*

A!HD

! ! & ! &

*

AO2F D ! & ! 0 # !

4 "$ , ") $

* A!OHD G 0 !+

! !

"$

" $

)

!

"$ ,

FACTS ABOUT SMALL VALUES OF AVERAGE GENUS

* / ! ! & -

! ! & ! # AI ;! D

* *

! 1

! # AS HD

AO7HD ! & 1 "* 0$

© 2004 by CRC Press LLC

656

Chapter 7

TOPOLOGICAL GRAPH THEORY

* AO7HD .! !

& & (

1 2 ! & ! # ! ( ! !

Figure 7.3.14 1 8 ! - , 6 6

! 6

* A!OHD . 1 + ! ! 0 ! &

! ! 4 %¿ + ! " + ! ! !& &

K C

F

& 2 K ! # ! ! %& ! ! ! &

Figure 7.3.15 6 '

( 6

"* 2 H !

% ! &

! * A!OHD

! ! 0 ! #! & 4

Figure 7.3.16 '

( 6 .

FACTS ABOUT CLOSE VALUES OF AVERAGE GENUS

* AO7HD / ! ! 0 !

!& ! &

© 2004 by CRC Press LLC

Section 7.3

*

A!OHD 2 !

&

!

*

657

Genus Distributions

*

A!OHD 2 !

!& &

!

*

*+

% 0 ! !&

*+ % 0 !

FACTS ABOUT LIMIT POINTS OF AVERAGE GENUS

*

AO7HD

! !

&

&

*

A!OHD

!

& & 0 ! !

*

A!OHD

!

& & 0

! !

*

A!OHKD G # &

REMARKS

1

*

2 & +

!

1

*

/ & & AHCD+ AHHD+ AHKD+

AHKD

7.3.7 Stratification of Imbeddings 6 !

M

! ! #! ! # & !

!

DEFINITIONS

:½ : ! ! ! ! & 0 Æ : : * # ! ! ! ! ! ! & ! 0 : : * 2 ! + ! ! ! ! 06 ! 06 "* ! ! ! !

! * / ! ! & ! !

*

#

! &

M & !

! !

FACTS

*

!

© 2004 by CRC Press LLC

"$

658

Chapter 7

TOPOLOGICAL GRAPH THEORY

"* AO HD ! ! % !+ ! + !

* AOHD ! !& ! * AOHD I +

!

0 + & ! ! ! #

* AO HKD 2 & & ! % ! + ! ! !

! & ! ! REMARKS

1*

! #! A !FHD+ #!! & ! ! I 0

1* ! #! A2OOFFD+ #!! ! 0 0 ! 1"* AO HKD

! # # ! #!

!& 1 M ! % !

! !

! + ! !! 0

References A/<EHD O . / # + <1

+ E + / !

!

0 + &" ! " & K "HH$+ PKK A!HD < ! + / 0 !

! ! & + &" ! $ " "HH$+ CPC A!OHD < ! < G O

+ G & "$* 0 0 ! + ! % KK "HH$+ FP- A!OHD < ! < G O

+ G & "$* 0 0 ! "#! < ! $+ ! % KC "HH$+ -FPH A!OHD < ! < G O

+ 7 # 10 ! & + % K "HH$+ --P

!

A!OHD < ! + < G O

+ O + '& 0 + $ " F "HH$+ PH A!OHKD < ! < G O

+ I # & + % 5 5 & 5 / + (6)5 "5 +227*+ FPH+ ; 0 + I # B 1+ HHK A!OHKD < ! + < G O

+ O + G # ! & + ! % H "HHK$+ FPHC

© 2004 by CRC Press LLC

Section 7.3

659

Genus Distributions

A2OFHD G 2 + < G O

+ + O #

! + ! % C "HFH$+ PC AO2F D < G O

G 2 + Q ! 0 &

!+ ! % "HF $+ -KP- AOHD < G O

O + G 0 % ! + % K "HH$+ KHP

!

AO7HD < G O

+ . ; 7 O + ' ! & !+ H "HH$+ KPC AO FHD < G O

+ + ; 1 + O 4 + ! % "HFH$+ HP-C AO HD < G O

; 1 + G ! + & 8 & H "H H$+ KPK AO F D < G O

; 1 + % % + & + -- 2 . + ; 0 + HF AO HKD < G O

; 1 + % ! + " "HHK$+ PFC

$

A<F D <1

+ ! + #! + % & " HH "HF $+ FKPF- A7 O D < 7

Q O + + & CC "H $+ FCPFH

! &

A7#7G HCD < Q 7#1+ O 7+ < G + ! 0 & + $ " KC "HHC$+ P - A7#G HD < Q 7#1 < G + . ! + K "HH$+ HPK

$ "

A7#!-D < Q 7#1 Q !+ 4 + $ " F "--$+ HP-F AHCD 1 !+ ! #! & + $ " KH "HHC$+ FKPH- AOF D G / O !+ !

! + 0 N& + HF

A FHD : !+ / ! + "HFH$+ KP

$ "

F

A;!FFD : O + O + / ;! + . 0

! + & " - "HFF$+ P- AH-D O + !

© 2004 by CRC Press LLC

! + ; ! N& + HH-

660

Chapter 7

A D O +

" % + 0E + H

TOPOLOGICAL GRAPH THEORY

AHKD / 1+ ' ! !

+ & "HHK$+ HPHF AHHD !(+ & + ! % C "HHH$+ PC

AH-D !+ ! + $ " F "HH-$+ K P F AHD !+ 0 * . 0 ! + ! % K "HH$+ HPF AHD !+

! + $ "HH$+ FPHH

" FH

AHD !+ ' ! ! +

! "HH$+ HPC AHKD !+ : ! & ! & 0 ! + $ " "HHK$+ KPK AHKD !+ ' ! & ! !+ ! % - "HHK$+ PF AH D !+ ' ! (

+ C PC- A --D . Q KPK

+ O +

A !FHD !

+ "HFH$+ KCFPK C

! " $ "

! ! I 0 +

C "---$+

! & -

AS HD I Q S + Q # ! !+ % C "H H$+ PK

© 2004 by CRC Press LLC

H "HH $+

!

Section 7.4

7.4

661

Voltage Graphs

VOLTAGE GRAPHS ! " E O!

I E + G O + I / !

E O!

& #! E O!

K ! 7!! M E G# C E O!

O!

F G E O!

H / E O!

Introduction ! & ! + ! #! " =& 0 >$ % ! #! / ! %

! 0

0& ! " = 4 >$ ! + & ! ( !

7.4.1 Regular Voltage Graphs ! & ! ! / + & ! ! ! & ! !

& + ! ! 6 DEFINITIONS

! & ! # # AO D ! % ! ! K C &

*

G 3 " $ ! / ; * ! ! #! & ;"$

! * O! ! ! ! " $ ! * !

#! & & ! 3 " $ * % # * " $ 3 3 + ! " $ 3 3 ! & & + ! ! 3 " $ ;

;

;

! &

© 2004 by CRC Press LLC

#

#

3 "# $ ! &

3 " ;"$$

662

Chapter 7

TOPOLOGICAL GRAPH THEORY

E ! & ! % ! + ! ! ! ! & & ! ! + + ! & + " $ !

!

! ! &

& 0 +

* ! / " $ #! ! !

& ! & ; #

! & ! /! ! & " + ! $ &

+ +

# # !

! C+ ! #

EXAMPLES

*

2 ! # ! # ! !

! K K ! # 0

Figure 7.4.1

# 6

,

*

%

! *

2 "$ & ! ; * ! & !

Figure 7.4.2

+ 2

"$

# 6

, 6

.! 0 ! & ! 6 # #0& + ! ! 0

& # ! & + !

© 2004 by CRC Press LLC

Section 7.4

Voltage Graphs

663

& 0 !

! ! & # & + ! ! 0 ! & !

#0& 0& ! ! 4 ! 0 + & - TERMINOLOGY

0 6 6

/

& ! &

0

& ! ! ! 0 ! & ! ! + ! & & #! ! · ·

!

! ! ! ! + #! " $ ! REMARK

1*

! & ! # ! + #!

Fibers DEFINITIONS

*

G ! & ! & ! 3 " $ ; * ! & 3 * ! $% &

+ ! 3 * ! $% &

* G 3 " $ ; * & ! ! ! ! & ! ! & ! & ! & & + ! " ! + ! 6 & = >$ EXAMPLE

7 * 2

+ ! #¼ #½ #¾ " $ "#! # # ! =#0& >$ ! & % & # ! ¼ ½ ¾ " $ "#! # # ! =0 >$ ! % & FACT

*

! % ! & ! ! ! & 0 ! & ! % + ! #! & + ! 0 ! & ! % + ! #! & Bouquets and Dipoles

2 + ! ! #! # &

© 2004 by CRC Press LLC

664

Chapter 7

TOPOLOGICAL GRAPH THEORY

DEFINITIONS

* *

! . % ! 0& ! #! 0

! " ! # 0& ! #! 6 ! # &

EXAMPLE

E ! ! & ! # #!

+ ! ! %

*

2 ! # ! 2 + ! & !+ % &

Figure 7.4.3

6

, 6

2 "$ & #! 0& % + 2 ! + # !& =

! > ! & % ! & !+ ! #

! & + # !&

! ! & ! 2 "$ & #! 0& " N ! #

& ! ! & ! & ! ! + #!

+ ! ! & ! ! ! % 6 & #! M ! 2 "$ & ! #

! ! ! #! 0& " ! ! / AO F D AOB HHD FACTS

*

! ! #! "$

0 ! ! & ! % ! 4 % #! &

& 0

* ! ! &

! & ! ! ! 4 % " !

2 $

* ! ! &

! & ! ! ! 4 % +

! ! ! &

© 2004 by CRC Press LLC

Section 7.4

665

Voltage Graphs

* ! ! &

&

- ! ! ! "

* ! 0 ! &

! & ¾ ! ! "

7.4.2 Net Voltages, Local Group, and Natural Automorphisms Net Voltages DEFINITIONS

* / #1+ ! & ! # ! !

0 ! ! ! &

* ! . #1 = 3 ¼ ½ ½ ¾ ! 4

& ½ + #! 3 ;" $ ;" $ ½ + #! ! & ! # 1# + &

"

* ! & ! ! ! & 4 EXAMPLE

7 * ; ! & ! 2 &

Figure 7.4.4

+ !

# 6

, 6

! #1 = 3 # # # ! & , - , 3 ; & ! ! #1 #¼ ¼ ½ ½ #½ ½ ¾ ! & !+ #!! # ! #1 = + ! & % & ! ! K The Local Group DEFINITION

* !

& & ! 3 " $ ; *

! ! ! & #1 !

& G

© 2004 by CRC Press LLC

666

Chapter 7

TOPOLOGICAL GRAPH THEORY

FACTS

*

A/O CD ! & + ! ! ! & & 0 + ! ! & 6 ! ! &

* A/O CD 2 & ! ; * + !

! & ! 4 ! A * G D ! ! &

*

A/O CD ! ! & !

!

EXAMPLE

7 *

! 2 "$ ! 0 #! & R ! ! # ¿ ! + ! #

! & ! REMARKS

1*

#

& * ! !+ ! ! ! & ! ! & * "! + ( ! + & 0 $ & ! & !

1

* ! ! ! # & A/O CD ! Natural Automorphisms

! ! ! ( & ! ! & ! DEFINITION

* G ; * & !+ > * & !

#

#

!

! + & # & ! !+ ! & # & ! & ! FACTS

AO F D

"*

! % & ! + ! & ! & ! ! & + & + #! ! % " ! ! % $

*

! & ! & #! ! & % & ! #! ! %

*

G ; * & ! / ! & ! ! & ! % & ! ! > G

© 2004 by CRC Press LLC

Section 7.4

Voltage Graphs

667

EXAMPLE

7

* ! ! >!" 2 "$ 0 ! & ! ! & R

7.4.3 Permutation Voltage Graphs ! & ! AO D ! 6 ! + ! ! ! ! & ! ! "2 K$ DEFINITIONS

* G 3 " $ ! / Y ; * Y ! ! #! !

! ; Y

O! ! Y ! ! ;"$ ! * ! $Y %

#! & ! 3 " $ ; * Y % # * " $ 3 3 + ! "$ 3 3

! & # & ! ! 3 " $ ! & # 3 "# $ ! & 3 " ;" $$ EXAMPLE

*

2 K ! # Y0 & ! ! & !

Figure 7.4.5

9 : # ;¿'6 4 9 : 6

! % & 0

& # ! ! ! "# # # $+ ! & "$ ! % & 0

# 6 " $" $+ ! & "$"$ & # & ! !+ ! ! % &

! & % & # ! & % & + ! & #! ! & "$

© 2004 by CRC Press LLC

668

Chapter 7

TOPOLOGICAL GRAPH THEORY

FACTS

*

AO D .& ! & %

0 & ! 4 % "

A1FKD$

*

2 :5 R ! A:5 FD " 2 C K$+ # ! & 0 0 " , $0 ! %

& ! 4 % + 0 ! % &

*

/ ! & &

& & ! # ! ! ! !

REMARKS

1

* / ! + ! & 0 0 Y0 & ! & & 0% + ! #! & + 0% + ! #! / + 6

1

* 2! & ! & AO F D AOB HHD

1

* & ! # #! + #! ! ! & & & 2 & + ! & ! % 2 & + ! X &

1

* +

! 4 %¾ #! ! # ! "#! ! :6 $

! ! & ! ! 0 :6 ! ! & ! ! &,,% 3 !

7.4.4 Representing Coverings with Voltage Graphs & + & & ! %

& ! & 0 & ! & + ! ! !

+

! ! & ! & & ! ! & & % + ! Coverings and Branched Coverings of Surfaces DEFINITIONS

*

G 9 + 9 * 9 + ! ! ! # ! * .& ! ! !

? ! ! ! 9 "? $ ! ! 9 ? ! 9 * 9 ! 9

© 2004 by CRC Press LLC

Section 7.4

669

Voltage Graphs

*

G 9 * 9 & 6 2 ! ! &

+ ! 9 ½" $

*

9 % 9 ! ! G 9 + % 9 9 9 & 6 ! ! ! 9 * !% ! 9 * 9

! 9 ! %9 !

! ! !

EXAMPLES

*

! & 6 ! ! ! % & ! #$ #$ & + ! ! & ! 1 ! + #!! - ! !

* * *

!

! & ! A/-D .& ! & ! !

! ! 3 " $ @ * , , 3 - ! " $ " $ ! 4 ! ! ! & 6 ! 6 & & + ! ! & 6 ! # ! =

> ! = > & 6 !

5

REMARK

1*

! ! & ! ! + ! !

!

Using Voltage Graph Constructions

/ # & ! & ! FACTS

G 3 " $ ; * & ! !

* * *

"$ 3 "$

! ! #

"$ 3 "$

! % & & "$+ & & ! !

/ ! ! % ! & !+ ! # * & & ! % & & "$

! "/ ! #! 0

!& $

*

9 AO HD G + ! !

0& + !

& ! +

© 2004 by CRC Press LLC

670

Chapter 7

TOPOLOGICAL GRAPH THEORY

EXAMPLES

*

! ! & ! !+ # & ! ! ! ! - & K ! & &

- K K+

2 C+ # 1 ! #! &

! ! ! #! 2 + ! ! 0& + #!! # !

# ! '

! # + ! "¿ + 0 : 2 F+ ! ! ! ! & ! + ! # 2 C+ !

!

% $,

Figure 7.4.6

# 6

, ! /

; 1 0& ! !+ ! K : 2 H+ #

! & - ! 2 C ! # !

"

* : + # ! ! !

& & & ! ! & C +

! & &

: !

. H+ # # !

# ! !+ # # !

& /

- 0

! % 0

/

! & ! 0

0& C0 ! #! # 6 + #! 0 + Action of the Group of Covering Transformations

! % ! #!! & ! &

& ! 4 #! ! DEFINITIONS

* G 9 * 9 & 6 / ! ! 9

9 3 9 '

! ! ! ! + % ! %

"*

* / ) & & 6 9 * 9

) ! ! % 9 / ) & & 6 9 * 9

! % &

*

/ & 6 9 * 9 ! ! ! & ! + ! 9

© 2004 by CRC Press LLC

Section 7.4

671

Voltage Graphs

! ! = & ! > &

( ! ! 9 ! ! ! ! 9 ! & 6

+ + 0

#! (

EXAMPLE

! ! #$ #$ & ! + &

! % & & ! ! + ! & 6 !

"7 *

FACTS

"* AO D G ; & ! & !

! ! & !

* AO D G

( ! !+ 9 & ! ! ! ! ( ! & ! &

"=.& & ! ( &

>$

* AO D G ; & !

& !

! ! & !

* AO D G ( ! !+

9 & ! ! ! ! ( ! & ! &

"=.& & ! ( &

>$

* G ; & !+ #! & !

! ! & !

! ! & ! ! Y0 & ! ; Z + #! Z ! ! ! Y "=.& &

&

>$

* G ; & !+

! ! & + ! & ;"$ & ;"$ + ! ! & !

! ! & !

* G

9 * & 6 ! + ! ! ! ! & ! 9 * ! & ! 1 + #!! - ! ! ! - ! ! !

7.4.5 The Kirchhoff Voltage Law ; ! ! 7!! M & #

! ! & ! ( ! ! + & ( + ( ( !

© 2004 by CRC Press LLC

672

Chapter 7

TOPOLOGICAL GRAPH THEORY

DEFINITIONS

* G = 3 #1 & ! ; * + & 4 G ! ! #1 =

3 " $ " $ " $ "

! #1

*

=

$ " $ " $

G = 3 #1 & ! ; * Y + A A & 4 G ! ! #1

! #1

=

" $

A

A

" $

A

"A" $$

" "A" $$$

A

" "A " $$$

A

* G = #1 & ! ! & = ! ! & + ! # ! 0 $12% ! = FACTS

*

AO + O D G = #1 & ! ! 7!! M & # ! = + ! & = = #1 ! & !

* AO D G = #1 & ! ; * + #! & G !& ! & ! ! !

4 = =% =% =% #1 ! & !

N ! ! ! 2 F+ ! ! #1 = 0 ( 4 + ! + #!

#1 ! & ! ! #1 ! 0 =

* AO D G = #1 & ! ; * Y + #! & A G A !& ! & Y ! ! ! 4 = =& =& =& #1 ! & !

N ! ! ! 2 H+ ! ! #1 = 0 ( 4 + ! + #!

#1 ! & ! ! #1 ! 0 = + !

7.4.6 Imbedded Voltage Graphs & ! ! + ! + ! ! & ! & ! " $+ ! " K$+ " C$

© 2004 by CRC Press LLC

Section 7.4

673

Voltage Graphs

DEFINITIONS

* G [ ! #1 ! & ! ! ! [9 ! + #! [+ !

* G ; & ! ! ! ! ! " $ ! ! ; ?

+ ; ! + !

;

=

*

=

9 ! #1 ! & ! 0 G [ & ! ; ! 0 ! % 9 ! #1 [ "#! 4 ! ! ! ! #1$ ! !

* G ; & !

! 6 * ! + ! 6 9 ! ! ! "#! ! #1 ! $+ #! 2 F ! !

9

* "*

/ #! #1 ! ! 4 / #! #1 ! ! 4

FACTS

"*

G ; & ! ! ! & + ! & ? & + !

+ !

! &

* AO/ + O D G ; & ! ! 7!! M & # ! ! #1 & ! + ! ! 6 9 * 9 & 6 7EG ! + ! ! 6 ! &

EXAMPLES

*

2 ! # & ! #!! ! ! ! 4 %¾ + ! ! + ! & ! B ! & ! ! ! ! & ! ! + 7EG ! #1 ! + ! ! & ! ! ! + ! 6 & 6

Figure 7.4.7

9 : ,& 6 4 9 : 6 ,&

© 2004 by CRC Press LLC

674

Chapter 7

TOPOLOGICAL GRAPH THEORY

*

2 F ! # ! & ! #!! ! ! ! 4 %¾ ! & ! B + ! ! ! ! & ! ! !

Figure 7.4.8

# ,& 6

%

! ! + + # + 7EG !

! #1 ? + ! & ! #1 !

K ! B ! + ! ! & ! K

! +

! # K0 -0 ? ! 6 ! & 6 + #! ! ! ! . ! ! & 3 K -,+ # ! ! &

*

AO D .& \0 #! 3 1 3 1 01 3 0 0 1 0

! ! / & E ! !

!& & #! ! & !

7.4.7 Topological Current Graphs ! & ! # ! % " AO/ D AO D$ ! --0 0B

AB CFD "

A D$ ! 4&% ,2 $ , AQ *FH-D+ #!! ! !

& ! ! & & ! ! !& DEFINITIONS

* G ! #! & 0 0 + / ; ! & ;"$ ! ! ; + !

* G 3 " $ ; * ! ! & ! #! + ! ! "#!! & & & ! +

$ 2 ! + # % ; " $ 3 ;"$

&

* ! ; * ! ! & + ! + ! & ! ;

© 2004 by CRC Press LLC

Section 7.4

Voltage Graphs

675

* G & ! ! ! ! + ! # ! 0 $/2% ! " + ! ! ! ) # 0 + ! ! ! $ EXAMPLE

* 2 H+ ! # ! # + #! ! ! ! !+ ! ! ! & #! ¾ # +

! 1 ! ; & ! 7EG ! ! ! & ! #! + 7G ! ! & ! !

Figure 7.4.9

6 ,& !,

REMARKS

1*

O R AOCD+ ! # 0 ! #! & # 1 #! & #1 ! & + #! # 1 #! 0 B O R = > "B R $ & !+ ! #! % + #! & ! ! &

& ! Q #

! 0 & # ! ! & ! #1 # " $ ! % 0 !

1"

* ! ! ! AO/ D ( ! & 0 ! 1 H % + 6 ! + # & ! #! 0 + % ! ! & ! !

& ! AO D ! ! + #!! & 00 4 ! + . H ! ! & ! AO D !

© 2004 by CRC Press LLC

676

Chapter 7

TOPOLOGICAL GRAPH THEORY

1

* ;! ( ! 6 ! & & !+ ! 6 ! ! & !

! !+ #!! & ! A CD ! 0 ! ! & ! ! ! # A <F-D AO< FD / M & & & A/HD

7.4.8 Lifting Voltage Graph Mappings ! # ! + ! 0 ! ! # # !

! # ( ! # # & &

! M ! DEFINITIONS

* G * ) ! ! ! + ! ! "/ ! ! ! $

* G * ) ! ! ! !

! ½ "$ "/

! & 4 ! !

& & ! & +

! ! $ ! * ) ! & + 1 & & ! !

* G * ) ! ! ! ! ! ½ "$ "/ 1 ! 4 !

! 1 & ! & +

! ! $ ! * ) ! + 1 & ! ! FACTS

*

AO!HCD / ! ! ) ! & ! & ! )

* A/O&H D+ A/1H D / ! ! & ! & ! )

*

AO!HCD / ! ! ) ! (

! & ! & ! ) REMARK

1*

AH-D

'! AI 1--D+ A--D+

© 2004 by CRC Press LLC

Section 7.4

Voltage Graphs

677

7.4.9 Applications of Voltage Graphs : ! ! ! #! & + & ! !& 4 & ! REMARKS

1* ! & ! ! ! #! ! & A/OFD+ A/7#G --D+ A2 7#-D+ A7 H D+ A7#Q G --D+ A7#G HHD+ AG FD+ AH D+ A--D+ A-D+ A-D 1* 2 & & & !+ A7#G -D 1* / ! ! & ! & ! A.#Q HD+ A2 7#7G HFD+ AQ HD+ AQ HD+ AQ HKD+ AQ HKD+ AQ 7#HD+ AQ 7#G HCD+ AQ 7#G HHD+ A77G HHD+ A7#!G HFD+ A7#Q G --D+ A7#G HD+ A7#G HD+ A7#G HFD+ A7#7G HCD 1* 2 & + A/G-D 1* 2 ! E

& 1 & + A -D 1* 2 # & ! ! + AG FD 1* 2 # & ! # 1 + A -D 1"* 2 ! & ! !

! ! & + AHD A-D

Applications of Imbedded Voltage Graphs and Topological Current Graphs

& ! ! !& 0 #! & REMARKS

1* 2 #! +

! ( !+ A FKD+ A FCD+ A F D+ A FFD+ AI 1H D+ A F-D+ A F-D+ A FD+ A HD

1* 2 ! & ! ! + A7#!G HFD+ A7#7G HCD+ AG 7-D+ AG 7-D+ A F D 1* / & ! ! ! " K$ AO D+ AOG F-D+ A<;!F-D+ A HD+ A FHD+ A D+ A FD+ A H D+ A ;HD+ A F-D+ A FD+ A FD 1* E ! !& ! 1 A/ KD+ A:QF D+ AO HD+ A;! FD+ ! 1*

! & ! ! % " H$ ! / ;! A;!-D

© 2004 by CRC Press LLC

678

Chapter 7

TOPOLOGICAL GRAPH THEORY

References A/ FD /0 !

+ . ! + % "HF$+ KP A/-D < ; / + I + -P

& "

A/ KD / + # ! + ! & F "H K$+ -P-

!

C "H-$+

% 5

A/O CD / < G O

+ ! & ! + ! % 5 - "H C$+ FP- A/G-D / / I G+ ! & * O ! ! &+ "--$+ PF A/OHD / / O + 2 & ! + ! % 5 - "HF$+ FPF A/HD /! + ! ! & 0 + - "HH$+ P

$ "

] + A/O&H D /! + O& (61+ < ] ! + " ' "HH $+ PH A/7#G --D /! + < Q 7#1+ < G + B !+ : & ! + $ " "---$+ KPC ] + 1 & + ] A/1HD /! + : ! + < ] :! &0 ! ! + & ! H "HH$+ -HP 5 A:5 FD 2 :5 + N \ 5 1 '+ " # ' - "HF$+ KPF A:QF D I : ; QM+ & 0" + + $ #! 0 + ! F "HF $+ KPC A.#Q HD .# Q + ' & ! ! #! & + & K "HH$+ F PHC A2 7#7G HFD ^ 2 + < Q 7#1+ < 7+ < G +

!

! & + &" ! $ " "HHF$+ CKP A2 7#-D B ^ 2 < Q 7#1+ % 0 ! + ! "--$+ KKHPKCK AO HD : G O+ E ! !

1 + % "H H$+ KPC

© 2004 by CRC Press LLC

!

Section 7.4

679

Voltage Graphs

AO D < G O

+ E ! +

$ " H "H

$+ HPC

AO D < G O

+ .& ! & ! !+ % 5 "H $+ P

!

AO/ D < G O

/ + ! ! ! + % 5 "H $+ FP

!

AO!HCD < G O

< ! + / % # 1

& ! ! + " % H "HHC$+ KP - AO< FD < G O

+ : <1

+ 1+

+ ; & 0 ! ( + K "HF$+ P AOG F-D < G O

< G + / ! 10 + ! % "HF-$+ CKP AO D < G O

; 1 + ^ ! * & ! Q #

0 + 9 ! " KK "H $+ HP- AO D < G O

; 1 + O ! & &

+ $ " +3 (+2::*+ PF AO HD < G O

; 1 + 2 & ! ! + & 8 & H "H H$+ KPK AO F D < G O

; 1 + % % + & + -- "2 . + ; 0 + HF $ AOB HHD < G O

< B + HHH

% &+

+

AOCD+ ; O + ' ! + "HC$+ P K

& "

- ! "

"FH-$+ PF

AQ HD Q + ! & 6 + "HH$+ PF

&

AQ *FH-D < Q #

+ 0 ! +

CH

AQ HD Q +

! ! ! & + " HF "HH$+ KPF

$

AQ HKD Q + . & 6 + $ " F "HHK$+ KPC

&" !

AQ HKD Q + O! & 6 % & 0 & % 0% + $ " "HHK$+ F PH AQ 7#HD Q < Q 7#1+ & #! ! ! + ! % "HH$+ CPC

© 2004 by CRC Press LLC

680

Chapter 7

TOPOLOGICAL GRAPH THEORY

AQ 7#G HCD Q + < Q 7#1+ < G + ! & #! & !& !

! + $ " F "HHC$+ FKP-K AQ 7#G HHD Q + < Q 7#1+ < G + : ! #! % + & " KH "HHH$+ KPC A< FD : <1

+

+ 1+ / ! ! + ! % K "HF$+ KKP A<;!F-D < / ;! + ' ! % + ! "HF-$+ PK A77GHHD Q 7 7+ < Q 7+ G+ I

!

! & !+ K "HHH$+ PC A7 H D < 7 !&+ / 1 # 1+ < / + & ! + ! % 5 "HH $ A7#!G HFD < Q 7#1+ < Q !+ < G + . ! &

!& % & + &" ! $ " "HHF$+ PFK A7#Q G --D < Q 7#1+ Q + < G + B !+

#! & !+ & K "---$+ HPC A7#7G HCD < Q 7#1+ O 7+ < G + ! 0 & + $ " KC "HHC$+ P - A7#7#-D < Q 7#1 B 7# + ! !

!& & ! + , & & C "--$+ HHP F A7#G HD < Q 7#1 < G +

! 0

! + $ " -K "HH$+ P A7#G HD < Q 7#1 < G + . ! + K "HH$+ HPK

$ "

A7#G HFD < Q 7#1 < G + ! 00 & + $ " F "HHF$+ HP A7#G -D < Q 7#1 < G + . ! & + ! - % % + < G O

; 1 + N&

+ -- A7#G HHD < Q 7#1+ < G + B !+

! 1 & + ! 6 % ; F "HHH$+ KP- AG FD O G + / ! & ! % % + $ " C "HF$+ CP

© 2004 by CRC Press LLC

Section 7.4

681

Voltage Graphs

AG 7-D < G < ; 7+ . ! ! \" $0 &

+ ! "--$+ KPF AG 7-D < G < ; 7+ ' ! & ! ! + F "--$+ HP AG FD 2 G ! + 2 & ! + "HF$+ PF

! % 5

AI 1--D / + I + 1 & + G ! ! &

+ ! "---$+ H PH AH D Q ( + !

& ! + & K "HH $+ P A--D Q ( +

! & ! + & K "---$+ KPC A--D Q ( + \ ! & + ! F- "---$+ PK

% 5

A-D Q ( + G0 ! &

+ $ " KC "--$+ KP A-D Q ( + G0 & ! + "--$+ PH

!

A FKD : !+ /1 #! & + $ " K "HFK$+ PH A FCD : !+ / & ! 0 + ! - "HFC$+ HP-C A F D : !+ ! 1 + ! "HF $+ P

% 5

% 5

A F D : !+ ! +

! % 5 "HF

A FFD : !+ :! 0 & +

$ "HFF$+ HPF

$+ CFPFC

A -D Q & + # # 1 + < C "--$+ KP AI 1H D I 1 & + &

0 + ! F "HH $+ F- PF A F-D

+ ! + ! -P-

% 5 H "HF-$+

A <F-D

+ 1+ : <1

+ # 4 0 & ! + $ " F "HF-$+ PK

© 2004 by CRC Press LLC

682

A F-D

Chapter 7

TOPOLOGICAL GRAPH THEORY

1+ O ! +

% "HF-$+ P

!

A FD 1+ I ! + ! % C "HF$+ HP- A HD 1+ ' 4 ! + " -H "HH$+ -P-K

$

A FHD 1 ; 1 + ! #! + ! - "HFH$+ CHP K A HD

1 ; 1 + F "HFH$+ K PC

$ "

! # 1 Q +

A ;HD 1+ ; 1 + ; + + "$+ PF

!

A -D G !+ E ! + # ! + C "--$+ F P A D E 7 + & + H

!

% + !

A FD E 7 + ' ! 0 + % "HF$+ KPKF A D O +

" % + 0E + H

$ "

! + N0

& "

CC

AB CFD O < ; B + ! Q #

0 0 + & 1 & C- "HCF$+ FPK A-D +

! & ! + + C "--$+ PFC AHD +

!

! & + C

$ "

&

F "HH$+ P

A1FKD < 1 & + ^ ! & + ! % 5 F "HFK$ PK A HD !+ 0 ! + "H H$+ HP-

! % 5

A;! CD ! / ;! + O

! + $ " "H C$+ HPHC AH-D : # 1+ G ! ! ! + ! % 5 K- "HH-$+ KPH A CD / ; 1 + :! & + & " FKHPFC

© 2004 by CRC Press LLC

"HC$+

Section 7.4

683

Voltage Graphs

A F-D ; 1 + I & + % & "HF-$+ C P H A FD

;

"

KF

1 + 2 0 ! +

% 5 "HF$+ FPHF

!

A FD ; 1 + / % Q#( ! 0 ! + ! % 5 C "HF$+ PCF A;! FD / ;! + : 1 ! + K "H F$+ CCPF A;!-D /

;! +

© 2004 by CRC Press LLC

! % 5

+ I !0Q + --

684

7.5

Chapter 7

TOPOLOGICAL GRAPH THEORY

THE GENUS OF A GROUP

#! $ K O!

K ! 0Q#( .4 Q#(R

K O G # O

K O 2 O

KK I

!

Introduction ;! ! " $ % + ! + ! !+ & ! + ! + # ! ! & ! ! ! + !

! ! !

! ! ! !

! ! !

! $ $, ;! A;! +;!FD % ! + ! ! : A:D ! 4 + ! & + !

7.5.1 Symmetric Imbeddings of Cayley Graphs ! ! " $ + ! 1 #! ! ! & 0 & ! !

! # ) ! + DEFINITIONS

* ! / " $ #! ! !

& ! & ; #

! & ! ; ! & &

+ ?

# # !

!

* ! + #!! "$+ ! ! !

! ! !

© 2004 by CRC Press LLC

Section 7.5

685

The Genus of a Group

* ! ! ! ! " $ !

! ! & !

" $ ! " ! ! ! + "$ 3 " $$

* ! ! ! 1

& ! &

* ! ! ! ! & & %

* ! % !

! ! !

!

* ! &

! ! ! & !

* / ! " $ ! ! " $

* / ! " $ ! ! " $ 0 & "*

! " & + $ !

+ C"$ " & + C "$$+ ! ! !

! " & + $ !

/ ! !

/ ! A:;! HD FACTS

* *

! % ! "$

! ! &

C"$ C "$

! ! " $ & 0

* AKFD / ! ! ! !

!

! + ! ! ! & 0 &

*

AO F D / 0 & ! ! & "! ! & & ! $+ ! ! ! + ! + ! !& !

! & & & + &

* 2 ! ! ! " $+ # & ! ! &

* ! & ! & ! 4 + #! !

& ! & + & ! ! " $ .& ! ! #

© 2004 by CRC Press LLC

686

Chapter 7

TOPOLOGICAL GRAPH THEORY

* AO F D / 0 & ! ! &

! & ! ! !

& + ! ! ! & 0# ! !

+ ! # ! ! & !& ! & !& ! + & * E # ! & !

" # & $ & ! '

#! & ! 4

! ! ! & / ! #! # 0 & ! !& 0 & & ! ! ! # & * AO F D ! & ! # + ! !

& + ! ! # ! # &

& + ! ! "$ ! & + ! ! & ! # + !

"* A FD / ! % ! ! " $ ! & 0 + ! ! ! C"$+ & C "$+

! ! ! + & + & +

* + ! C"$ C"$ C "$ C "$

+ !

* A: D + ! ! 0 ! + "$ "$

EXAMPLES

* E # ! !& #! & G ! H- ! !

G ! ) ! ! ! 1 ! & ! ! ! !

& 0 & + ! & + ! 3 ! + ! & ! & + ! !

*

¾

3 3 3

3 " "

* G ! & + ! F-

! & ? ! ! ! +

! & # ! ! ! ! & 0 & + ! &

+ ! 3 ! + ! & !

& ! !

*

3 3 3

#!! ! ! F

© 2004 by CRC Press LLC

Section 7.5

687

The Genus of a Group

REMARKS

1*

! & + ! & & + #! ! % ! % + ! & &

? ! ! 1 ! ! & !& ! & ! !

+ ! &

!& ! ! & +

! & ! &

% ! #

1

* ' 0 & ! ! % + ! & + & ' !1

+ #! ! # * ) 2 + !+ # & + ! ! !& ) ! & % ! & :

! ! !& !

! ! &

! ! !& + & & % + ! ! & ! "1 ! $ AO F D

7.5.2 The Riemann-Hurwitz Equation and Hurwitz’s Theorem O& & ! + ! . ! ! 0 ! & + ! ! & + ! . ! + ! ! & ! /

& 0& & ! + 0 ! ! ! # DEFINITIONS

* ! ! ! " $ ! ! + ! & ! & ! ! 4 % ! ! ! 0 & ! !

*½ * ! ! . ! "+ $ ! 3 -" . "#! ! ! $*

"+ $ 3 " " $

*

! 4

*

*

!

! " & !$ #!

& ! * ! ! &* ! ! & +

! = > + #!! %

*

" &

!+ ! ! 0Q#( 4 $

* / 4 ! ! 0 !

& ! ! & + ! !

© 2004 by CRC Press LLC

688

Chapter 7

TOPOLOGICAL GRAPH THEORY

+ ! 4 ! !

& ! ! # & + ! ! 3 -" .

"+ $ 3

*

" " $

&*$

/ % #!

* *

'

3 ( 3 " $ 3

! !

! ! ! & #! 8&9 8 D 8&* "! ! ! + .

! + #! ! ! !+ 4 +

! 8$

*

/ 1 $, $#.#%

*

!

* *

'

3 ( 3 " $ 3

! !

*

¾

3 ¾ 3 ¾ 3 " $ 3 " $( 3 " $ 3

) ! 8&9 8&D 8&*

*

/ 1 $, ! * ¾ 3 ¾ 3 ¾ 3 " $ 3 " $ 3 " $ 3 $#.#% (

* / "9 D *$ + #! & + "9 D *$ ! ! # " ! ! # $ FACTS

*

" D *$ + ! ! 0Q#( 4 !

. ! 3 3 , 9 D 9 D

3 )¾

3

9

D

* + " D *$+ ! !

3 9 D

*

"Q#(R ! AO F D+ AQ* FHD+ A F-D$ !

C "$ 2 + !

!

F"C "$ $

#! 4 " $ ! C"$ 2 + ! CF"C "$ $+ #! 4 " $

* A F-D / 2 K+ ! % &

!

© 2004 by CRC Press LLC

Section 7.5

*

689

The Genus of a Group

A F-D " 0! & Q#(R ! $ "$

#! 4

2

+ !

CF" "$ $

" $

* A F-D / 2 + ! % & !

*

<& ',

,

A:HKD+ A !HKD * ! % & 0 & ! & 2 + ! % ! & 2 REMARKS

1*

! Q#( ! !& ! + ! ! # C"$ "$+ ! & # # !

! & !

Q#(R ! !

! 0Q#( 4 #! "+ $ - ! % & + #! & C"$ C "$ 2 + 2 F-"C"$ $+ ! " $+ " $+ " F$ 2 % Q#(R ! AO F D+ A FD

1*

! ! & Q#(R ! ! ! . ! ! ! ! & & (

+ #! ! & ! ! . ! ! 2 !+ ! ! ! !

+ ! & 00 #! = ! > ! % & + #! & + Q#(R

! + ! ! !

! ! " AO F D+ A FD$

7.5.3 Groups of Low Genus 2 # + !! 2 + ;! R ! ! 0 ! 4 ! ! + ! ! ! +

! ! ! ! *

! ! ! + . ! "& # ! . $ DEFINITION

* /

! . !

! ! ! ! ! "! & ! ! !

& ! % $

! ! # ! $ + &

© 2004 by CRC Press LLC

690

Chapter 7

TOPOLOGICAL GRAPH THEORY

FACTS

2 & ! !

"* ! . +

! + 0 ! # 01 # A F-D * " AHCD+ AO F D$ ! -

+ # + + $ + ! - ! ! #! ! #! " ! + !

( ! !

2 + "$ 3 C"$ 3 - "

* " A:CD+ A D+ A FD+ AO F D$ . ! + "$ 3 C"$ 3 & + C"$ 3 %

4 ! . ? ! " A F-D AO F D$ ! ! + #! "$ 3 C"$ 2 !& + F+ F

"$

3 * 3 3 3

"$

3 * 3 3 3

"$

3 * 3 3 "

$

3

* "O # A FD$ !

HC ! " F$ *

3 *

! ! "$ 3 !

3 3 3 " $ 3 " $ 3 " $ 3 " $ 3 " $

* " A\HKD$ ! * ! ! !

* " A\H D$ ! * ! " $ ( 6" $+ " $ ( 6" $ "

1 # 7 R

CF$+ ! " C$ " " $ * " # A\--D$ ! C

+ - + -

7.5.4 Genus for Families of Groups / ! 0 ! &

( "4 $+ ! # ! 04 + ! & ! !& ! * ! # !& #!

! ! ! 1 ! + #! ! ! & + # ;! R A;! D #! ! ! / ! ! #! 2 + " $ ! , &CF+ ! ! & Q#(R

!

© 2004 by CRC Press LLC

Section 7.5

691

The Genus of a Group

DEFINITIONS

"*

/ "

*

# 4 !

½ " "

!

"

#!

·½ 3 *

/ /

!

!

+ ! * !

* *

/ " $¼ -" / " $ -"

FACTS

* A<;!F-D ! ! !& "¿ ! /½ & ! 1 * 3 ! "$ 3 , "* $&+ #! & ! ! ! 4

*

A:4HD+ A 1;!F D ""¿ "¿ "¿$ 3 + ! ! & + M (

*

A FHD G % 1 # ! + ! " $ 3 , "* $&

"

* A -D+ A\HD

*

! 1 #

*

A F-D 2 2 C + ! + " $ 3 , X&CF 2 C

Q#(

*

/ ! C " 9 D$

9 D? ! Q#( / ! 2 C ! G ! Q#( AG HHD

* ! ! A ;; HD+ ! 0 A FKD+ ! 1 # I ! ! + ! C"$ 3 C "$+ ! #

*

A\-D 2 & + " $ 3 & C

C

"" " $ 3

/

+

C

""¿

REMARK

1*

2 + ! # ! ! C

1 # #! ! C !&

7.5.5 Nonorientable Surfaces

+ + ! + 1

© 2004 by CRC Press LLC

692

Chapter 7

TOPOLOGICAL GRAPH THEORY

DEFINITIONS

* . ! " $+ ! " $

!

" # !

% ! &

! ! !

+ ! & 1 ! ! ! ! 5

$

* 2 + ! 4 " $ ! ! + ! . +

+ ! .

* !

! + 9"$+ ! ! !

! !

*

! ! + "$+ ! ! !

.

*

! ! + C "$+ ! ! !

! .

*

/ ! " $

! " $

"

* !

+ C 9 "$+ ! ! !

!

* / !

! & ! ! &

! - + #! ! ! 0 &

0 & FACTS

*

/ ! " $ ! ! ! & !

& ! ! !

*

! % ! #*

"$ 9"$ C 9 "$ "$

"$

C "$

"$

"$

3 "$ 9"$

"&$

C "$

3 C"$ 9"$

* / ! 0 + ! + ! ! ! 9"$ "$ ,

*

+ ! 9"% $ 9"$+ :R

© 2004 by CRC Press LLC

! A:

D

Section 7.5

693

The Genus of a Group

* A FD ! ! ! + + ! C 9 "% $ C 9 "$ "* Q#(R ! ! & * A FD ! ! . ! + ! "¾ ! & + ! . ! 0

+ #! ! 0 & + C "$ "9 C "$ $ *

! # .

*

" "

"

0

3 3 A D 3

! . + ! 6 &

& + ! ! ! . . 2 3 3 + + ! + "$ 3 "$ 3 + "$ 3 "$ 3 ! #! C "$ 3 K ! $ ?

! "$ $ 3 K " A HD+ A\-D$

*

! #! 9 3 ! #! 3 - ! #! ! #

#! 3 ! #! C 9 3 ! #! C 3 - 6 ! ! #! 9 3 '! #

A HD+ A\-D

*

!

1 #

* Q#( + ! 9"$ 3 , &F + 9"- $ 3 , &F 2 C A FKD

References A: D G :+ ! + KP-

! % "H

A:HD G :+ E 0 & ! & 0 & + ! K "HH$+ KF PC A:1D :1 + ! ! + CKPCCH

$+

%

& ! "

K "H$+

A:;! HD I : / ;! + + " , + N&

+ H H A:4FFD O : 4 + ' ! H "HFF$+ P A:D ; : + H A FKD +

% 0 +

+

!

N&

+

! + HPFC

% 5 H "HFK$+

© 2004 by CRC Press LLC

" " "

!

694

Chapter 7

TOPOLOGICAL GRAPH THEORY

A H-D . + Q#( * & + "HH-$+ KHP -

& "

A ;; HD . + / ;

+ / < ; + !

+ & " C "HH$+ CKPCC A F-D Q ; ' < + ; $ "! $+ 0E + HF- AO F D < G O

; 1 + % % + & + -- "2 . + ; 0 + HF $ 5 AQ*FHD / Q#(+ N !

!+ " & "FH$+ -P A<;!F-D < / ;! + ' ! % + ! "HF-$+ PK AG HHD / G! +

1 Q#( + ! & H "HHH$+ KPKC ACKD !+ / ! + , " K "HCK$+ CHHP A*FHCD Q !1 + KCPH

! % +

& ! "

F "FHC$+

A\HD G < \ + ! % + ! " "HH$+ --P A\HKD G < \ + O + " ! "HHK$+ KPH

=

A\H D G < \ + ! ! + ! " "HH $+ K PKH-

#

A\--D G < \ + O + % "---$+ PK A\-D G < \ + ! . ! 0+ ! " "--$+ P K

!

#

A\-D G < \ + ! & + , " + A 1;!FKD : !+ 1+ 1 & + / ;! + ! ! + $ " KC "HFK$+ F PFH A D E 7 + & + H

© 2004 by CRC Press LLC

!

% + !

! + N0

Section 7.5

695

The Genus of a Group

A FHD 1 ; 1 + ! #! + ! - "HFH$+ CHP K AKFD O

+ '

% 0 ! + "HKF$+ F--PF-

& "

H

A !HD !

+ ! ! 7 & 0 & ! % + % & " "HH$+ FHP-K A F-D ; 1 + ! & + KF "HF-$+ C P H A FD

;

% & "

1 + 2 ! +

% 5 "HF$+ FPHF

A FD ; 1 + ' R + F "HF$+ HP

!

! %

A FD ; 1 + / % Q#( ! ! + ! % 5 C "HF$+ PCF A FD ; 1 + "HF$+ CHP K

! # +

! % 5 C

A HD ; 1 + ! + -KP- % 5 5 & "7(

HFF$+ ; 0 + HH A;!FD / HF

;! +

5 +

A;! D / ;! + ' ! + -P

© 2004 by CRC Press LLC

& . + I !0Q +

% & "

"H $+

7.6 MAPS

! " #!! $ !% ! !! & ' (! ) &

Introduction * & ! + $ ! %+ , % ! '" & ! & $ '--. % #! ! % %, #! / ( 0! 1 + / 02 02 3 /% * 4 56 %! 5 & ! 789)": 78) : 7" : 75*' : 7*-: 73-:

7.6.1 Maps and Polyhedra Maps 8 ; ! ! !! & 2 * 4 & ! 2 $ DEFINITIONS

; # < ! 4 2 * & & !

; ;

+ = ;> ? @ * $ $

*

& ? @

; ;

! & ? @ * $ *

& $

© 2004 by CRC Press LLC

! "

#

$ $

½

%

¾

½

¾

&

"

" $

" $

$

"

&

$ $

EXAMPLES

' ()*

$ $ + $ $ " ,

3

1

1

4 1

2

2

6 5

1 2

2 3

1

3

1

3

4

6 5

M Figure 7.6.1

4 1

M*

' ()-

-& $ .& $ ' ) $

Figure 7.6.2

© 2004 by CRC Press LLC

698

Chapter 7

TOPOLOGICAL GRAPH THEORY

!

" "

Figure 7.6.3 REMARKS

# ! $

"

% % & "

"

$

% '( )**+% " % "

"% " "

" &

" " $ % % & "

, %

%

-

& % " %

" "

- % "

% " . $

-

% % " / % % " "

FACTS

%

#" #"

/

/ %

% %

%

¼ ½ ¾ 0

¼

1

1

½

%

1

" "

$+

" "

2

'

"

" & $ & "

¾

! " #$ % &

&

- %

! &

'

' 34+ - ! " "

%

½ ¾

© 2004 by CRC Press LLC

½

1

¾

"

½

¾

52

/ % " %

Section 7.6

699

Maps

REMARKS

!"#$%& ' %

(

*

Æ

!$& )

CONJECTURES

+

)

+

# , - , ,

7.6.2 The -Vector, - and -Sequences, and Realizations + ' # . )

+

'

Æ

)

# + / 01$0 " 2/ 0$

DEFINITIONS

)

3 4

. 5%

#

#

#

'

+

'

6

)

© 2004 by CRC Press LLC

6

6

700

Chapter 7

TOPOLOGICAL GRAPH THEORY

REMARK

!

"

EXAMPLES

# "

(

) &

* + '- ,.

01 +& ).

+' ) ).

$ %&' (

& )

01 +

* +, '- . /

# " $ %&-

&

'. *

01

$ "

)0"

$ %&,

Figure 7.6.4 FACTS

#

1

0

$ "

01 ( * " (

2

01

2

4 5 " "

+

$ 6" 7

-.

2 ' )

2

4''

+

-.

+&

3

+&

2

3

2 '

'.

+'.

" (

2 ,+

.

2 '-

+-.

8 +-. Æ

6 " *

2 -

1

. 2 '- 6

)

2 & )

01 " ( " $ 4 5

# " 9 % :) # 7 * 4 5 " (

44*;- /

" (

2

) )

© 2004 by CRC Press LLC

* 4

3

* 2

2

+

.

0 +

2 -

2 )

2 -

+

.

2

. 6 *

6

<

Section 7.6

701

Maps

!" # $!

% &' (

% )& *

+)

.

# 0 , +/ +-

" , -

+ /

1 " /

2

/

+ $/

1 3

-

-. . 4

,

-

+$

/

1

3$ +

,

/

+ /

1

#

2

1 8

8

8

!"#$

# 0 +

+""

/

$ 5 %# 67 &"

8

1 $ .

/

9 $

$

2

1 3

+""

/

1 "&

!&!0

:.

5 2" 3 "

- ( :

- .

. .

< &!

+

/ ; . .

:.

;. !$'

Figure 7.6.5

!"#$ ! " % " " "!&

© 2004 by CRC Press LLC

702

Chapter 7

TOPOLOGICAL GRAPH THEORY

7.6.3 Map Coloring

DEFINITION

Æ Æ ! "

! # FACTS

$ %&' ( ) $ *)' ( & $+,&-' ! # .

( % / )01 1) 2 3 4

) 53 " 6 $ 0*' 1 53 $ 0%' 7# 53 2! *3 893 $+:: 0&' ; )3 REMARKS

!

<-51 : =

6 >

6 $ %&' Æ 7 $+:: 0%'

! !

! $<-0?' #

4 ! 4 + , $+,&-' © 2004 by CRC Press LLC

Section 7.6

703

Maps

EXAMPLES

!

"

#

$

% !

#

$ &

#

#

$

$ & ! '(

#"

' $ 1

2

3

4

5

6

7

1

3

4

5

6

7

1

2

3

Figure 7.6.6

) * +,- " .

# ' $ / !

Figure 7.6.7

7.6.4 Minimal Maps )

0 1! * !

'

! 2 ! 2 3 # $

"

4 ! !

)

DEFINITIONS

)

)

#

$

"

! ! , )

!

! !

5

© 2004 by CRC Press LLC

704

Chapter 7

TOPOLOGICAL GRAPH THEORY

e

Figure 7.6.8

EXAMPLES

! ½ ¾ " #$%&'

( ) * + &,- )( )! .

a

Figure 7.6.9

b

FACTS

/ ¼

¼

&0

")

)"

¾ ¾ ¿ $ ¾

+ )% #$%,' 1 2 %

© 2004 by CRC Press LLC

Section 7.6

705

Maps

! " # $ $

# % %

&

'$

( ) *

!

REMARK

* (

! # %

½ $

7.6.5 Automorphisms and Coverings

$

$ % ! + %,

- % #

" % $ $ % $

&

% # $ . . /0

$

12

DEFINITIONS

!

½

-

-

¾

/

½

34!

¾ $ #

%

+ ( $

½

5

½

6

½ ¾$

½

% !

5

½

½

5

¾½ $

¾½

½

7!

! # %

¼ ½ ¾

¾¼

6

¾½ 6 ¾¾

¼ ½ !

6

6

½ ¾ !

¾ ¼ !¾ 6

6

'!

EXAMPLES

& 32 % %

& 324

7( $ $ % #

© 2004 by CRC Press LLC

706

Chapter 7

TOPOLOGICAL GRAPH THEORY

Figure 7.6.10

FACTS

!

"#

"#$

%

& # ' !( %

&

& &

'

& *

"#

+,-./

Ï Å

*

.

Å

&

'

Ï

)

Å

0

Å

$

1 2 2

&

#&

2 2

3

1

Figure 7.6.11

© 2004 by CRC Press LLC

2 2

2 2

Section 7.6

707

Maps

#$%

*

½ ¾ !

'

!

"

"

& ' ( )

$+

( * - ! " $+ . / / ! 0 -

,

" . # 1

$/ & '

'

'

'

2

! "

REMARK

3 4$ /5

6

! "

!

"

. 6

! - - - "

.

7.6.6 Combinatorial Schemes -

0

, )

DEFINITIONS

,

"

!

7

¾ ´µ

(0*

!

"

8

½ 7 ! ¼ ½"

½ 7 ! ½ "

·½

7

½ ¾

7

"

!

( ! "

6

) )

2

"

!

9

© 2004 by CRC Press LLC

708

Chapter 7

TOPOLOGICAL GRAPH THEORY

Æ

!"# !

$

%

&

' '(

¼ ½

½

)

* "

#(

¾

+!,

(

- * " . - -

-

REMARKS

/

01234 05674 0684 019634 0:674 09 3*4

+!,;

0<3'4 = 0> 624 0+684

0?34

?

½

#(

EXAMPLE

@ > 62"

½ ) " '

© 2004 by CRC Press LLC

¾ ) " # 7

¿ ) 7 # '

Section 7.6

709

Maps

· · · · · · · · · · 1

ï 2

2

ï 5 ï 3

0 0

ï 4

+ 2

+ 3

+ 4

Figure 7.6.12

ï 1

1 2

+ 5

1

0 2

+ 1

0

2 1

1

2

1

1

0 2

1 0

0 1

1 2

1

0 0

0 1

2

FACTS

! " ! ##! Æ ´ µ Æ # ! $ Æ Æ

# %&' ! # ! Æ # %&' ! ( # %&' # 7.6.7 Symmetry of Maps

) ! $ ! * # +" # DEFINITIONS

¼ ½ ¾ ( ! # ,#

,# © 2004 by CRC Press LLC

710

Chapter 7

TOPOLOGICAL GRAPH THEORY

!

!

"

#! $%&

REMARKS

'

"

&

(

!-

)#*

"&

+

,

EXAMPLES

!

"

' .&

/

/

/

.

.

"- &

/

/

0

0 / -

$

$

1

2 , $-

"' 0%&

#! %0

,

$

.

/

.

. /

/ 0

0 /

/

/

$

/

' ! ' 1 ' %

$ .

.

/

341%5 , ' 0 6 -

$ %0

. /1

- 7

0

' %/

( ' 1

0 / "%$ . 8

& &

/

0

0

/

Figure 7.6.13

© 2004 by CRC Press LLC

"%$ 8

"$1 8 &

Section 7.6

711

Maps

$! % $

' '"%'

'

( , -.

"

' ' '

& ! "

!

"

"

/ * ) 0 '

Figure 7.6.14

3 4 '

(

) * +

"

!" #

&

1 2

/ 5

( 6 5

!

! !%

5 $

6 '

0

0 4 3 2 1 0

Figure 7.6.15

0

- 3 )

$! ! !"%

3 ) 7 $ ! !"%

/ *

'

2 )

' ' !#

*

)

( ) *

'

; ;; ' ;;

/ ) 9 &&

&& :"

!"

. ) * ) * !'

'

8 3

7' ( ) ! 4*

'

,

$ %

) <= $! %' ( ) *# ) 4

%

'

) *# ) , 3 '

© 2004 by CRC Press LLC

3 >

3 * ? $# @ ) 4

712

Chapter 7

TOPOLOGICAL GRAPH THEORY

FACTS

½

½

½

½

!

!

!

! !

" ! !

! ! ! #

!

$ ! ! %

! ! ! & '

()*'+ (,-*+ . ! ! ! &# !/

!

$ !

!

.!

0 /

! 1 2! !

()*'+

½ ½ 5

!

½¾

-34

/ !

6 ($ 7809+ ! !

#

(,*:+ 1 !

(<-9+ 1

!

!!

()*' *'+ 1 !

!

%

= =!#

/ > !

4-

; & ' 3

! !

!!

!

> !

. !

'

!

¼ ½¾ ¼½¾½

1 ! !

!

> ? (>?4-+

© 2004 by CRC Press LLC

¿

! -399 !

4 4

!

Section 7.6

713

Maps

¾

"#$%"&#$% '

¾

!

(

) $ !

7.6.8 Enumeration * "+,% $#+)- .

/

"0 1,(% "2##%

! 3 "4#5%

DEFINITIONS

6

(

EXAMPLES

'

!

7

' 5+$+ !

( (

8

' 5+$+

9

Figure 7.6.16

© 2004 by CRC Press LLC

7

!

9

714

Chapter 7

TOPOLOGICAL GRAPH THEORY

FACTS

!" # $ %&'

(

) (

+ +

) $

*

·¾

¾

7.6.9 Paths and Cycles in Maps * *$* ,$

- .

/ 0

* $, $ / $ ,$ " $ $ 1 2 , "

DEFINITIONS

3

"

3 / , $, $ / , * . ,

* " 3

4 $ "

¼

¼

,

$

0

* ,"

© 2004 by CRC Press LLC

Section 7.6

715

Maps

EXAMPLE

¾

¾

Figure 7.6.17

FACTS

! "#

$

$

%

$

&"'# !

%

$

$

$

% $

! ( )

*+ ,'#

% -

$

! ( )

./",# 0

0 0 ) 1

2 3

%

./"# 3 4$

(

'

© 2004 by CRC Press LLC

716

Chapter 7

¼

!

#$ %

&!

¼

,

$

'

!" &

!"

!" !

¼

()#* +

!"

TOPOLOGICAL GRAPH THEORY

- ,

-

-

!" . /

, &! $

( 0 ! " 1

! ! ! $& ! $& ! $&

½ ¾

2#3 #$ .

!

4 -,

!

-

&

!!

! !

!

5 + 3*

½

#- " ! ! Æ ! " 5

½

! !" !

6

6

REMARK

&"

! !&!&7

8 !

/ !

References %#' 8 % 9 ! 9

3 -##' -':-#$

%3- % % 9! 5 ,& ! '&

¿

!

- -#3- ,--:,$

%(* % % ! ; ( < / !

=$ -#* -$3:-=3

%(3 % % ! ; ( / ! ,& ! 0

,

%

'= -#3 ->':-,'

%3> 0 ) %! < ! / -#3> ''=:'3

© 2004 by CRC Press LLC

8 -

Section 7.6

717

Maps

!"

#

! $ % $

&%

'( !!"

)#)

*! + * , - ./- - &% - ./- -

' !! " '#

* $ * -- $ - -% -% - 0 - --

1

!"

#

* ! $ * -- + / % 2 - - &-

!!"

)#

*34 ! $ * -- 5 3-6 7 4 % 8 %

'

!! " !#)11

* 9 *&& -%

&%

! "

)# )

* 9 *&& :% - -- -%

( !" )1)#)1

*+!' ; 5 * &- ; ; +--

< & /, & 9 = >

!!'

*!1 ? * % 5%

=-% =

& : - &% -% 8 =%

!!1"

')#

! &

!!1

*<!' ? * % 3 <% 7 6- -% - -% 0 -

!

!!'" (!#'

"

# $ ;7; 5 * 7- @+ !!

*<%! ? * % <%- 5%

)('#)'

*!) ? * % A B 5%

, * ')'#''( 9-%/ -

"

# %& $ !!)

*) 3 *= - /

'

!)" '#'('

*:( 3 *= : :-- 8 -% - -

/

!(" '#'

*B7! 7 * - * B% 7 * 7%- < - & - &

0 - %- &%

&

!!"

1 #)

*<-11 *& $ <- - 7 6- & -- %

-

)

)

© 2004 by CRC Press LLC

( 111" (

#''

718

Chapter 7

TOPOLOGICAL GRAPH THEORY

!"!

#$ % & '

#

(#$

$# ) * + , - .

* /0 1 ($#

2( 0 2 '

3 45 (

2 267 8 4 2 . ) 26 % * 79 %

/

: ( ;";

2 : . % 2 8 '

!

# (: :!:

/= ' " # ;1 (#: $":

<#: < & 5 >

?

/

<# * <9 3' +

! (#

!#"!

<@#( ) <' , @ ' 3 = '

$

(

(#( (#":

< ;! < 9 8 +

: (;! ;:;";:(

A #( A 8 ' 5 ;/ 5

" #

:8 (#(

!!"!!(

A)3 ! . % A . )' % 2 3 B 8 ' '

$ (! ;("!#

A. ; @ A . 9

% .'

, C

* D6 E 9 (;

A ; A 5 3'

2 ' F '

;

(; #!"(

A 3# . 4 A 3 , 39

.' , C *

D6 E 9 (#

A G # 1 A G ' +

()*&+,

&'

;"

A G :# 1 A G

$ -

@ (:#

)( . ) 6 / '

.

! (

;;";;

)( 8 ) 65 & ' A 3 '

© 2004 by CRC Press LLC

! ( !;"!!

Section 7.6

719

Maps

!"

# $ % %& %

"

''

(( ) * + , - . / % $ 0 1 2

((( "'!

) * 2 30/ $ 2 $

"(

) * 2 +/4 $ 0/2 )

!"!(

5 1 5 5-2 -$2

!

60710/ 5 '"'

8 8 )07 2

" #

"

83 . 8 . 9 3: 02 $ 0

$

"

6 1 66 .; $ /

%

"!

6 1 66 . /0 :%

"(

66< 1 66 + 6 = < . 2 $2

' !'"!!

6 + 60 > - $ 2 /2 2

!"(!

630( + 60 > 302 % / 1

& '

30 0 ?- @7

((

6<(( + 6 = < .; $ 2 $ /

' '

&

' ((( '!"'!

4 4 A-( . A 6 -% . 2 $ ,0 ,0

((

'"

1 B ? 1 - B A% 0 $

&

! ! " !

1 B ? 1 - B A% 0 $ ,0 ,

"

.C ) . < 3 C $ 0 ?, 27 7 2

( )

© 2004 by CRC Press LLC

( '"''!

720

Chapter 7

TOPOLOGICAL GRAPH THEORY

!"#$ %"%&%

'#( ' )Æ *+ , + ,

!"##($

-"& -

'#- ' ' *+

-. !"##-$ %&

#" / 0 ,

" !"##"$ "%-&" #

#1 / 0 ,

!"##1$ "-&"

2 ) + 3 4 2++ / + + +5 6+7 8+ 9 1 -

1% !"# $

1#& -

2( ) + 3 4 2++ : ;7 9 ,+ <= +

, 1

(. !"#($ % 1&%(%

'#( ' > ? ,+ 0 * ,

%1 !"##($ 11-&1#

." 3 @ A ;0 , , , + , 0

" !%.."$

!

-&-%

. + * B + , +5 , ,

% # !"#.$ "%#&"

"" !"#.($

"# $ %

1 !"#%%$

C .( ) 5 >

)+ +

+ (&

%% ) 5 + + , "&"1#

'#. ; ' ) , , = + +

!"##.$ "

&"--

'#" ; ' '+ , 8+ + 0 7* 0 , D7

'#1 ; ' 60*+ ,

1%1 !"##"$ (. &(1

#

!"##1$ #&".

'#- ; ' ;+ * + , D7

-. !"##-$ (-&"..

'- 2 '

& '!&

> 0

E 3 "#-

'1 ' 2 '< 6 , , , ,

© 2004 by CRC Press LLC

1 !"#1$ %&#

&

Section 7.6

721

Maps

! " #

% ! %"

$$

$

% ! %" & ' "

$$

% ! %" (' )

% ! %" * + ' ,

$ $

& - . /+ )

$$$

0

. ! .

1 .0

/ 2.' ! $

. $ / . + 1 '.

$

$

- 1 0 '

- & #" ) '

$ $

$

- 3 + ' #

$$$

4 5 4

67 !" " 2. 8

4 9 4 :; . ,' "+" ' 73

<) ! <) 3 *= ' > "" "

© 2004 by CRC Press LLC

$ $

722

7.7

Chapter 7

TOPOLOGICAL GRAPH THEORY

REPRESENTATIVITY

! "# $ % & ' ( & #& & "'

Introduction ½¼¼ ½¼¼ ) ' *) + , ) + ' !- ' , ' . . ' "

+ )

,

' ' / + .

+ + ' )' '

+ . 0

) " ) 1" 223 . )

) . 1 )2 3 1& 43 ' ' ' ' '

+

7.7.1 Basic Concepts + 5

) . ) ' # )'

) DEFINITIONS

6 / 7 8 9

: '

' ' ' 7 :

6

) ' '

) . ;

6 / . ' # <)

6 6

/ ' ) ' + '

/ ' )

' . ' 7<*)+

. + + , = . )+ :

© 2004 by CRC Press LLC

Section 7.7

723

Representativity

"

!

"

! !

#$

!

!

"

%

!

!

& ' # ! ! $

EXAMPLES

( ))* ! ! &

! !

! + !

! , !

Figure 7.7.1

( )), !

!

© 2004 by CRC Press LLC

-

724

Chapter 7

TOPOLOGICAL GRAPH THEORY

Figure 7.7.2

FACTS

"

!

# $

"

%

!

!

$

"

%

%

$ '

%

%

'

&

&

'( '

#

$

$

&

( $

'

© 2004 by CRC Press LLC

Section 7.7

725

Representativity

!

"

!

#$% & " !

7.7.2 Coloring Densely Imbeddable Graphs ' ( "

) * *

" " & + ", " " & "

-!, " * " "

, !" * ! ., " *

' & ! *

*, ! " & & " " & ! &Æ"* *

Coloring with Few Colors / 0 " ! 1 " ! ! 2 ! ! ", 3& ! 0" *

FACTS

' " ! $( 2

(,

$. 2

&"

1 "

&"

,

1 " 4% &

"* ! " ,

! " 5 ", * "

( "

&" * 8 "

4#$( "

6& &

7

$% 4 " 0 & "

& "

* "

& " 1 "

REMARK

2 "&" * & " ! "* ! "

5 ", 4 " 8, 1 " 4 " 1

Coloring Graphs That Quadrangulate / " " & ! " , "0" *, ! * "

9 " "

:; 9", !" * * ( . "

© 2004 by CRC Press LLC

726

Chapter 7

TOPOLOGICAL GRAPH THEORY

DEFINITION

FACTS

!

"

#$%&'

(

!

#)*%+'

+

+

#,%-' .

/

+

#,%-' 0 /

+

#$112%%' #*"%%' )

0

+

REMARK

/

0 ! "

!

+

Coloring Graphs That Triangulate DEFINITIONS

3

4

FACTS

! 4

#$5"%%'

© 2004 by CRC Press LLC

4

+

Section 7.7

727

Representativity

! ! "#

PROBLEMS

$% &

! "

'

() *

,, -

*

, !

# .

+

+ ,! ! !

-'

/

! ,'

0'

, ! 40

1! 12 3

Æ#

7.7.3 Finding Cycles, Walks, and Spanning Trees ! $

$"2 - , , ! # 5 6 ! , ,#

DEFINITIONS

6 # , 6 !

!#

FACTS

, ! , , )6#

7 )#

& ,

!

6+

, !!!

822 0 , , , !!!

0#

819." 0 , , !

:

, !!! 0#

. 0;

¿ < ; ; (+ , ¿

!# & , + # &

-#

=% >

-(;

0 , ! # & , 06 ,

) +

!!! - #

=% >

-(;

- , ! # & , ;6 ,

) +

!!! 0+ 1- #

© 2004 by CRC Press LLC

728

Chapter 7

TOPOLOGICAL GRAPH THEORY

!

"# $ ! % &'

% #(##

CONJECTURE

)! (

& !

! ! *$ # %!

!#

7.7.4 Re-Imbedding Properties +! $ +!,, %* ! *$ , %! ! $ - # ! %

! %* # !# # %!

* $ . / 0 $ % %! !* # %% 1

LEW-Imbeddings DEFINITIONS

2 #

&! & ! $ ! ! ! ! $ &'

2 %! # & ! ! * (

& ! * (

& !

FACTS

)!3 2 %! ! + # # !##%!# $%

4 ! # ( ! ### # ! %!

)!3 + # , %!

! $ ! $ !

!

)!3 )! %$# # !# ! * , %!

! + #

Ë

567/ . !

#

!*

! ! # (

!

Ë

)!

! ! # 8(

&

(% ! & ! # # $

Imbeddings and Connectivity DEFINITIONS

& ! $ 9%% !

%! * $ & ! % !

½ ¾

& %!

!

!* $ & * ##

# )! & % ! #

#%

$

¾

& ! # # !& . , & )!

© 2004 by CRC Press LLC

# $

Section 7.7

729

Representativity

Figure 7.7.3

FACTS

½

½

¾ !

¾

"##$ %

"&'$

"&'$ (

()

!

#

()

!

Imbeddings and Genus

Æ

FACTS

.''

"*&+$ (

!

)

,-

)

"% &$ /

0

,1 / 23

-

Re-Imbedding Results DEFINITION

(

4

Æ

,

-

5

¾,- 6 2

4 !

5

½,- 6

½2 7 ¾2

© 2004 by CRC Press LLC

½ ¾

730

Chapter 7

TOPOLOGICAL GRAPH THEORY

FACTS

! " #

!$

"

½ ¾

%

&

"

'

! " " " #

() !) *+ , #

" # ,

"" -

./ ! " #

"

" + 0, "

" # # , " 0 "# # 1 "

2 " #

3" " # " , "& # "

" # # , " " " , " " .4

CONJECTURE

53 , "#

"

& " #

+,

7.7.5 Minors of Imbedded Graphs ! # "# " " + " + " - # " + " 6 + # # 7 8" 9 * : " ;))

* " + # " " " " .)$

Surface Minors DEFINITIONS

*

"

# + " - #

" "

* + < "

+ < "

*

" " + #

" #

" +&

* +& "+" "" # "

: # " #

" #

" + " - # " " & ,

=#

" " +

"

" " # "# "& " # -

"

FACTS

"

53 "#

+ >

"

7 - " "

© 2004 by CRC Press LLC

? " +

Section 7.7

731

Representativity

!" #

$

#

#

%

&'()"

" *

#

# #

+ #

#

,

Finding Imbedded Cycles #

$ + + #

-+$

+ # # .

DEFINITION

/

½

#

0

#

FACTS

(1" '+ #

,

1

)

2 ,# 3 2 ,#

6 7

45(!" '+ #

,

# 9 ½ (1" *

2 ,#

# #

8 67 6

: (;" * # ' # #

8 67 6 7 $ 9 ½

7.7.6 Minor-Minimal Maps <7

$

,

= . , #

$

$ #

,

# ,

, / #

,

, #

, +

© 2004 by CRC Press LLC

732

Chapter 7

TOPOLOGICAL GRAPH THEORY

DEFINITIONS

FACTS

#

!

"

$ %&

$

'

)

¼

¼

%

' (

%

¼

'

*+,-.

/

*01-.

/

%*0,!. *2,/.' -

$

*01-. - / *3. 45 $

-

Similarity Classes on the Torus DEFINITION

6

REMARK

© 2004 by CRC Press LLC

3

Section 7.7

733

Representativity

FACT

!

¿

¿

"

Kernels # $% & '

(

( ( ( )

! ' (

DEFINITIONS

' ( '

()

*

,

' !

+

-

( ! %

!

¼

!

# '

(

!

(

.

)

¼

)

'

)

FACTS

/ £ £

!

0 ( 1

)

!

¼ % . ¼ ' ' ' 3

- 2

¼ !

/ 0 (

(

%(

References #"4 5

6

#' ) 6 7 ' -)

)

#"- 5

6

' )

' 2

8

9 ) + ) 4"4

#' +

:

3) * (

" 4"-) ;<

# - = #

) = '

( )

4-)

4>;>

# ?@@ 6 =

#

)

A

7

? ) #

@%)

B 6) 9 ' 3 ( ) >< -884) 488;44

© 2004 by CRC Press LLC

@ ()

734

Chapter 7

& '

TOPOLOGICAL GRAPH THEORY

!" #$ % "

! !!(

& ' #

) !!

*( + , - . * # ,# # '

'

( )())))

+/ *) * % / 0 + # - , / 1 * * 2 " %$ 3 % 4 $

) ) !!

/* * % / * * 2 & #

#$

!5!

+ ( 6 + * # 7 ,# 3 8"

9 , $$

55 *

( ( !)!!

2 2 ": !555

+( / 0 + # - 2

%

(

;<%**) = * ; = 6 <% > * * 0 *$ ?#% $ % 3 & '

!5 ) !5

;/( 2 ;> / ? 8% "$%

5 ( !!

( / 3

%' %3 6 @ ' 3 ,#"

# ,# # (

% = . % >

"7

' !55

A % $%# ? & % ( % , #

$

6

< < +# B

1 3

#$

!5!

< C < / "# # " ' % / B C># 0 @ ' C>#

<%) = 6 <%

" #$ %3 8 3

' "

) ) ))

<%*2 ( = 6 <% * * 6 2 #% ? +% " %

© 2004 by CRC Press LLC

( !55! !!)!

Section 7.7

735

Representativity

!" # $

*(& " * +, -

%& !'($ ')'%

.& !'((&$ /)0/

*1' " * -

2 - 3 4 5

6 *- 2 6 7 2 % !%11'$ '.)'0/

*8(0 " * 7 8 4

0 !'((0$ &)'''

*8( " * 7 8 9 -

/ !%11'$ /)&

*2(( " * 4 : 2- +

*;1' " * ;-

. !%11%$ /1')/'1

5< =

4

%11'

8(& 2 4 8 *- -

%

!'((&$ '/)'0/

820 7 8 4 : 2- -

>? #9

.' !'(0$ (%)''.

82 7 8 4 : 2- -

>@@? :

. !'((1$ %)%

8;(' 7 8 8 ;- A - B

' !'(('$ .1&).'(

8>(1 7 8 8 > 8 - %(/) /%

!"#!

" B 6 6C D 5 4 E -

2 2 +> " '((1

2(% 2 A F

<

!'((%$

'.0)'01

2(/ 2 - -

!'((/$ '.&)'

2(. 2

G -- +,

0' !'((.$ %'&)%/0

2;(0 4 : 2- 8 ;- =F

-

%/ !'((0$ //&)/.(

;(( 8 ;- 8 9 - - %1')%%%

%&&& 4

'(((

© 2004 by CRC Press LLC

$

: 6- : 4 - =

736

Chapter 7

TOPOLOGICAL GRAPH THEORY

!!"##

$ %& ' (

$

"!

%& ' (

)* "$$

+, -. $'

( (

/ &

)* *)"*#

!

$"$

# ' 12 (

& 0 !

# # )#"

(!) 3 ( 0

* !)

")

4* 5 4 *'

$' & / &

! * "*

3$$ 6 3 *'

7) 8 0 7( ' / &

!! $$ *!"*!

* ) *"

**#

7(# 9 7( 8/

-

:

$ # $$$"$!

© 2004 by CRC Press LLC

Section 7.8

7.8

737

Triangulations

TRIANGULATIONS

Introduction ! "#

$ ! ! !

# # % & # ' # (

7.8.1 Basic Concepts )# # & ! % & # # # & ' *&#* % ! + ! ! ' DEFINITIONS

, ) # % # #

! ,

!

# * % & # !

# ¿ # #

, ) # ! ( * -" ! !

, % # # ' ! % # * # .

, #

, # ! & & -.

# ! -. # !

, ) # ! #

, ) / # 0 ! # # ! # # ! # '

© 2004 by CRC Press LLC

738

Chapter 7

TOPOLOGICAL GRAPH THEORY

!

"

#

½

¾

$ ! %

½

½ ½

&

¾ ¾

¾

½

&

¾

' !

(

))

NOTATIONS

!

!

*!

!

´µ ¿

+

! +

!

!

EXAMPLES

, !

© 2004 by CRC Press LLC

Section 7.8

739

Triangulations

Figure 7.8.1

!" #

! $

Figure 7.8.2

% & "%

% '& & & ( !

! !" & ' % #) " "%

! "!

% & *"% !"

& & ! & !" ! & ( ! % # !" " "%

*

! "!

FACTS

+ "! ! &

%

+ ! , ( &

%

+ !&

&

- - ! !

+ !&

! !

& - - ! ! -"

( "% ! !&! ( "%

&

© 2004 by CRC Press LLC

740

Chapter 7

TOPOLOGICAL GRAPH THEORY

!

" "

#

!! $

# "

!

%

!

"

!

"

!

( & & &

! ! " ! ! !

"

¿

'

") "

!!

& '

%

£ !

& & &

*

£

!

!

"

+ "

" &

&

EXAMPLE

%

%

"

¼

" ' ,

¼

&

'

"

Æ &

- " "

'

. /

'

7.8.2 Constructing Triangulations 0 & && & 1 ( & ( Æ "

Triangulations with Complete Graphs 2 3 &4 &

!

&

!

)

&

! 5$

6 !

!

! '

"

& & " & && (

© 2004 by CRC Press LLC

Section 7.8

741

Triangulations

DEFINITION

½

FACTS

!

$

"

"

%&

!

'

# $ %& ! '

*$

$

"

"

# % $ (

) !

(

+ ,

-.#% ! /

01 %&

- 22.## !

& &

¾

¾

%0 $(

%0 33 %#4

EXAMPLES

5+60

!

-.#%

!

%

-+03

!

& 1

REMARK

7 8

9 +:0(

#

Minimum Triangulations ; ' "

© 2004 by CRC Press LLC

! <

742

Chapter 7

DEFINITION

TOPOLOGICAL GRAPH THEORY

FACTS

#

"

#

%$ ! !!

&'

(&

(

) * +

# ,

" !+ +

#

! !! !+

# '

-

. ', / ! !! !0 . ', / ! !! !0

&

(&

!+1

EXAMPLES

/ ! !! + +2 +

!0

+

/ * + * ! !! 3 0 + 1

!+ !

Covering Constructions 4 5+ 6 / ! + +

7

+ 80

* : ! " ! !0 !+ +

5+ 6 / ! 0 0

! + 0

9

$&

/ 0

!

DEFINITIONS

;$& # +

1 ! + / +

!

/

+ +

!

+ 0 -

/

00 3< !0

+

0 - +

© 2004 by CRC Press LLC

Section 7.8

¾

´ µ

½

¾

½ ½

743

Triangulations

!

½ ¾

¾ ¾

½

´ µ

""#

$

FACTS

% &' !

% &' *

¿

+ ,

( )

-./ *

/ 0 1

´ µ

-./

! 2/

´

µ

'

´ µ

"

´ µ

#

3

0

"

EXAMPLE

4 5 &.0

7

¿ ½

+6*

&

! 5 /'

5 Æ

Figure 7.8.3

"

¿ ½

© 2004 by CRC Press LLC

!

744

Chapter 7

TOPOLOGICAL GRAPH THEORY

7.8.3 Irreducible Triangulations

!

Edge Contraction DEFINITIONS

" # $ % & ' (

Figure 7.8.4

" "

) ) " ) * FACTS

"

) ! ! + " ,-.(/ ) REMARK

"

) 0 + 1

© 2004 by CRC Press LLC

Section 7.8

745

Triangulations

Classification and Finiteness in Number DEFINITION

FACTS

! "

# $

%&'! " # (

# ) *&+ " $

,

Figure 7.8.5

-&*! " '.

# ) *&/ # # 0 ".

" ". "'.

" $

-12*! " '+ 3 " 0 # 3.

3'. ) *&* # 3. 3 ) *&& # 4 5

% &2! " 0

162+!

.*.7'

7

88

#

7

8

9

*'

EXAMPLES

"

© 2004 by CRC Press LLC

746

Chapter 7

Figure 7.8.6

Figure 7.8.7

½ ½

TOPOLOGICAL GRAPH THEORY

¾

¾

¾

½ ¾

© 2004 by CRC Press LLC

Section 7.8

747

Triangulations

Figure 7.8.8

¾

REMARKS

Other Irreducibility ! !

! "

# !

$

DEFINITIONS

%

%

%

&

FACTS

'

$ #

(

# $

)* +, -

)**+, -

).* +&, $ -# / #

© 2004 by CRC Press LLC

748

Chapter 7

TOPOLOGICAL GRAPH THEORY

!

REMARKS

"#

# # #

$ % #

! "& '(&)

*

+

,-.&/

7.8.4 Diagonal Flips + 0 1 23 + Æ 4 5 2 '&*

DEFINITIONS

+ 0 ! 67& 8 2 9

Figure 7.8.9

$

% 0 2

+

$

2

¿

67: ;

Figure 7.8.10

© 2004 by CRC Press LLC

! <

$ 9

;

Section 7.8

749

Triangulations

!

FACTS

" # $% &

& ' '

( " # ) &

*

& ' ' ( " +,$ &

& ' ' ( ' ' " # ) &

- & ' ' ( ' ' " . ( / ' ! &

½

¾ & ½! 0 ¾ ! ! ( ' ' " 1, . ( / ' 2 ! &

½

¾ & ½ ! 0 ¾! 2 ! ( " )3 4 ½

¾ &

& '

' (

35 !!

&

' ½

¾ ( " 6

EXAMPLES

" . ¼ ( *

½ ( ½

- ¾ ( & & '"

¼ ! 0 ½ ! 0 % ½ ! 0 , ¾ ! 0 5 "

7 8 ( &( "

3

9 $ :

,

% 5 9)

93

" ; ´µ

( 6

REMARKS

" ' & (

& !

& '

( & & 8 ' < ; ' ' '( '

' &

)3 © 2004 by CRC Press LLC

750

Chapter 7

TOPOLOGICAL GRAPH THEORY

!" #

$%&&'

( ) $%&& '

*

!& (

+

Estimating Bounds

DEFINITION

,

-

.

FACTS

$%&!' , /

$%&!' 0

/

Æ ÎÌ

12

-

$%34' 0

( /

*

5

4& 4"

$%&"' 6

7

1

1

$8% 93.' ,

:

$8% 3.' ,

/

.3

;

<

"

$6 = 34' ,

7:

EXAMPLE

, / <

+

© 2004 by CRC Press LLC

Section 7.8

751

Triangulations

REMARK

Catalan triangulations

!

" " ! #$%& '

( # ))& ( ( (

DEFINITION

*

(

FACTS

# ))& *(

( -

! "

+.

+,

/

/

#01%2& *(

34

5

#

$ .. & *(

5

#

$ ..& *(

6

5

#

73 $ .-& 7

/

½ ¾ ½ 8 ¾

5

Preserving Properties *( ! ( $ ( +

- " (

(

DEFINITIONS

*

/

9

*

(

*

© 2004 by CRC Press LLC

!

!

!

752

Chapter 7

½

TOPOLOGICAL GRAPH THEORY

FACTS

!"# $

% &

'

%

(

%

)

%

*

% &

)

!"# $

'

( %

!!# $

& % &

+ % %

*

'

!!# % & %

+ % %

( (

EXAMPLES

(

, (

-

-

-

.

.

REMARK

+ % ( *

© 2004 by CRC Press LLC

!!#

!"#

Section 7.8

Triangulations

753

7.8.5 Rigidity and Flexibility

Equivalence over Imbeddings

!"

DEFINITONS

½ ¾

½ # ¾ ½ ¾

½ # ¾ $ %

# $ % $ %

FACTS

& '

" $ % ( "

$ % ) ) $ % $ % * " $ % $ % Uniqueness of Imbeddings

&

+ + Æ + © 2004 by CRC Press LLC

754

Chapter 7

TOPOLOGICAL GRAPH THEORY

DEFINITIONS

FACTS

!"#

$# %

$# &

Figure 7.8.11

&

$%# '

(

(

$% # &

)

)

Figure 7.8.12

&

Re-Imbedding Structures *

+ ,

© 2004 by CRC Press LLC

Section 7.8

755

Triangulations

DEFINITIONS

½

FACTS

!""#

!!"$# !!"$# %

%

%

!!"$#

&

' %

"# ( ) *' ' +' ,' - *

% )

./+# ( *' ' +' ,' 0' -' 1' */' *' *,'

*-' ,' ,1 */ %

EXAMPLE

%

'

3

4 *

&

4 *

'

'

2

' )

'

4 */

4 +-

)

REMARK

%

' Æ ' 2 1+

5 ' &

6

7

% . !!"$#

© 2004 by CRC Press LLC

756

Chapter 7

TOPOLOGICAL GRAPH THEORY

Imbeddings into Other Surfaces

DEFINITION

FACTS

!

! " #$%% & ' ()*" +

Figure 7.8.13

$$,- .* ' " / $$,- .* $ 0 / $-.. ! 0 / 1 ). )2 - 1 ). / -.* 3 24 5 4 5 " / ½

EXAMPLES

! / ! / 6

+ ! & !*" & &) 7 (% ! / 8 . ) % 4 )25 ! 9 :; /

© 2004 by CRC Press LLC

Section 7.8

757

Triangulations

References

!"# $ % $ ! & " % '( '

)* # +#

! + ! , ' ( -

** + *

! . + / ! % . ' ( ' 0 '

)1 + * +

6 ! ,,2 3 4 ! 5 $ , 2 ,' $ 27 6 .8 ( ' ( ' ' ( '

1+ ) +

! + ! 3 ' 9 ' ( : ; < :

* + *) *#

! +9 ! 3 ' ' ( ' : ( '

) + #1 1

!2 $ ! = 2 '; " ' '

* +

!" ") = ! " > 2 " <' ( ' ' '( ' ' '0 '

)+ )

# *

3 " 3 3 7 ' " > <' ' '

) 1 +*

3 " 9 3 3 7 ' " > <' ?9 '

#

3 ,5 " 3 3 7 ' 3 , 57 @; " > <'

' ( '( '

# )* 1

1* ? : / A' ( ' '

1*

*

. =1 4 B . & = 3 ' ( 5C

9' 9

* 1 * *

D" " 2 D'> ! 5 = " 5 ' ( -

D* 4 D > '8 9

© 2004 by CRC Press LLC

+ # *#

) * *)* *)

758

Chapter 7

TOPOLOGICAL GRAPH THEORY

!" #$%#!&

'($! ) ' ) ( * +, ,

!&

#$$!" .&%/&

&. ) 0 1 ,

!&." #2%#.

3 4 35 6 7

/$

!" !/%!/

3! 4 35 8

!!

!!"

!./%!//

3 && 4 3 9 : * ; ',, ,

"

" ,,/% 6 :9 49 '9 <; !&&

)$ 6 ) 6

!./

!$" !#!%!/.

< 44 3 < : 49 4 * +, = .

&

!" %#

0& 0=9 > 2$

!&" /#%#

/! 4 /

? 6 @/&$$#? A @

!$" #/2&%#/.2

0# 0=9 > ; , , B; ,

/.

!#" !%#$

04& 0=9 4 <

&$

!&" #/%#!

04 0=9 4 8 , <

&&

!" #!!%#!

04(. 0=9 4 : > ( > , = , , !2/

!." 2&%2

66 # : 6C 6

/

!#" !.&%!.

64/ : 6 4

4" .

-

!/" /&%&

6 4D$2 6 : 49 < D * +, 3 ,

,

6 4$2 6 : 49 * +, 3 , B; , ,,

© 2004 by CRC Press LLC

Section 7.8

759

Triangulations

!""

# $% #% &

!"

' ( &% )

)) ")!")

+ '" ' * ', + - &%% . % $$

/" ' 0 . % $%

)) /" 1!/

/) ' 0 .2 %2 $ 345$

"1 /) /!"

/) ' 67 15 5 $%

51

/)

8 ' ' 8 #

) !11

) ' # 9$

" )

!"

:1 ' ; : & < 25%

)" 1 !)

; ' ; ; =5

))

)!

/ ' # 9$ $$

) / "!)

' # 9$ 42

' >

) $ !)

) $ !

' # 9$ $ 5

) / !1

' ' ? '> 56

$ . %

)

/ !"

= ( = 8 % % @A % A % & # >

© 2004 by CRC Press LLC

!"

" " !"1

760

Chapter 7

TOPOLOGICAL GRAPH THEORY

!"#!" $ ! "! ! !" !

%% & ! ' ( ) "! !! (!# ! "! ( * +# !*

,%%- ./.%

! 0 ! !1 ( 2 ( " " 3 + ( (

* (

4

" 5 "1"6 "! # ( "$# 7# 8"( "! ! "$# !

). 9 ) 4 6" 1" $+ + +! ,.- :./:

© 2004 by CRC Press LLC

.

Section 7.9

7.9

761

Graphs and Finite Geometries

GRAPHS AND FINITE GEOMETRIES

Introduction !

" ! # $%

&

% " ! '

7.9.1 Finite Geometries % ( % ) ( ! DEFINITIONS

* + , & & %

* *

+

! "

, "

&

+ ,& +%% , + , " - " *

. /

. /

. % /

* * *

+ ,&0$01

-

+ ,&0$01 "

- -

+ ( ½ ¾,& 2 ) 0$01! / % / ! " " &2 + 2, % / ½ + ¾ , +%% ,

© 2004 by CRC Press LLC

762

Chapter 7

TOPOLOGICAL GRAPH THEORY

! " #

½ " $ ¾ #

! " #

%

& #

&

#

'

Æ

( $

& $ (

(

)

( $

( $

(

( * #

FACTS

+

,+,- ! " #.,+,- ! " #,-

¿

&

/

0 ( 0

'

* #

(

1

&

'

2 3 /

4 ) $

'

5 )6 )#" ( % 1 )

7,8209 +

# ' 2

$

1 )

' ¾ ¾ * * # #,+,- : ¾

( * * # ( Æ

© 2004 by CRC Press LLC

Section 7.9

763

Graphs and Finite Geometries

EXAMPLES

! " Æ

"

'

#

'

#

#

(

#

$

$

#%%&

$"# !

"# !

) * * * * +# !

, , + +- "#(%%& . /0 1 ¼

¼

¼

¼

¼

¼

¼

¼

¼

¼

¼

¼

¼

( "

&

* * * +# !

+ +- "#(%%& . /0 1

¼

¼

¼

¼

¼

¼

¼

¼

¼

¼

¼

¼

&

"

2 ! Æ $ " ¾ ¾ 3 3 "#%%& * 4 $ " # Æ +# " $# Æ $ $" 5#

! 4 6 " * $ " ¾ 3 3 ¾ 3 3 3 3 "#%%& * 4

2

$ " # +# " $#

4 6 Æ 6

¼

6" Æ

$ " $ "

© 2004 by CRC Press LLC

764

Chapter 7

TOPOLOGICAL GRAPH THEORY

!

""# $ !

%

Figure 7.9.1

&

%

!! ' % (

) *%

" *

+ ( % ! , % * ! ! -. ! & *% ! % %

!

! )! * * ! /

,

! * %

0

1 *% 2 * ! &

+

)

%

+ * % % 1 1 4 & & 5 1 & - . 1 4 & - * ! ! + ! 3 !

! ( ¼

& - . .

¼

¼

# !! ! 3 !

+6

)! / & ! ! ! !

+ 7)

! - & % # !

+ % +

! *

+

/

/ +

+ / &

+ - /

-

&

+ +

/ -

+

& /

& +

-

-

+

+ /

7 * * ! 8!!

+ +

© 2004 by CRC Press LLC

Section 7.9

765

Graphs and Finite Geometries

7.9.2 Associated Graphs DEFINITIONS

½

!

"

! "

FACTS

# $#$%

&

"'" (

&

!)" *

! "

)"

+**,

- . * ! /

-

- .

) 0

/ - . * ! /

-

- .

1 . !$% "

# &

EXAMPLES

!

# &

/ /

2! (

2 /. &

/ / 3 "'" 4 * ! "!

2!

4 1 . / ! "

4! 5 ! "

& * 6 / 4 1 . !7$#$% 87 )" 9 5" 2:/ ! & 4!)"

© 2004 by CRC Press LLC

766

Chapter 7

Figure 7.9.2

¿

TOPOLOGICAL GRAPH THEORY

¿

´¿µ

7.9.3 Surface Models DEFINITIONS

!

"

#

!

!

$ % " & !

# #

#

#

% # %

%

% %

'%

FACTS (

$ % % %

# # #

! !

# )

© 2004 by CRC Press LLC

!

Section 7.9

767

Graphs and Finite Geometries

!" #$%&

!' ())& * +

+ ,

- -

!(./& 0

-

¾ 1 1 2

1

3 1 ¾ 1 1 ¾ 1 1

3

)

- 3

¾ 1 1

¾ 1 1

4

¾ · ·½

3 , 5- 3

¿

, !()& ,

3 3 3-

¾ 2 ) *77+

6

8

9 * ,

: 3 5- , 3

3 , 3 3 ; - 3

-

3 ,

, ; 8

- 5- 3 3 3 ;

EXAMPLES

:

3 4

-

<

'

<

3

<

' 6.

- " = 3

© 2004 by CRC Press LLC

¾ ¿

3 3

768

Chapter 7

Figure 7.9.3

TOPOLOGICAL GRAPH THEORY

¯ ¯ ¯ ! "# Ü$%& ' ( ) ' * + ( , - .

# " ' -

, $% ,

, $% , /

Ü$%" , $%

Figure 7.9.4

References

012%3 2 - 14 - 5 2 ' 6

" "%% 778% © 2004 by CRC Press LLC

Section 7.9

769

Graphs and Finite Geometries

!"# $"%&$'$

( ) * ( + , - .

+

!%/ # $'&$$0

+ 1

2 3 % !%/

# ' &''"

/0 4Æ . 5 3

!'# 6 !%//0# ''&00

%

7

, + % "%

" !""# $6/&$/'

© 2004 by CRC Press LLC

! "$$

770

Chapter 7

TOPOLOGICAL GRAPH THEORY

GLOSSARY FOR CHAPTER 7

!

© 2004 by CRC Press LLC

"

"

"

"

771

Chapter 7 Glossary

½ ·½

! "

#

$

$

Æ

¾ ½ ¾ ¾

%

© 2004 by CRC Press LLC

772

Chapter 7

TOPOLOGICAL GRAPH THEORY

½

½ ¾

½

¼

¾

!

"

¼

!

!

!

#

½ !

$

¾ !

$ $

$

!

!

$

© 2004 by CRC Press LLC

!

½

$

$

773

Chapter 7 Glossary

$ %

$

(

'

$ &

"#

½

!

)

$

*

$

+ $ !

!

$

/

,

© 2004 by CRC Press LLC

&

Ê

/

,

- .

,

/

/

774

Chapter 7

TOPOLOGICAL GRAPH THEORY

! " ! " ! ! # ! # $ % ! % ! ´ µ !! ! ! ! ! $ % ! % !

´µ !!

$

½

$ &'

¾

( *

)

) +

,

)!

,

!

,

-

-

)

. /

! %

!

!

© 2004 by CRC Press LLC

775

Chapter 7 Glossary

¼ ¼ ½ ½

¼ ½

¾

!

"#

$

%

½

'

'

(

'

'

)

%

)

%

&

(

© 2004 by CRC Press LLC

776

Chapter 7

TOPOLOGICAL GRAPH THEORY

½

!

"

# $

!

%

!

!

&

½

& ' &

! & '

(

(

*

)

+,-

© 2004 by CRC Press LLC

777

Chapter 7 Glossary

¾ ¾ ¾

!

! " # $

$ !

½ ¾ ½ ¾

¾

% &

¾

% "

© 2004 by CRC Press LLC

778

Chapter 7

TOPOLOGICAL GRAPH THEORY

¾ ¾ ¾

!

"

!

# !

!

! $ %

& '

(

$ Æ )

$ $

!

'

$

$ (

© 2004 by CRC Press LLC

$ *

779

Chapter 7 Glossary

!

!

"

## " $ $ $ $ $ $

% & ' (

)

© 2004 by CRC Press LLC

780

Chapter 7

TOPOLOGICAL GRAPH THEORY

½

·½

!

"#

½

!

"#

$

%

%

& '

*

&

(

$

$

)

· ·

&

· ·

&

+

+

,

© 2004 by CRC Press LLC

781

Chapter 7 Glossary

½

¾

½

½ ! ¾

! ½

! " "#

# # © 2004 by CRC Press LLC

782

Chapter 7

TOPOLOGICAL GRAPH THEORY

!

! "!

$

# # !

$

%

Ê $

! &

' '

¼

Ê ¿

!

© 2004 by CRC Press LLC

¾

¾(

¾ ( ¾ ( ¾ ) "

783

Chapter 7 Glossary

!

"

# $

$ $

$

&

%

!

'

(

$ ) $

$

!

*

½

&

&

#

+

-

© 2004 by CRC Press LLC

,

+

½

!

+

½

784

Chapter 7

TOPOLOGICAL GRAPH THEORY

! ½ ¾ ¿ " ! " # !

$ % ½ ¾ ¿ ½¾¿ $ % "

© 2004 by CRC Press LLC

785

Chapter 7 Glossary

½

·½

¾ ¾ ¾ ¾ ! !

" ¾ ¾ !

!

#

¼ ½ ¾

$ %

$

& '

' (

& ¼ ½ ½ ¾ ½ & ½

© 2004 by CRC Press LLC

786

Chapter 7

TOPOLOGICAL GRAPH THEORY

© 2004 by CRC Press LLC

Chapter

8

ANALYTIC GRAPH THEORY 8.1

EXTREMAL GRAPH THEORY

8.2

RANDOM GRAPHS

8.3

RAMSEY GRAPH THEORY

! 8.4

THE PROBABILISTIC METHOD

" # GLOSSARY

© 2004 by CRC Press LLC

788

8.1

Chapter 8

ANALYTIC GRAPH THEORY

EXTREMAL GRAPH THEORY ! " #$! %!&' ( ) ! * + !& & , -!! & & . / 0 ' !1 2!! . 3 4 !& ! 5! ( ( &

Introduction / ! && %! ! 6!! / &! / !7! & ! 6!! ! 4&&!/ 8 ! ! %! 8 ! & / 7 8 ! & ! !9 & : ! ! ! 7! % % &! % ! !& % / 8 7 / !/ & ! &! !& ! ! 7 8 ! %!& % && !& / !7! & !' 8 ! !! / 8 & &!7! 8 &!6 8 ! & 8 / ; & & ! NOTATIONS AND CONVENTIONS

2 !&! 8 / 9 7 <= > ?@ / %! 7 !& ? @ / %! 7 !& ?A B@ / ?A # B@ 8 A! ! > ?@ %! 8 7 7 / 8 ! -!!B 7 / !! / ! -!!

?@ ?@ 7 !& / ! /

© 2004 by CRC Press LLC

Section 8.1

789

Extremal Graph Theory

C?@ Æ ?@ ! !! / ! 7 / 8 D ? @ ! ! !/8 D

? @

! ! / E

%!& /

½

½

E

/

? @ D

¾

!F! /8

½ ¾

? &! @8 !8 !

% / F!!/ 7 7

D? @ !

? @ >

? @ % ! ! &

! !

/

¾

? @

?@

½

7 7

½ G ¾ $

¾ / %!

&! & /

!

DEFINITIONS

:

!7 /

8 %

! !

: 4 ! & / 7 !& &!6 /

:

!7 /

:

:

4 / !

?@

& &

!

8 ?@

7

& &

?@

?@ 6

8

! ! / &!6

! &!6 ! &

! ! %

& &!7!

/ !!&

/

! 7 !& 7 !

! !

&

EXAMPLES . ! !H & % %! % ! ! / %! %

:

.

?@

!! !/ &7 /

& / %!

7 !&

?@

!

6! !H ! ! ! ! / 8

:

! A 7 & & / %!

! /

&! & & %! !

7 !& B

! !

7 !& ! ! ! &

! / %! % ! !: ! ! ! !77!/ 7 !& 8 !! / 8 & &!7! 8 &!& & ? ! / & & @

8.1.1

Turán-Type Problems

6! ! ! / ! %!/ 6 ! : /!7

I

8

© 2004 by CRC Press LLC

?@

?@I

% ! ! !' /

6!7 8 % ! ! !'

&!

790

Chapter 8

ANALYTIC GRAPH THEORY

DEFINITION ! / !'

:

½

.

/

?@

7

? @ ! &

&!

/

! & 8 !!

! 9 ! &

6 & !! /8 J? @

J? @ %!&

½

?@

? @

>

&

? @

6 & &

½

! / !

!! ! /

Turán’s Theorem and Its Extensions !7!/ & /

!

&

DEFINITIONS

:

.

! /

?@

! &

!

/ % & 6 !

:

?@

! !' /

?@

FACTS

:

G

>

?3

@8 ?@

8

& &!!

?@

>

"

?@

<53,= /

:

!

?@

>

:

?@8

<# $"0=8

? ?@@ 7 8 > ?@

:

%!

:

<"= /

> ?@

!

"

8 >

?@

!

K !

?@ "

8

> ?@ ! 8 ?@ > ? ?@@ &

<+= /

>

?@ ?@K

?@

/ 8 ! ! / ?@8 & ? @ ? @ 7 ?@ ! & /8 & ? @ ? @ ?@

? @

!

> ?@

:

& &!!

G

>

?@

<,3= L 7

>

!"##

7

<M,,=8

>

# $

?@

© 2004 by CRC Press LLC

% 7

?@

!

8

Section 8.1

791

Extremal Graph Theory

: "# %& # ' # <5+*= /

½ /!7

¾ ´ µ ¾ ´µ

> ?@ ! ·½

¾ ´

µ

¾

: (" # ' # ?@ ! ·½ ! ½ /!7 ' 8

¾ ´ µ ¾ ´µ

G

¾ ´

µ

¾

¾

¾ ´

?@

µ

. ½ ' ! ½ 6! ! ?@ ! ! ! ! > 8 ½ > > 8 7 !& ½ &!6 ! Structural Properties of the Graphs ? ?@ G @

1 / 7 ? ?@ G @ &! ·½ 8 !7 !/! 7 ! & / E % !& &! ! NOTATIONS

?½ @ & ! / %! & !' ½ &!7 / · ?½ @ ! ! ?½ @

!/ / 9 &!9 &8 ! 8 ½

L Æ ? @ ! 7 & 7 / ? @ &!6 %!

? @ : Æ ? @

FACTS

): (*
? ?@ G @ 7 %! / ? @ ? G D? @ !& ½ ?? @@ /

@ ½ &

+: (
: (*
> ? ?@ G @ % 3 . 7 ! / ? @ > ? 8 ? @ > ? @ 8 D? @ ! ! 8 7 7 D? @ ! F! 7 7 D? @

: <+= L 7

3 ! > ?@ 3 & 7 ? ¾ " G @ &! ¾· ? / ½ @

: &# # <!,= L 7

? ?@ G @ &! · ? @

: , <%,,=8 <%, = 8 Æ ? @

© 2004 by CRC Press LLC

Æ&! / 8 7

¾

?@8

792

Chapter 8

:
ANALYTIC GRAPH THEORY

@"8 ?@8

¾?

Æ ? @

?@

REMARKS

$:

8 L& ! ! 8 !& ! ! 9 / && !

/ > ? ?@ G @ · ? @ ! ! Æ&! / -% 7 8 L& // ! ! ! 8 / % %

& !

$: $:

N !$!7 % L& * 7

! L& + &9 &F & N 3 ! !Æ& 6 ! ! Æ ? @ ½ ?@ ?@ ? 8 /8 <P =@

Books and Generalized Books

$ % !!! ! 0+ <+= & & H !& 7 8 $8 & ! & 8 / DEFINITION

):

L 8 ! /

!/ & &!6

´ µ

&!!/ !!& ? G@ &!6 8

7 ! & $K % %! ´µ ?@ !' / $ ! / E & $ ! %! ?@ ´¾µ?@

FACTS

: # - ? ?@G@ &! ·¾ %! / 7

: , <5=8 !"## . /##0 <M!,0=

¾ "

G

+

! ! 6! ! ! ! ! 7! % %!/ / . > +

!! <= ! + !½½ 8 !½¾8 !½¿8 !¾½ 8 !¾¾8 !¾¿ %! !½½ > !½¾ > !½¿ > !¾½ > !¾¾ > !¾¿ > G L " F! 7 7 ! 7

7 ! 8 " > F! 7

7 !½ 7 7 !¾ !' !/ / ! / ¾ " G 8 ! $!' ! + G

© 2004 by CRC Press LLC

Section 8.1

793

Extremal Graph Theory

: . . $ <L(0"= ? @¾ Æ&! / 8

´µ ?? @@

! !

" ? G @

L 8 3 9 / %! ?@ > < = <= <= F! % 7 !& ?½ #½ $½@ ?¾ #¾ $¾@ ! ! ½ > ¾ #½ > #¾ !/ >

>

% 7 > ? @8 8 ´µ ?@

>

¾

): . 12# <L0*= ! + & ! > ?@ Æ ?@ 8 ?@

Vertex-Disjoint Cliques

% ! ! / !&8 -F ' ! 8 / % !!9 N !
! 7 !F! &! / !

+

: . / & ?@ > ? @ 7

! !

FACTS

+: 3 #%4 5

<-+= . %! > 8 > ? @ > ?@ Æ ?@ 8 &! 7 !F! & & / G 8 / !&8 ! Æ ?@ 8 &! ¿

4 5 %&"#

: <-',3= > ?@ C?@ 8 ?@ & !! ! G ! & !' % !H

8.1.2 The Number of Complete Graphs 4 &!!/ !Æ& ! ! ! ?@ /!7 7 ?@ ! % !!!/ %8 / ! / 7

© 2004 by CRC Press LLC

794

Chapter 8

ANALYTIC GRAPH THEORY

FACTS

: $ - <+=

"

G

! ! 6! ! !

: 6 " &# # <.! = % 8

"

G%

%

! ! 6! ! !

: . 3 % / ! ! !/ % !F! / / & / ?@ ?@ ! /!7

G &

?@ > ? %@ > %

: <+0= L 7

3 % 8

! > ? @ 3 > ? @8 & !

?? ?@ G %@@

? %@

! ! 6! ! !

: # ? @ " ?@

0 ,? @

! ! ! '? @

: / &, <+=8 <55+= > ?@ ?@ 3 ?@ ? @ ??@ @ ?@

: (*

. &! (?@ 9 ! & 7 > 8

?!@ (? ? ?@@@ > ? ?@@K ?!!@ ( ! ! ! ! 7 < ?

© 2004 by CRC Press LLC

?@@ ? ?@@=

?@

(? ?@@

Section 8.1

8.1.3

795

Extremal Graph Theory

˝ Erdos-Stone Theorem and Its Extensions

& ! & *3 K ! %! &! ! & 7! % &! ! 1 : !/ ?@ / ! / / ·½ ·½ ?@ ! / 6!7 8 7 !& 9 ! ! / NOTATION

L 3 8 )?

@ ! & 7 ? ? G @¾ @ &! ·½ ?@ Æ&! / FACTS

): %& <"+= L 3 8 &! )?

@ !9! % !9! !& ?@ ! &!&8 ! !! !7! <!++= ! > ½ ! 9 ! / G > ! *? @ 8 ? @ >

¾ G +?¾@

+: (*
½ ¾ 3 &

½ / )? @ ¾ /

: (* . &# #
/

)? @

/ ½

3 & !

! 3 & ! 3 )? @

/ / ½

: 3 &"# <' = )? @

: (* & !

/ *33 / ½

,
3 ,

3

7 / Æ&! / %! ?@ ? G @¾ &!

·½ ? %@ & ? , @

© 2004 by CRC Press LLC

/ / ½

? , @

/ /

% ½·¾

796

Chapter 8

ANALYTIC GRAPH THEORY

: 7#8 # <3= ! & - 3 & !

3 ,

3

7 / Æ&! / %! ?@ ? G @¾ &!

·½ ? %@ & - ? , @

/ / ½

- ? , @

/ /

%

The Structure of Extremal Graphs

& / ! ! % ! & 9 ! ! / 5 7 8 !! !7! /!7 !! & / %! ! /8 7! !' ! & ! FACTS

:

&# # <!+ = L 7 ! > ?@ & ! % 7 ·½ / ? ?@ %@ ! &!&

:

*? @ >

<+ =8 &# # <!+ = . / %!

G L 7 3 ! Æ 3 & ! ! / ?@ ?? @ Æ @¾ ! ?½ @ %! ½ G G > & ! &/!/ ¾ /

: <+ =8 &# # <!+ = .

> ½ 9 / ! %! G > ! *?@ ½ ? G @ &!/ ! %!& & & &! 7 !& ? @ >

? @ ¾ G '?¾ ½@

J? @ ! & ?@ > ? @8 Æ ?@ > ? @ G +?@ %!/ : ?!@ 7 !& & !! !

& & !' G +?@K

?!!@ & 7 ! F! 7 !& ! % & & L 7 3 7 !& F! 7 !& ! % & ! +?@K ?!!!@ '?¾ ½@ / F!!/ 7 !& &

: &# # <!+ = . 9 ! / ! Æ&! / 7 / > ?@ %! ?@

? @ G ? @? @ G

&! ·½ 8 > ½ G ? @

© 2004 by CRC Press LLC

Section 8.1

797

Extremal Graph Theory

8.1.4 Zarankiewicz Problem and Related Questions #$! %!&' ! & 1 !! /K ! !Æ& R ! 7 % 7 !& ! NOTATION

. $ ? @ / !' !! / &!!/ & !! / ? @8 $ ? @ > $ ? @ FACTS

): ? ? @@ $ ? @ ? ? @@

+:

2 . & <M*"= 8

$ ? @ ? @ ? G @

G ? @

$ ? @ ? @ G '?@

: $# <( !* =

?!@ $ ? @ ?@? G " @K ?!!@ 7 > G G 8 % ! % ! 8

$ ? @ ? G " @ > ? @? G G @

?!!!@ !

$ ? @ >

: . $ # & <(++=8 (, ? @8 % 7 G ? G @ ? G G . @ ? G @ G !

?@

? . @ >

/!7!/ % ! ?@ 7 !& G G ! F &!7 / ? @ 7 9 8 % ! ? # $ @ ? # $@ F! ! ! G # # G $ $ > 3

: 2 #
? G G . @ ? G @ ! ! % ! 8

? G G . @ > ? G @

© 2004 by CRC Press LLC

798

Chapter 8

-Free

ANALYTIC GRAPH THEORY

Graphs with Large Minimal Degree

0, ! & !!! / %! / !! / & ! &!!8 &!& & / ! 8 !7 !/! % 8 ! & ! % ! Q ! ! !7 H / 6 ! ! 8 & /!/ %!& ! &F & 7 > ?@ %! Æ ?@ ! " &!& DEFINITIONS

: E / ! / ! ! !/ : ?@ ? @ & ? @ 0 ?@ !! ? ? @ ?@@ 0 ? @

: . ! / %! ?@ > <= ?& " @ 0 ?@ ! ! & " >

:

! ! & %

EXAMPLES

:

<5 &**= Q 9 6 & / 1 1 %: 1 > K 1 ! 9 8 ?1 @ > <= ?1 @ > < G =8 0 ?1 @ !

0 ?1

@ ?& " G @ : ?& " @ 0 ?1

@ ? G &@ : &

L 7 &8 / 1 ! *?1 @ > & !&8 / 1 ! . K / 1 ! $%

: ! <4+= ! > 7 & ! & ?& @ % . L 7 &8 / ! ! *?1 @ > !&8 / ! ! . K / ! ! $% FACTS

9 0 #. & <4,"= / > ?@ ! %!

: !! /

Æ ? @

! &!&

: %4 5 %&# # 8 : <!,= 38 2 ! Æ&!

/ 8 ! / ?@ %! Æ ?@ ? @ *?@ 2

: 42 88# <-/ = 7 / > ?@ %! Æ?@ !& ! > . !9 ! 6! *?@

!

: 42 88# <-/ = L 7 8 ! " &!&8

8 3 / / 0

: ;# <S!0*= 7 > ?@ %! Æ?@ 30 ! !& / ! !9 ! 6! *?@

© 2004 by CRC Press LLC

Section 8.1

799

Extremal Graph Theory

) 3 . ;#

: <SM0,= 7 &!& > ?@ %! Æ ?@ ! !& / ! *?@ "8 1

+: (
"

42 88# ;#

: <-S!0 = . > ?@ . Æ ?@ "8 ! !& . " ! ! 8 !

8 . 8 / ? @ ! !& .

8.1.5 Paths and Trees O 7 ! / !

& - 5-:

%

7 / ? ? @ G @ &!

&F & ! &! &!! /8 / & ! NOTATION

/

FACTS

: %1 # <*0= 7 > ?@ %! ?@ ? @

&! / > ? @ G 8 ! / > ? ? @ @ &!!/ /

: &-: G ? / @ >

G

3

8

/ $%

: ( *
: & - <" # <E0,= 7 . / ? ? @ G @ & !

: < 8. 6# 6# <E..33= 7 / ? ? @ G @ % / & ! /! * &!

:

*

@ G @ % / & !

? "@ &!

© 2004 by CRC Press LLC

800

Chapter 8

ANALYTIC GRAPH THEORY

8.1.6 Circumference DEFINITIONS

: ?@

:

/ ! / ! / & & !

/ ! / ! & &

NOTATIONS

& & / / ! . ?@ ?@ ! / 7 ! . ?@ +?@ ! / ! . ?@ FACTS

: 1 # <*0=

?? ? @? @ G @@

): (* 42 88# ?@ Æ?@ 8 ?@

? @ ! ! 6! ! !

+ 8 , #

: </5! 0= > ?@ ! ? @ G ?@ % 7 % F& 7 !& 8 ?@ ? @ ! ! 6! ! !

: # - ?@ ! & & %! Æ?@ > Æ 8 ?@ Æ

: = 8 ?@ ! & & 8 !! / %! Æ ?@ > Æ

8

+?@ Æ ?@ Æ

8.1.7 Hamiltonian Cycles -!! / ! / - % 7 % $% %! A B ;7 DEFINITIONS

: :

4 / > ?@ ! !

"

! . ?@

/ ?@ ! ! && !7 F!!/ 7 % F& 7 !& %! ? @ G ?@ FACTS

:

# - ?@8 8 Æ?@ 8 !

-!!

© 2004 by CRC Press LLC

Section 8.1

801

Extremal Graph Theory

: &# <!0=8 (* (#8,
> ?@8 3 7 !& / 3 8 ! & & ! !& 7 7 3

: > ?@ %! 8 ! ? @G ?@ % 7

% F& 7 !& 8 ! -!!

: ? < += . > ?@ %! 7 %! ? @8 7 !& / & !/ ! 8 7 !& / ? @ & ? @8 ! -!!

: 3 6 0 ( 3
: 3 <7,= . > ?@ %! %! 7 /

?@ ?@ 7 ? @ ! ? @ ? @ 8 ! -!!

): 3 % <,= ?@ ?@8 ! -!!

+: 42 88# ?@ Æ ?@ *,8 6 .

&!

8.1.8 Cycle Lengths : . ?@ % & & / ! / - &F & %!/ % FACTS

: 1 0 .

. &"# <M "= ! 3 &

7 / > ? @ % 7

:

. ?@ /?@

: 1 0 . ?2. &"#. =#8 < P *= ! Æ&! / ? G @8

:

. ?@ ?33 / @ ½

Cycles of Consecutive Lengths

! &! % 7 Æ&! &!! ! !' !! / ! & / ! 7 ! . ?@ & & / FACTS

: ( %&# #
© 2004 by CRC Press LLC

802

Chapter 8

ANALYTIC GRAPH THEORY

: = 2

1 8

:
G . . . . > ? & @ . .

: 1. 4 @. &- <-3= L 7 3 ! & > ?@ & ! > ?@ Æ ?@8 &! & & 7

7 <" ?@ = 7 < +?@ = Pancyclicity and Weak Pancyclicity

0, N !& && & &!&! & !& ! !7 E % % $% DEFINITIONS

: ):

4 / > ?@ ! & ! . ?@ ! ! 7 4 / > ?@ ! & ! . ?@ > < =

FACTS

: ( ? "@ ! -!!8 ! & &!& > ? @

: ( ? " G @8 ?@ ? G @ !

% $ & &!&

) 9 . # . #. 1

: <4LL = 38 > ?@ ! -!! Æ ?@ *8 ! & &!&

+: &# <! += *38 > ?@ ! -!! 7 % F& 7 !& 8 ? @ G ?@ "*8 ! & &!&

: ( . 1 " + , ?@

( . . 1

:
& &!& !!

(

(*

© 2004 by CRC Press LLC

Section 8.1

8.1.9

803

Extremal Graph Theory

Szemerédi’s Uniformity Lemma

2!! . ' !8 % % 7 !! & 7 !' 8 ! $ ! !& !& .

8 ! / 7 / 9! / & 8 %!&8 !!/ 8 ! ! && &$ !Æ& &!! N% . M $T' ' !8 & !7 2!! . 8 7 !Æ& / !/ &F & L & !7 7 ! <M!0+=8 <M00=8 <M33= DEFINITIONS

+:

. 3 4

! &!

% !! ! !

: 4 !! / %! & ! ! & ! 7 ! 7 4 ! 5 %! 4 ! 5 % 7 0 ?4 5 4 5

@

0 ?! @ !

: 4 !! / %! & ! ! & ? Æ @ ! ! ! ? @ Æ ?@ Æ ! % 7 7 !8

!

FACTS

: &"# A #0# 6 <' ,+= L 7 3 ! > ?@ > ?@ & 7 / ! !! ?@ > ! !/ &! !:

?!@ ?@K

?!!@ 8 ?!!!@

:

> > K

! ? @ !

&! ?@ ! ' !1 2!! . ! 7

% /

1, *

: <%0 = ! & 3 3 & 3 8 ! / % 7 !& & !! &&!/ &! ! ?!@ ?!!!@ ' !1 2!! . ! / % /

: (,%: 6 . . & 2". &"# <M0,= L!

/

%! ?@ > < = L 7 Æ C 3 ! 3 & %!/ L! !F! !' Q 9 / ?@ %:

© 2004 by CRC Press LLC

804

Chapter 8

ANALYTIC GRAPH THEORY

?!@ ??@@ > 7 ?& " @ 0 ?@8 & / %

?!!@ ?@@ > 7 ?& " @ 0 ?@8 & ? Æ @ ! ! %

?@ C? @ C

Applications of the Uniformity and Blow-up lemmas

2!! . N% . % ! / E % ! !&! 8 ! &8 &! ! &! ! !/ % FACTS

):

. & 2". &"#

<M0 = L 7 3 ! > ? @8 & !

> ?@ Æ ?@

G

/ &! % -!! & &

)+: 9 B <4U0+= L 7 3 2 ! > ? 2@8 & ! ! / 2 %! *? @ >

> ?2@ Æ ?@

G 2

/ &! &

):

. & 2". &"# <M3= . / 2 *? @ > ! > ? @ > ? @ & ! > ?2@ Æ ?@

2 G

&! &

):

. & 2". &"#

<M3= L 7 38 ! & 8 7 / > ?@ %! Æ ?@ ? G @ &! 7 & C ? @ /

8.1.10 Asymptotic Enumeration 4 !!/!/ 6 ! ! % / %! /!7 ! L & ! ! !$ A ! B A !& / !!& B !& % 7 ! DEFINITIONS

:

4 ! / ! & !!

© 2004 by CRC Press LLC

Section 8.1

805

Extremal Graph Theory

: 4 / / ! & /

! / !! / 7

: 4 / / ! & !& /

! / !! / 7

:

L / / 8 / > : / ?@ > / ! 7

?/ @ > ?/ / @

: . 3 ! / 4 / ! & ? @ ! ! 7 !& & & ! &8 7 !& & %! & & &!6 & 8 ! %!

: ?/ @ ! / ! / ! / & 8 ! / &! 7 ? @ & / NOTATION

!7 ! / 8 ? / @ ! / ! / > ?@ %!& ! / > ?@ %! ? @ > ? @ 0 ? @ 0 ?@ > 8 7 / %! / /

FACTS

) . # . $-#

: <M(,= / ! /!7 !&

/ >

G +?@

): . . $2 <L( += . / %! *? @ > 2 / ! /!7 !&

/ 2 >

)

G

G +?@

##. ?2. $-#

: <M ( ,= L 7 / ?@ 2 &!& / ?@ !

2

>

):

4 / ! ! ! ! 6 & / & / ! & &! / 7!/ / !!&

):

4 / ! ! ! ! ! 6 & / & / ! & &! / 7!/ !& / !!& &!/ ?/ @ ! & ! & 3 ! ? @ &

© 2004 by CRC Press LLC

806

Chapter 8

) : ?2 &8

?@

%

> 3 8

< 0= !

.

%! !&

>

> ? G

ANALYTIC GRAPH THEORY

&

! /!7

¾ +?@@

)): ?2 &8 < 0=8 < 0= L!

8 B

/

! A !& / !!&

? / @

!

>

/

?/ @

++: 9 <4 0=8 (*
!

>

?/ @ ?/ @

!!

?/ @ >

!

?/ @

!

+: &-# # <&#!0"= L 7 !

/8

%!/ ! : ?!@

/

Æ&! /

! ! !& 3 K

?!!@ / > V?@ ! / ?!!!@ ?!7@

38

38 /

/

K

K

+: (*
?/ @

>

? / @

!

!

>

/8

?/ @

+: ( 8. (* . <# #-
!

/

%!/ ! :

?!@ ! & &! ! /

/ >

6 ?@& K

?!!@ ! / ?!!!@

8 / > ¾

/

?!7@ ! /

8

K

/ >

6 ?@

&

Æ&!

K

¾

8.1.11 Graph Minors / ! % !!! !7 &F & -%!/ 7

&!& /

! -% 7 8 !

7! %8 ! ! % &!!/ !

© 2004 by CRC Press LLC

Section 8.1

807

Extremal Graph Theory

DEFINITIONS

):

. / E ! 8 % %!

! !F! 7 ? @8 ? @8 & 7 ? @ !& & & / ! 8 7 ? @ 0 ?@8 ! / % 7 ? @ 7 ?@

+:

. 8? @ !! 8 & ?@ 8?@ !!

FACTS

+: <5+,=8 <5+ = 8? @

/

-

+: (* . 3 # .
+:

.

<M =8 <M "=8

< "=

8? @ > '?

+:

/

/ @

<3= ! !&! & > 30 & 8? @ > ? G +?@@

+ :

, ? @ > !

/

<5 3= !7 / 8

¾

9? @

%!

¾

>

% 9? @ /!7 !/ 7 !& 8? @ > ?, ? @ G +?@@

8.1.12

/

Ramsey-Turán Problems

( ! & %! %! !&! ! & L & !7 7 ! !& <!3=K % % !/!/ NOTATIONS

.

9 / . ? ?@@ ! !' / ?@ %! ?@ ?@ % / & & ! & ! ! & & E %! ? ?@@ ! ? ?@@ % !

. ?@ ! ! & & & / & / ! &8 ! &!& !/ & ! & ! ? @ & © 2004 by CRC Press LLC

808

Chapter 8

ANALYTIC GRAPH THEORY

FACTS

+): 8 : <+= L 7 ! 3 & ! ! Æ&!

/ ! / > ?@ %! /! )?@ ! &

?@

+: & <,3=

G +? @ % & %!/ /: $ / ?@ & ! & & 8 % ! !' & ? +?@@ >

: (* ?@ %! ?0 @ ? @ " 7

?0 @ 8 ? +?@@ G +? @

: &"# <' ,= ? +?@@

G +? @

: . 4 5 . & . &"# <- = ? +?@@ >

+

* G +?@ "

7 % &! %!/ /: % > "? @ $ 0 G ? %@8 & ? %@ & 8 % ! !'

: & <,3= ? +?@@ >

G +? @ "

: . 4 5 . &. &"# <- = ? +?@@ >

?@ G +? @ ?@

: . 4 5 . &# #. &. &"# <-0= ? +?@@ > ? +?@@ >

*

G +? @ G +? @

6 ! / ? . . +?@@ >

© 2004 by CRC Press LLC

G +? @ "

Section 8.1

809

Extremal Graph Theory

References <4,"= N 4 !8 P 8 O & &! % &!& 8 ! &!6 !! / /8 ?0,"@8 3*W <4LL = Q 48 L!8 L! 4 8 & &! ! -!! /8 ) ?00@8 W <4 0= P 4 $ 78 (/ 7 ! & / ?( !@8 ?00@8 " W*, <4U0+= 4 ( U 8 ?00+@8 +0W

& ! /8

T <4+= N 4!8 2 ! ! ? @8 ?0+@8 ""W"**

© 2004 by CRC Press LLC

810

Chapter 8

ANALYTIC GRAPH THEORY

8 4& !& 8 0,

! 5 !&8 .

/ + ?00,@8

© 2004 by CRC Press LLC

Section 8.1

Extremal Graph Theory

811

?0+@8 ",W"
¾8 !

<%,,= %8 / 7 / !/ ! /8 8 ) ?0,,@8 3W3 <%, = %8 / %! / 7 / 8 ! 1 2 8 -8 4 % U$8 0, 8 0W3+ <5= %8 4 % / !/ %! & / 8 5&! <L( += 8 L$ P (T 8 !& / & !!/ 9 / / 7!/ 8 ?0 +@8 W <L(0"= 8 ( L ( 8 / !' / 8 ?00"@8 0W* <L0*= 8 ( L T!8 O $ !' / %! / !! / 8 + ?00*@8 *W"+

© 2004 by CRC Press LLC

812

Chapter 8

ANALYTIC GRAPH THEORY

</5! 0= U /% 5! 8 / & & ! / %! !! / 8 ?0 0@8 *+W+ <M(,= 8 Q S M ! N . (&!8 4 !& !

/8 1 0 3 %4 ()*5, 008 0W, ' 6 (* 63 8 ( 8 0,+ <- = 8 4 -F8 P ' !8 5 ( 8 ?0 @8 +0W <-0= 8 4 -F8 5 !7!8 P ' !8 (

! !& & 8 ?00@8 W*+ <+= 8 !! 8 ?0+@8 "+W* <+= 8 O & / &! ! & ! /8 " 0 =778 4 ?0+@8 "*0 "+" <+= 8 O ( & 8 0 ?0+@8 W, <+= 8 O & ! /8 0 ?0+@8 *+W+3 <++= 8 & ! / ? @ ,W3 ! $ %0 4 ()77,8 N &8 % U$K Q8 ! <+ = 8 O % ! 6!! && !/ ! /8 ,,W ! $ %" 1 ()77,8 4& !& 8 0+ <+0= 8 O & / &!&! &! ! /8 8 "8 ) ?0+0@8 03W0+ <,3= 8 O / ?-/!@8 ?0,3@8 "0W* <*0= !8 O ! &!&! /8 + ?0*0@8 ,W*+ <(++= 8 4 ( ! P 8 O / 8 ?0++@8 *W* <!++= 5 !7!8 4 !! ! / 8 ?0++@8 *W*,

<!,= 5 !7!8 O 7 & ! / 8 ?0,@8 W" <,3= P 8 $ ( 1 1 8 0*W"3" ! 00 %" 1 ! $- ()7), -8 4 8 0,3

© 2004 by CRC Press LLC

Section 8.1

813

Extremal Graph Theory

< = P 8 O / !'! 1 / 8 W * ! " 8 N!$T 8 0 <"+= 4 - 8 O & ! /8 ?0"+@8 3 ,W30

< P *= 4 8 - T 8 ' ! N P!/8 O &! & & & / ! /8 ?0 *@8 "W* <%0 = E % 8 .% % ' !1 !! 8 # ?00,@8 W, <-/ = ( -T//$7!8 O & & &!9 / ! !! /8 0W00 ! % ()9(,8 - 5 8 8 -8 0 <-S!0 = ( -T//$7! S!8 %! /! 7 !/ !! / 8 ?00 @8 *W+ <-',3= 4 -F ' !8 &F & T 8 +3W+ !

0 00 %" 1 $- ()7), -8 0,3

© 2004 by CRC Press LLC

814

Chapter 8

ANALYTIC GRAPH THEORY

<S!0*= S!8 !/ &!& /8 ?00*@8 *W ,3 <3= U !/!8 &F & N M $% T 8 ?33@8 W*" <M,,= MX'!!778 6!! / ?(!@8 4 + ?0,,@8 ,0W,0+ <M00= S M 8 % 8 " ?000@8 +W,+ <M33= S M 8 !!/ 8 + ?333@8 3W <M = 4 M&$8 !! -%!/ / %! /!7 / 7 !& ?(!@8 3 ?0 @8 ,W* <M "= 4 M&$8 4 % -%!/ / ! 7 / / 8 ?0 "@8 3,W+ <M!0+= S M 5 !7!8 ' !1 /! ! !& ! ! / 8 " : 8 P ?M ' 8 00@8 0*W*8 N ! & 5 8 8 S N ! 5 &8 N 8 00+

<M ( ,= M!!8 - T 8 N (&!8 ·½ /: !& & 3 %8 + ?0 ,@8 +,W+, <M0,= S M 8 $T' 8 ' !8 N% 8 ?00,@8 30W <M0 = S M8 $T ' 8 ' !8 &F & / /8 ?00 @8 "W+3 <M3= S M 8 $T' 8 ' !8 4 U &F & 8 ?33@8 **W+0 <M3= S M8 $T' ' !8 !/ ! /8 " + ?33@8 0,W"+ <.! = . .7' 5 !7!8 O & / / 8 "*0W"0*8 N!$T 8 0 <5+,= E 5 8 -! !/ & ! M !& 7 ? @8 ?0+,@8 +*W+ <5+ = E 5 8 -! T ' T ? @8 ?0+ @8 *"W+ <53,= E 5 8 8 - 7 $8 E 5 8 S ! ! 5 8 L & E 4 E H8 ; /' + ?03,@8 +3W+ <55+= S 5 . 5 8 O 8 <2 0 <- 3 , ?0+@8 W +

© 2004 by CRC Press LLC

Section 8.1

815

Extremal Graph Theory

<5+*= 5'$! 8 5 ! / % 8 ?0+*@8 *W*"3 <5 &**= S 5 &! $!8 &!/ / ?L &@8 1 8 ?0**@8 +W+ <5 3= S 5 4 8 &! & !8 ! <+= N %8 !/ ! ! /8 ?0+@8 W"
. 8

<! += ( !8 O &!! & &! -!! / ?! @8 ?00@8 ,0W03 <!0= ( !8 !/ -!! &!!8 ?00@8 +,W,

<!+ = 5 !7!8 4 7!/ ! / 8 !! 8 ,0W0 ! $ %" 1 ()77, 4& !& 8 0+ <!,"= 5 !7!8 / %! !& / 4!! &!& &!!8 ?0,"@8 "0W,+ <! = 5 !7!8 / 8 +W33 ! ? . E N ! $ ( S E!@8 7 8 4& !& 8 0

<' ,= ' !8 O / &!!/ & / %! " 7 !& ?- /!@8 ?0,@8 W+ < "= 4 8 4 &! &&! /8 " " ) ?0 "@8 +W+*

© 2004 by CRC Press LLC

816

Chapter 8

ANALYTIC GRAPH THEORY

<3= 4 8 &! & !8 ?33@8 W <"= 8 O ! / ?-/!@8 2 #3 ?0"@8 "+W"*

= 0 ?0,,@8 *,W,3 <E..33= 5 E/8 .! 4 .!8 4 T &F & 8

?333@8 W, <# $"0= 4 4 # $78 O ! ! & ?(!@8 6 CD ?0"0@8 +W

© 2004 by CRC Press LLC

Section 8.2

8.2

817

Random Graphs

RANDOM GRAPHS

( 5 L&! / Q / 6 & " !! * 5 5 ! ( + ( ( / , O ( 5 ( & ( &

Introduction 9 / & ! ! % %! ( ! ! 0*0W+ ! ! & ! ! !!!& 8 % & / % 7 ! & / %! ! ! 8 / !7 8 ! !/ / 7 % !&7 L !& 8 / / & ! ?! &! @ 7 ! &! ! 9 / 6 8 ! %! &! / & /8 %! 7 / / %! & 4 / & ! 7!! !!/ & &!&! %$ 7 / 7 7 F & ?
8.2.1 Random Graph Models

L 8 !!7 ! / 6 8 3 6 8 > 6 4 7 ! / 7 > <= > / ! & / ! : > ¾ !! &! ? 8 &!7 8 7!& !

=

DEFINITIONS

- % &! % & /

: L 3 6 8 ? # @ 8 ? 6@8 ! !! & % !/ ! 7 / !!

&! ! ! &! !/ / && ! %! !! 6 & 6!7 8 !! /!7 / %! / ! 9 6

© 2004 by CRC Press LLC

818

:

Chapter 8 L 3

?

:

8

ANALYTIC GRAPH THEORY

@ 8

?

@8 ! ! !! & / %! &

/

7 !&

7 / / ! : ½ ! / %! !! !! ! ? 6@8 % %! ? 6@K !7 % %! ? 6@ &

:

4

8 !!

$

! / ! %!& 7 ! !&! !7

;

/ !! & 8 7 /

!/ / %!

:

4 /

;

9 7 ! % 8

! 7 !

;

; ! ! / ; % 7 ;

! !/ /

:

4 / ; ! $ ! ; % 7 ½ ¾ ½8 ¾ 7!/ ; %! 7

:

%

:

& !& !/ !

! !& !/ & !/

EXAMPLES

:

.

;

/ A! & B

?; ?

:

.

;

A7

7

?;

?

?

:

6

6 ?

@8 %

½

·½

! ! % 7! & &7 8

!! / :

@ >

@ >

/ B

! A

@8 % 7

@8 % 7

?;

?

! ! B L N ! /

?

L ( ! /

@ >

6

!

B ! /

8 A

! & B ! ! !& !/ & !/8 7 / & B ! &7

A

! &7 8 % A

REMARK

$

:

! & / && & !

6 ?

@ !! &&!

!!

Asymptotics 5 ! ! / ! ! !& 7!

6 > 6?@ > ?@ &!

© 2004 by CRC Press LLC

8 %!

Section 8.2

819

Random Graphs

DEFINITIONS

:

4 7

!

8

!

?

?@ ! ?!@ ! 9 6 & !! &

! ! 7

8 &

N ! /

( ! /

?

@ %!

? 6@ %! 6 > 6?@ > ?@

&!

()8 &! ?@8 )?@8 (?@ (?@ ! 8 % %! > '?)@ ( 3 8 % %! > +?)@ !7 8 ) ) > '?)@ ) > '? @8 % %! > V?)@ ?@ > ? G +?@@)?@8 % %! )

& +?)@ ! > +?) @8 &7 ! ! ' ?) @ V?) @

3

)

:

!!&! !

.

3

3 B

! 6!!

4½ 4¾ 7! < 3 & < !

E

&

?4 ?

9 ! / &!

+

:

38

4

L 7!

>

@ >

< Z

:

E

4 > 4

? 4 ? [

!! @8

%%

!

@ >

G

!

<

>

!

= 4

!

=

½

¾ ¾

% 8 !

%%

9 7

8

? 7 @8

&

?%! !/

? @ ! !

O 9!!

' ?)@8 > V ?)@ > +?) @8 &!7 >

4

+?)@

! 8 6!7

9!!8 &&!/ 7 !

>

= 4 8 $ !

%! 3

+?@ G

4

+?@

! ! !

4[ > ?4 4 @

%!&

! 6 & 7! !77!/

!8 7 8 % %! A

()

&!

>

+ ?)@

>

L !& 8 <SY .(33=

'?)@8

)

> V? @

FACTS

; / 3 6 > 6?@ 6: > ?@ ! !!7 ! / &!8 9 > ? @

> 6: G 6: ?!@ ; ! ! ? @ % 7 ! 8 ; ! ! ? 6@ :

?
&

© 2004 by CRC Press LLC

820

Chapter 8

ANALYTIC GRAPH THEORY

?!!@ &7 ?!@ ! ! ; ! &7

: ?L%!/
$:

L& % % &! ! ! L !& 8 ! ! L& ! ? @ %! ¾8 & 3 L ! 8 % !! 7 !/ ! F 8 % & !/ ! L&

$

: !& /8 !& ! /8 !! ! %! K ! ! % ! ! & &!6 8 &! %! ? @ L 8 !7 !& / %! 7 !& 8 / !/ 8 ! !/

!! ? @ !/ & & 6 !/ % ! ! L !! % /

8.2.2 Threshold Functions DEFINITIONS

: 4 ; ! ? 6@ ! &! ?@ & ? 6@8 %! 6 > 6?@8

?? ;@

3 ! 6 > +? @ ! > +?6@

!7 8 & ! ! %! 3 ! &/

:

4 &! ; !

?? ;@

! 7 9 > 38

3 ! 6 ? >@ 8 ! 6 ? G >@

? %! 3 ! &/ @ FACTS

:

:

+?/ @ &!

© 2004 by CRC Press LLC

? 6@

&

Section 8.2

:

821

Random Graphs

))

<4&L = L 9 8 !/ & &! ! ? 6@

: < = . ! ! ! 9 /8 !8 !/ 7! 7 !& 8 !/ 6! F& & !8 N & &!7 8 6!9 8 L !! 8 3 8 ! ? 6@ %! 6 > 8 ! ! !

: <( +3= L ? 6@ 9 8 ?/ @ ! &! !! 7 / !/

:

/ 7 ! L& ,

/!7

REMARKS

$:

6

L /!7 8 & !&

$:

/ L& " ! / !! & && !/ ?%! &!!8 ! & L& "@

$:

&! $% ! 8 ! %!& && !! ! $% ! &

$

: L& + ! ' % ! /K <SY.(33= <3= & ! !&

$ :

L& , / ! ?/ @ ! &! !/ / & & <( +=8 7!/ &!/ %!& ! ! & ?! ! 7 @ 7 ?! ! @ <( ++=8 7!/ -! & & <M' =

$)

: &! 9 ! ? @: ! &! ; ! ? ;@ 3 > +? @ ? ;@ > +?@ ? %! 3 ! &/ @ &! 9 ! 7! % L L& 8 ¾½ / ! &! ! ? @ ! ! ! L& , 8 !8 7 8

?

?

8.2.3 Small Subgraphs and the Degree Sequence DEFINITIONS

:

/ !

8?@ >

0 ? @ ? @ : ? @ 3

: 4 / ! ! ! ! ! ! &! 7 !6

>

© 2004 by CRC Press LLC

822

Chapter 8

ANALYTIC GRAPH THEORY

FACTS ABOUT SMALL SUBGRAPHS

! 9

\ @ / %! / 8 4 ?4 / ? 6@ ? ? @8 &!7 @ !!&

-

): 4 ! !&

! %!

+: ?
: <( = 0 ? @ !!! 4 ! !& ! ! 6 ? 6@

: <S0"= 0 ? @ 8 8 : ?: @ 8 4\

! !&

À

: <SY.(03= L 7 6 > 6?@ 8 ] > ! ?4! @ : 0 ? @ 3

??4 > 3@ À %

: ??? @ ! !! @ L > 38 / "

\ " > 3@ / ? @ ! ? >@ / ??4

??4\ " > 3@ / ? @ ! ? G >@

: <E0+= L 6 > +? @8 ??4" > 3@ L > : > +? @8

??4\" > 3@

: < 0= L ? @8 ??4 > 3@ ? ! & @ !H & !& G &! & &!!& / ?! 8 / % !! & &!& @ FACTS ABOUT THE DEGREE SEQUENCE

7 !& / ! / ! ? 8 ! & !/ !/ / 7 !&

: <( += . 9 8 9 8 > ?@ > # Ü 8 ?/ G / / G G +?@@ ? @8 ?? > G @ > G ? ! !! 7 / @

: ?@ 8 9 > 3 > 6 > 6?@ > ? ! ? 6@8 7! ? ! !&

! %! ? ? > 6 ? 6@ @

): ?@ ! ?!@ 6 3 8 ?!!@ 6 ! % 3 ?6@ 6 8 7! ? ! !&

! ! ? 6@8 ! 7!! !& % ? > ? ?@ ! 7! @ > @ ?@ 3 ? & @ ?@ > '?@@

© 2004 by CRC Press LLC

Section 8.2

+:

823

Random Graphs

7 9

8 ! ? 6@

!

? 6 #@ > 6 G

%

:

6? 6@ ?/ @ # ! 9

?? ? 6 #@@ > #

6

?

6 6

!6 7 !6 7

@

/

?

$ Z

$ % G

@ !6 7

!! / !H

6? 6@

! / !6 7

/

!H

! /

6 ?

@

!! /

! ?@ 6 & ! !! 6@ ?%!& ! !!! / /!7 7 ! ? 6@@ ! / 6 & " ?@ ? 6@8 " ?@ ? @ . ! 7 9 6 & 8 ! / 6? 6@ ! ! 6? 6@ / ?!@ > 6: ! % ! / 8 !!! 7 ! ! % " ?@8 ! ?@ !& 6 & %! 8 !H +?@ ?!!@ 6 !!! %! 6 7!& 6? 6@?: @8 & ! ! 7 ?3 @ !!! 7 ! ! % " ?@8 !¼ ?@ !& 6 & %! 7 8 !H +?@

:

<5&E0,8 += .

7! 8 & N!?

REMARKS

$+:

O /8 !! !& &!

%

! S ? <SY .(338 +=@ - ! & ?! & @ L &! 7 !F! &! 8 /! ! & ! & 8 %! !/!9& ! !/ ! <03=K !&! ! <SY .(33= &7 !/ 7 7 &

% /!7 & <03=8 /

!/ ! !/

7 !& L ? 6@ &! &!& & /!7 /

7 &!/ /

8 $:

<(T (0*= O !!! ! ! !!!

4

4\ 8

!

7 ! % ?
$:

L& , ? !! @ %! !!!/ !!

!& % 3 8 &7 !/ % /

L ! 8 !

!! !!!/ !!! 6 3 L 8 L& , ! % %!

> ?/ G / / G @8

?? G @ $:

6 ?

5 ! / 6 &

@

?

" ?

@

@8 &!7 8 % L&

!&!

6 !

?? G @ 3

6 &

% 7

© 2004 by CRC Press LLC

!

" ?

@ /

! &F & ! <5&E0,=

6? 6@ / 8 %!& &7

824

Chapter 8

ANALYTIC GRAPH THEORY

8.2.4 The Phase Transition ( ! !!! / ? @ 77!/ F &8 /%!/ 8 !& & / !/ & & / / 8 9 & / % > : !// ! ! ! ½¾ 8 % !& !/ +?@ & &/ !' / & '?/ @ & ! - % ! ? @ ? 6@ ! / !/ ! 5 L& E /! %! ! % ( !8 !! %!& '

:

<( +3= L! 38 9 > ?@ ! G & > ? L!/ @ . A 7 !& ! / & ?& @ ! ? @ % >

A>

'?/ @ ¾¿

! 8 V? @ ! > 8 ? G +?@@ !

Figure 8.2.1

8, 0 8# E

5 &! !! N / 7 ! ! % !! $ & !! % 7 % 7 %! ¾¿ % & > G '?¾¿@ &!!& 8 ! &!!& ! &!!& ( &!7 8 !/!9&& A FB ! ! !!&8 ! ! ! % 6!!/ 9 & ! DEFINITIONS

: :

4 & & / ! % & & @

:

%! 7 !& / !

! ! & ! ?

! / / %! ! 7 /

FACTS

L 8 &!9& $%

© 2004 by CRC Press LLC

Section 8.2

825

Random Graphs

Throughout the phase transition

: <S0= L ?@ ¾8 / ? @ & & %! % 7 !&

:
: <( +3= 3 > ! 9 8 !' / & ! ? @ ! ? G +?@@ / % > ?@ > ? / @

Figure 8.2.2 #" 0 8 -: E Subcritical phase:

: <( +3= 3 ! 9 8 !! ? @ ¾ ! ! !& 8 & 7 !& /!/ & & % ? @ ? @8 / & ? G +?@@?@ /? @

: ? <SY.(33=@ .

9

): <SY.(33= L 8 !! ? @ &! & & ! ? @ ? ! 3 ! &!!& @

+: 8 3 8 > ?@ 8 ? 6@ 7 !& ! !& &!& & 6 8 / / & & !

: <SY.(33 &! *"= . B / & & !

?

L > ?@ @ !9 B B

8 /

Critical phase: > G '?@

: <SY.(33 &! **= . > G '?@ & ? @ %! & L > ?@ 8 8 7 !& ! & & ? @ !

@ 9 !!

! ? @ & & & & & / & ! ! ½ ¾ Z " Z ? @Z G '? @ Z Z

: <SMY. !0 *= . > G '?

© 2004 by CRC Press LLC

826

Chapter 8

"

ANALYTIC GRAPH THEORY

"

G G G ? ! & @ & !!

@ # 30* & & ! & G '?

%

>

: G % ! & ?3 0@

/!7 !' & & & ? @8 ! 3 ! ! & !/ %! &8 6 & ? 3 0 @ ? 3 0@ &7 / ! !!! 6 & ?4 4 @ ?4 4@

: ? <SY.(33 &! **=@ > G '?@ > ?@ / B / & & ! ? @ !9 B B Supercritical phase:

! G & > ! ? @8 ! & & %! G '?@ 7 !& 8 %! 7 & ! & 7 !&

:
/ & & ! % ?+ G +?@@ ?,"0+ G +?@@

:
? G @ /

: ?@ 8 / B / & & ! ? G @ ! / & !9 B B8 / & & ! / &

): > N L& + % / & ! !6 . 5 7 !& ! & / & ? @8 5 7 !& ! / & ! & 8 5 & / & L > '?@ %! 8 5 ? @ 8 5 5 ? G @ L 8 & 7! ! !& !! 8 5 G 5 ? @ 7 !& 5 G 5 G 5 ? ? G @@ / ! / & REMARKS

$: .

? @

/ ! !/ / &

?8 ! ! / & 8 @ L L& 08 ! % > '?@ 8 !& %!& ! ? @ % ? *" ! <SY.(33= &! @ - > 8 ?3 @

$: O ! !/ & !! & 7! %!/ / & ? &! @

8.2.5 Many More Properties of Random Graphs

& &!7! / ! ?@8 / & &!7!
© 2004 by CRC Press LLC

Section 8.2

827

Random Graphs

FACTS ON CONNECTIVITY, DIAMETER, MATCHINGS, CYCLES AND PATHS

& !! !/ & & 8 ! &!&! %! 7!/ !! /

+: <( += L 9

3 > ?@ > ?/ G / / G G +?@@8

??? ? @@ > @

# Ü

? ? @@ > G

: Æ ?@

6 9 8 3 6 8 > L ? 6@ > > ? & @8 % ! 3 7 !& !! / ! 3 > 8 ! !6 ! 8 & 48 ? @ > Æ ? @ &8 Æ ? @ Æ ?@ G 3 &

:

: ?@ ! / 8 % 9 6 6

> /?@ 6 ?/ @ 8 ? 6@8 ! ! ! G 8 !! ! ! % ! ! : ! 6 / ? 6@ ? 6@ ! K ! : / ? @ !

L& "" "* !& ! %!& 7! & &!!8 !! / 8 ? L& ,@ 9 ! &!& &! & & ? 6@ %! !& 7 ! /

: <( ++= L 6 > 6?@ > ?/ G G +?@@8 !! & &!/ #

Ü

: 6?@ > ?/ G G +?@@8 !! ? 6@ & ?! 8 !/ / & Ü % & ! !!& @ # %! !& ! / !7!!

L& "+ && %!& 7! & &!!8 !! / 8 ? ( $ 0 L& ,@

: <M' = L

6 > 6?@ > ?/ G / / G G +?@@8 !! # Ü

? 6@ -! & &

: <4FM' = . ?@ - & ? 6 > @ &! / - ?@ 3 8 ! ?@ > FACTS ON INDEPENDENT SETS AND CHROMATIC NUMBER

6 > 6?@ ? 6@ !9

Æ Æ 3 L 9 > 3 ! & ?@ ?@ %

:

> ? 6@

> /& /& /& 6 G /& ? @ G G ?@ >6

?/ 6 / / 6 G /? @ G ?@ >@ L & .' 8 ! & ?@ ? 6@ !9 ! 6! ?@ 7! .' 6 > 6?@ /

):
> 3 > 6

© 2004 by CRC Press LLC

828

Chapter 8

ANALYTIC GRAPH THEORY

+: 6?@ 8 ? 6@ *?@ / ? / 6@

: <40=8 <M33= . > 3 6?@ ' 8 ? 6@8 % 7 *?@ * ?@8 % * ?@ ! &!& ? ! &!& @

: < !E0+= L 9 8 ! & & ! ? 6@ &! 6 > % > ! ( 8 ?8 @8 ?8 @ > 8 &Z ? & / ! ! / !! / @

: <50+= . 7 > : *?? @ > " ?L L& *8 % 7 8 !& / %! & & ? @ & @ FACTS ON PLANARITY, GENUS AND CROSSING NUMBER

: <( +3= L ! ! ? @8 > ! &! 5 &! G ?% ! &@

: <40*= 6? 6@ ?/ @ 8 / ? 6@ ! ?G +?@@ 6

: <(T0+= L 7 ! / & 8 ! ? 6@ ! ? G +?@@?&"?& G @@6

6 8 /

: <3= & 7 &!/ ? 6@ ?! 8 !! &!/ ! ! %!/ ! 8 / &!/ !@ ! +?6 @ ! 6 V?6 @ ! 6 > 9 FACTS ON EIGENVALUES, AUTOMORPHISMS AND UNLABELLED GRAPHS

: ! ? 6@8

< 6 < > 6 G '? / @

): <E,3= & ! ? @ !H !? : @ ?/ @ & % ! ! % /

! : 7 ! /

+:
/ 8 3 8 > ?@ ! & ! : ?/ @ . ? @ & / 7 !& / ! ?? ;@ !H ?? ;@

8.2.6 Random Regular Graphs ! & ! !/ / ! ? 6@ ? @ O & ! !&! ? @8 / %! &!9 / 6 & ( / / &! & !

© 2004 by CRC Press LLC

Section 8.2

829

Random Graphs

L 3 7 8 ! !! & &!!/ F / / 7 !& 8 !/ 6! FACTS

& & ! ! N 9
: L 9 9 / %! / 7 !& 8 / / &! / !!&

:
: <MPE3=8 <L( 3= L > ?@ "8 / / ! & &

: 3 ! ??@ ! 6!!

G / ½ G / ½

/ +

??@

!9

G / ½ ?? G >@ / @

: <5&E "= > ?@ > +? @8 & ! ?& L& *0@

: <M!P3= !

> ?@ "8 !7!

: <MPE3=8 <L( (!3= > ?@ 308 8 ?@ / /?? @@ *?@ ?@

: <(E0"= L 9 8 / / -! & & 8 8 / &!& 6 8 <M!E3= 9 8 !! ! / ! / ¾ -! & & ? 7 @8 ¾ ½ -! & & & &!/ ? @

): <MPE3=8 <L( 3= L > ?@ 8 / / ! !!

+: <5& = L 9 4 8 ! !/ 7 F& & ! %!& !

"? @ ¾ G +?@ =?¾ ¾@

:
F& & ! ! G / G +?@

:
?! 7 @ F& & ! ! '? @ 7! > +? @ REMARKS

$: ! !/ !! % / 8 ! &!/! ? <E00= <SY.(33=@

© 2004 by CRC Press LLC

830

Chapter 8

ANALYTIC GRAPH THEORY

$:

7! !' / ! 8 !!/ 8 &! ! 7 % ! ? ! <E00=@

$

: / %! /!7 / 6 & 8

! / / L !& 8 / %! /!7 / 6 & 8 %!& ! % )8 7 & &!7! 6 !! /
8.2.7 Other Random Graph Models 5 / 8 & !!!& 8 7 & !7 & ! !/ & & 8 % /!7 7 ! ! 8 ! & !/!9& F & !!:
8.2.8 Random Graph Processes 4 / & ! ! / ! ! / & ! & / 8 %!/ ! !& ! & ! !/ ! ! ! %!& ! DEFINITIONS

):

\ @ /! %! / ? % / ! 8 & & !

\ @ ! 5$7 &! ! ?% ! / @ ?L 8 ? ! & 6!7 / ? @8 ! ! & 6 & ? @ @

+:

/ ; ! ! : ? @ )

FACTS

\ @ / !9! 4 4 8 !& /!/ ? 7 7 %!& % 9 / /!7!/ / & ! ! 9 / %!& ?! @

:
:

© 2004 by CRC Press LLC

Section 8.2

831

Random Graphs

!/ / !F! -! & & 8 ! ! 8 &!/ !'

!F! & & 8 ! 6 !!/ ! 7!/ !! /

: ?@8 !!/ ! !/ & & ! 6 !!/ ! 7!/ !! /

: . / 9 & & ! \?@ <S ,= ¾ 9 " 8 ?? > " @ 6 > K ½ 6 8
#

: <SMY. !0= !! \?@ ! &! & & ! ? G +?@@ % REMARK

$): 4 % / & ! !& & %!& % !7 % %! %
References <4&L00= Q 4&! L! /8 4 &!! 8 4 " ?000@8 +W,3 <4FM' = 5 4F!8 S M 8 ' !8 / ! /8 ?0 @8 W <4 !33= Q S 4 N ! 8 O / %! !!/!/ 7 !& : / & /! & 8 4 , ?333@8 ,0W3 <40= 48 ( !& &!/ /8 W ! ' ())5 M E$ ?@8 !/ 2!7 ! 8 00 <4M!0,= 48 S - M!8 S & 8 & &!/ ! / ! /8 0 33 ?00,@8 ,W , <40*= Q 4& & Q4 8 / /8 " ?00*@8 W,
Q N &8 !/ &! / !/ 8 "+ ?33@8 0W**
© 2004 by CRC Press LLC

832

Chapter 8

ANALYTIC GRAPH THEORY

.

8 O 8 00

8 ! 5 !&8 !/ P /8 0,0
?0

@8

© 2004 by CRC Press LLC

Section 8.2

Random Graphs

833

<L0"= 4 L! ' 8 -! & & ! & ! & /8 + ?00"@8 *W+ <L( 3= 8 4 L! ' 8 N ( 8 ( / / & / : & &!7! -! & & 8 " ! ?33@8 "0W+ <L( (!3= 8 4 L! ' 8 N ( 8 O (!8 ( / / & / : ! & &!& 8 " ! 8 W" <( +3= 4 ( !8 O 7! /8 " 0 * ?0+3@8 ,W+ <( += 4 ( !8 O / & & /8 ?0+@8 +W+, <( ++= 4 ( !8 O ! & & / & & /8 , ?0++@8 *0W+

+ ?0 ,@8 W3 <S0= S8 / # $ 4! 8 5 ! 4 5 & *"8 4 5 &8 7! & 8 ( ?00"@

© 2004 by CRC Press LLC

834

Chapter 8

ANALYTIC GRAPH THEORY

<S0"= S8 5!& &!& & ! / & 8 4 ! " ?00@8 ,W " <SMY. !0= S8 Q M8 . Y &'$ N ! 8 ! /! & 8 4 " ?00@ W* <SY.(03= S8 . Y &'$8 4 (&! $!8 4 ! !! ! & &!9 / ! /8 ,W , ! 4 ?95 5 M$!8 S S%$! 4 (&!$! ?@8 E! 8 003 <SY.(33= S8 . Y &'$8 4 (&! $!8 4

8 E! 8 333

<S(3= S 4 (&! $!8 ! ! 4 3 ?33@8 ,W" <M03= ( 5 M8 !!7 & !/8 4 ?003@8 ,W0 <M!P3= S - M!8 N $78 P P8 O / / /8 4 ?33@8 +W" <M!E3= S - M! E8 ( &!/ %!& !& -! & & 8 !! &!! / /8 ?33@8 3W"" <M += P L M&!8 4 ! ! ! 5 !& /! !/ O!!'! % 8 &8 !&! Q!7!!8 % U$ ?0 +@ <M00= P L M&!8 4 & & ! 5 !& ! 4!& !8 * !/ 2!7 ! 8 000 <M' = S M ' !8 .!! !!! ! & -! &!&! ! /8 " ?0 @8 **W+ <M!33= 5 M!7 7!&8 &!& /8 " 0 ?333@8 0W+ <MPE3= 5 M!7 7!&8 N $78 P - P8 E8 ( / / !/ / 8 4 ?33@8 "+W + <MP3= 5 M!7 7!& P P8 !! ! /8 ! ?33@8 "W*,
© 2004 by CRC Press LLC

Section 8.2

Random Graphs

835

0

E! 8 0 * < !E0+= N ! 8 S & 8 E8 / & /! & ! /8 +, ?00+@8 W* < 0= - S T 4 / 8 !& / &!!/ 9 & &!!& /8 ?00@8 "+W", < 0+= - S T 4 / 8 O !& & !/ /8 ?00+@8 ,W* < 0+= - S T 4 / 8 !/ /8 ' % *" ?00+@8 W* <(!33= O (!8 !/ / /8 " 0 ?333@8 *W" <(E0"= ( E (! E8 4 / / ! !8 4 * ?00"@8 +W," <(T(0*= P (T 4 (&! $!8 &! ( ! 8 ! ?00*@8 *W,3 <(T0*= P (T ( 8 O / /8 4 + ?00*@8 W

© 2004 by CRC Press LLC

836

Chapter 8

ANALYTIC GRAPH THEORY

<( = 4 (&! $!8 E / / !! I " 4 # , ?0 @8 W3 <(E0= 4 (&! $! E8 ( / & %! / !& !8 " ?00@8 +0W 3 < = S & 8 # % /8 ?0 @8 0,W* <03= S & 8 &! ! 8 * ?003@8 +W3* <3= S & 8 $ 4 !&8 !/ P /8 N !8 33

4/! !

< 3= S & 8 !/ /8 4 ! ?33@8 ",W* <03= E 8 4 & ! ! 6! ! !! 7 !/ & / /8 4 ?003@8 W"
8 000 <E,3= 5 E!/8 %! /!7 / 8 + ?0,3@8 W0

^ ( & ! /

© 2004 by CRC Press LLC

Section 8.3

8.3

837

Ramsey Graph Theory

RAMSEY GRAPH THEORY

!

( 1 L ( !& ( " !' ( * !' ( + ( 5!! , !'! P!! ( &

Introduction / ! % ! % $% & / ! ! // / ! ! & / & / ! & ! ( N 8 ! ! ( !/ ? @ N !/ ? @ 5 7 8 ! &&! !

8 ! ! !! & ! ! &! & & / 7! L ( <(3= - 7 !!7 ! / 8 ! ! / & ! & / ! & ! ( N 8 %! ! ( N & ! / !

? @8 ! & ? @ ( ( / ! & & !/ / 5 / 8 & ! !& F %8 &!& / ! R F & /8 / !/ / & ! !& !/ &

8.3.1 Ramsey’s Theorem ( 1 !/! ! / !!&! !& &!!& / L &! ! ( 1 8 <(03=8 < , =8 < 0+=8 7

!&
8 ! !!7 ! / ? @ & 8 !! $ $ $ / & / %! 6 ? @8 ! & & / % / ! $

: ?$ <(3=@ !7 !!7 ! /

© 2004 by CRC Press LLC

838

Chapter 8

ANALYTIC GRAPH THEORY

DEFINITIONS

: ?½ ¾ @ ! & ?½ ¾ @

: !! / & / ! ! &! &!/ / %! &8 &!9& Ramsey Numbers for Arbitrary Graphs

( 1 !! ! & A&!&B & / ! ! & ! / &!/ Æ&! / & / !&

/ 7 !& ! !!& / 8 ! ! & 6 & ( 1 ! ! & ( ! / DEFINITIONS

: ( ) ?½ ¾ @ & &! / ½ ¾ ! !!7 ! / & / &!/ 8 ! & &!& & ! & &

: !7 / ½ ¾ 8 / ! ! ?½ ¾ @ ! / &!/ ! & &!& & ! &* & ! ! ?½ ¾ @ 8 ( ?½ ¾ @ ! / & ?½ ¾ @

: [?½ ¾ @ ! !' ?! 8 / @ / & ?½ ¾ @

: 4 / ! ?½ ¾ @ ! ?½ ¾ @8 / % REMARKS

$: $:

& &8 > 8 ?½ ¾ @ > ? ½ ¾ @

!& / ( %! & % & 6! &!& / ! & 8 %! / !' / ( !77 / !'! ! /

$:

! $!/ 6 ! & A!H B / A%B ( !! / &! ! +

8.3.2 Fundamental Results 7 F! ( / && &!/8 7 & ! ! & 6 & 9!! FACTS

: L ! / ½ ¾8 ?½ ¾@ > ?¾ ½@ 5 / 8 / ! ! /

© 2004 by CRC Press LLC

Section 8.3

839

Ramsey Graph Theory

: L / &!7 8 ? @ ? @

: L 8 ? @ > ? @ >

: &" <'*= L 8 ? @ ? @ G ? @ %! !& ! 6! ! ? @ ? @ 7 4 & 6 & ! ! G ? G G @

REMARKS

$: ' $ /!7 9! (

! 9! / & !/ 9! & & &! 9! /

$: 7 ( ? @ > 68 % $ 4 ! /!7 % / &!/ 8 ( N &!/8 ! ! ( N 8 ( N &!/ ! !! ! ( N $: & / &!/8 %! ( N 8 ! ! ! &7 ! F / 8 %!& / !& ( / 8 & N / EXAMPLE

: % ? @ > +8 7 ! L!/ ! ( N &!/ %! ! ! &8 7 ' $ ? @ ? @ G ? @ > G > +

Figure 8.3.1

? @ *E

8.3.3 Classical Ramsey Numbers Q !!/ &!& ( ! 6! !Æ&8 !7! &!& ( $% &! ! 7 !! O !7! !& ? &@ &!& ( ! $%

! !7! % & ( ? @ $%8 %!& ! / 7! & !Æ& ! !!/ (

© 2004 by CRC Press LLC

840

Chapter 8

ANALYTIC GRAPH THEORY

* EE - #- $ * ? %@E

% %

"

*

+

+

"

"

0

*

+

"

,

0

3

"

*

"

*

*0

+0

,

"3

"+

*

*0

++

,

+

"

+

"

*

"0

0

0 "0

",

*

"0

*+

+0

3

0+

"

*

, " +

+

""

3

0*

"

+* 0

"0*

, 3

,

3 ,

*

,,

* "0

3 ,*

*

" *

* 0

"

* + ,

,"

+

0

*"3 3 , + ,

3* + ,3 * +303

,

0

+ +*

+,,

0 *+*

* 3 **+

3 ,0

Ramsey Numbers for Small Graphs

&! $% &!& ( ? %@ / %! % 7 / / / 3 / ! !/8 / % / % & ( 7 & 8 ! 8

© 2004 by CRC Press LLC

Section 8.3

841

Ramsey Graph Theory

% ! L !& 8 ? @ > 8 ?" @ ! % *+ " & & & 7 ! ! / %! & % 8 % & ! !!& / % !& ( L !& 8 M+" ! ? +@ > 4 !!/ ( & & & ! &!& ('!%$! <(3= FACT

: 1 , 1 <**= ? @ > , REMARK

$: ! %$ !!/ &!& ( % %

<**=8 M <M +"=8 7 U&$ <U+ = .% % ! %! !&! !!& &!/ !/ / !& &!6 8 % ! !/ / &!6 5 & 8 5&M 8 ('!%$!8 / !&!8 7 &! & !6 8 % /!8 &!/ % ! / (

% -% 7 8 / % % ! 7 7

Asymptotic Results

! !& 7! ( ? @

! !& 7 !7 ! ' $ ?L& L*@ % ? @8 ! !& ? @8 & ! %!& !& 7 ! ! > < = 7 ? @8 M! <M!0*= 7 !9 !& % FACTS

: < = L 8 ! & &

G ? G G @

! !& ? G G @

: L & !!7 8

? @

): ?< =8 <M!0*=@ & & /

? @ /

!!!& &!6 7 ! & &!/8 &!9& & !/ !!

© 2004 by CRC Press LLC

842

Chapter 8

ANALYTIC GRAPH THEORY

8.3.4 Generalized Ramsey Numbers &!7! &! ! / !' (

! / ( % !! 7

7 &! 8 % %! 7! % F % !/!/

Initial Generalized Ramsey Results FACTS

+:

? %

/ %!

:

/ /

< +,=L !!7 ! /

?/

/@

/

/

8

@ >

G

>

%

!

&!

/ 8

>>

!

! !!7 7

&

!!7 ! / %!

½ ¾ @ >

? %

& &!/ ! /

?

>

! !!

!/ 7

@ G

?

> %! L

@ > G >

> 8

7 3 %!

REMARK

$ :

/ /

/

8 8 ! ! & 8 / / !

& %

%

( L& &!/ !7 ! & /

/ /

Ramsey Numbers for Trees ?

@

& ! %!& ( $% -% 7 8 & $%8

?

@

G

!

FACT

:

>

@8

! 7

@ > G

?

CONJECTURES

3: 3 5-

@ G

?

© 2004 by CRC Press LLC

?

%!

8

>

Section 8.3

843

Ramsey Graph Theory

3: %& 3 5- 4 / %! 7 !& ? @ G / &! ? @ / REMARK

$):

&F & !! F &

Cycle Ramsey Numbers

( & & / 7 && !!& ( CONJECTURE

3: ( 3 5- " EXAMPLE

:

L 8 &! / &!/8 %! ( 8 N 8 8 ( / ! " 8 N / ! !!& &! / % 9 % % & / ! ( 8 !!/ / L ! ( . 8 !& & ( / 7 !& 8 ! N . !& / !! 7 & & 8 ?. . .@ " " FACTS

:

?<(,=8 <(,=8 ? @ ?" "@8 ?. .@ >

G G

% ! 8 % 7 8 % ! 7 !

L / &!/ & & ( 7 !! ! 8 !! ! & !Æ&

:

:

:

<(U0= ?. . . @ > , * <.00= L "8 ?. . .@ ?" G +?@@

Good Results

(

: ?@ & ! 7 7 &!/ %! *?@ &8 7 & & 7 !&

© 2004 by CRC Press LLC

844

:

Chapter 8

ANALYTIC GRAPH THEORY

?@ ! & !

? @ > ?*?@ @? @ G ?@ FACTS

: ?( ? @? @ G % &F & ! <L(, = L& , ! & > > 8 7 !9 > % 7 !9 > " * +8 ! !! 7 ! <!3= " G

: ?3 <,,=@ L ! / 8

? @ > ? @? @ G ! 8 %!& & ! /8 / !' ! % % ! & 7 & / %! &!& & & & /

):

+

: ? @? @ G Æ&! / 8 ! ! ?@ > 8 ! *?@ 7 &!/ & / !& % & & ! / &!/

:

. ! / & & / !!7 & ½ ¾ & ! / ! ! Æ&! / ?!@ ¿8 "8 0 ? @ ?, G @*8 ?!!@ 8 "8 0 ? @ G ¾´ ½µ8 ?!!!@ <L( *= 0 ? @ G ½ 8 C? @ ¾ 8 ?!7@ .¾·½ 0 ? @ ? G ¿ @8 ?7@ ¿ > ½ G . ?% @8 ?7!@ .¾·½ > ¾ G ¾ EXAMPLES

:

! ( N &!/ ´ ½µ´ ½µ ! %!& N / ! 7 !F! &! & / ½ ?? @ ½@ ( / ! & /8 ½ ¾ ½ 8 % ½ > ¾ > ½ > ! N 8 ! & N & & / %! 7 !& 8 ! (

: / &!/ /!7 % ? @ ? .@ 5 7 8 ! / %! &!& ! ( / ! &!/ !! ! &!& *?@ > ! & & / 8 ? @ ? @? @

© 2004 by CRC Press LLC

Section 8.3

845

Ramsey Graph Theory

: L 6 > ?*?@ @? @ G ?@ 8 &! ( N / &!/

! %!& N / &! *?@ !F! & / & / ?@ 8 ( / ! & / ! N ! & & / ! ( ?@ REMARK

$+: 4 & !7 A/B & !
5 / !' ( 7 / % ! ! 7 - <-,=8 & <,,=8 - <- 0= L!/ !& / %! 97 7 !& 7 ! 7 !& / &! !/ /!& ! & G / %! 8 % ! / ? !6 @ ! ! ! !/ 7 & /

Figure 8.3.2 1 : 0

* ,# # E

REMARKS

$: /!7 !/ ( $% ( / 97 %! ! 7 !& 4!! !! & & ! <(3= $: ( 7 & / 7 ! 8 ! & 7 ! ! <(3= $: ( ! ? ¿ @ % 7 ! 6 !
+ !
! ! / 6 3=8 6 > , ! 0

$: !/ ( ? @ / %! 7 / %! ! 7 !& & ! <- ,= $: ( ! ? @ % ! & & / * ! ! % && !
© 2004 by CRC Press LLC

846

Chapter 8

ANALYTIC GRAPH THEORY

* EE 1 #" $ * 0 8 :E

[

/¿

¿ /

.

/ / " * " +

0 /

* + * 3 0 3 +

* + 3 . / , 0 * , * /

/ .

/

* 0 "

A

. * 3 "* .

.

/ 0 3 "

/ / ? / @ / / 0 **

*

? @ / /

. /

Linear Bounds

N !/ ' !1 /! 8 <( = 7 !9 %!/ &F & & 6 & A! B / / & / /8 ! ! !& !&! CONJECTURE

3: 3 5-
DEFINITION

)

: 4 / ! ! 7 !& & ! & %

7 8 & F& & && ! F& &! & ! FACTS

: <( = ! ! / %! ! / C8 ? @ C !!7 & ?! 7 !9 F & @

: <&0= ! / / 8 ! ? @ ! ! !

: <&0= ? @ ! /8 ? @ ! ! !

:

<(0+= ! / / 8 ? @ ! ! ! CONJECTURE

3: ( # 3 5-
/ ! 8 ! & > ? @ & ? @

© 2004 by CRC Press LLC

Section 8.3

847

Ramsey Graph Theory

8.3.5 Size Ramsey Numbers & ! ! !' ( [? @ % & !

8 <L(, = General Bounds FACTS

:

0

?@ G 0 ? @ [? @

: <L(, = L 8 ?!@ ?!!@

[? @ >

´ µ ¾

´µ8 ¾

[? ½ ½@ > G

REMARKS

$: 4 /

? @ 7 0 ?@ G 0 ? @ ? @ !! [? @ ´ ¾ µ

& / 8 & ´ µ

$: ! !7 !/ !! [? @ %! ´ ¾ µ

0 ?@ G 0 ? @ N !!!! && Linear Bounds

!' ( [? ½ ½@ ! ! ! 8 %! / ! & ( / ! ? ½ ½@ ! 6!& ! N &$ ?
:
&

! / C Æ&! / 8

?!@ ?!!@ ?!!!@

[?/ /@ 8

[?. .@ 8 [? @ C_ ?_ / @½¾

): <-M0*= L %! ! / C8 ! & & [? @ C REMARK

$ : 7! % &F & !
© 2004 by CRC Press LLC

848

Chapter 8

ANALYTIC GRAPH THEORY

CONJECTURE

3: (- 3 5- ?C@ & [? @ Bipartite Graphs

L & !! / 8 % 7 8 <L(, = X X! ( < (, = % % 7 ( <(0=8 !& DEFINITION

+

: 4 %! & & !/ %! / %! FACTS

+:

<L(, =8 < (, =8 <(0= L +8 ?+3@ [? @ ?@

:

@

> ? G @? G @

REMARKS

$)

: O !! &! 7 !' ( $%8 !& & !Æ& && / !' ( %! & 6 ! & / %!& $%

$+:

&! 7 !' ( / !! ! ! 8 ! &! & %!/ &F &
3: & 3 5-

8 > >

8

[? @ >

6

% 6 > G : & G " > Small Order Graphs

& !' ( $% /8 7 / ! ! ! !Æ& && 4 ! && ! !& !' ( 8 %!& ! ! !&!/ A%!/B / % ! (

© 2004 by CRC Press LLC

Section 8.3

849

Ramsey Graph Theory

* EE. : E &#" $ /*

"

*

+

,

0

3

"

*

+

,

0

© 2004 by CRC Press LLC

3

850

Chapter 8

ANALYTIC GRAPH THEORY

* EE. : E &#" $ /*

*

+

,

0

3

"

© 2004 by CRC Press LLC

Section 8.3

Ramsey Graph Theory

851

DEFINITION

:

L / 8 ? @ ! !! !' / ? @ & ? @ REMARKS

$:

8 %!& & !
$:

4 A%!/B / & !/ !' ( % !/ ! / 8 ! / 8 !6 &! ! & %! ! ! & <=
8.3.6 Ramsey Minimal Graphs DEFINITIONS

: L ! ? @ /8 % ? &@ > : ? &@ / ! % ? &@ ! ? @ : 4 / ! ? @ ! % ? &@8 / ! ! % : ( !! / ! % ? &@ %! %? &@

:

! ? @ ! * ! !!& / ! %? &@ ! !9! O %! 8 ! ? @ ! *

: 4 / ! *(

! ! ! & & & %! % 7 !& ! ! EXAMPLES

:

L / 8 & ? ¾@8 ! ? ¾@8 7 / - & 8 %? ' @ > 8 ! ? ¾ @ ! ( 9!

: O 7 / &!/ & & . %! 7 & &!7 / %! & & ! 8 . ?/¿ /¿@ 8 / . %! A%B ?/¿ /¿@ 8 ! ?/¿ /¿@ ! ( !9! 8 ! ! % %?/¿ /¿@ > . : FACTS

X X! (T < (, = !!! ( !9! ( 9! ! ! 7 !/! ! %$ % !
© 2004 by CRC Press LLC

852

Chapter 8

ANALYTIC GRAPH THEORY

: ?/!!# $2 < (, =@ ! ? @ ! ( !9! ! ?!@ *?@ 8

?!!@ ! * & & 8 ?!!!@ ! &!!/ /

:
:
: <.0"=

! ! &!/ &! & & 8 ! ? @ ! ( !9! REMARKS

$: O & 6 & L& * ! &!/ > ! / & ! %! / ! ( 9! !

$: L& * % 6 ! !
8 ! ( !9! /

$: 4 & && !'! ! ( 9! ! $%8 & ! $% ! / ! / CONJECTURE

3 : $ %F # 3 5- ! ? @ ! ( 9! ! ! ! ?!@

! &!/8

?!!@

!

8.3.7 Generalizations and Variations / !'! &!& ( % 7 &! ! ! 7 E %! ! ! ! &! 7 Graphs

!& ( / A%!/B 8 !& & !& / !& ! & ? @ % 7 !9 (T ! ! & ! <(,=8 % 7 !9 ! !&!

© 2004 by CRC Press LLC

Section 8.3

853

Ramsey Graph Theory

/ ? @ 7! / & ! <M (0 = DEFINITIONS

:

? @ ! !!7 ! / & ! / & / &!/ ?( N @ ! !& & ! ( !& & ! N

: L !! / & ? @ ! !! / & ? @

):

&

+

? @ ! /

? @ / !& & & ! & &

: L / 8 ? @ ! !! &!& &! ! / &!/ ( / % > ? @ REMARKS

$

: ! & !! ( & ? @ % 7 !9 ( <(*+= && !& / 8 8 & !! / ! !! 9!! &

&!& 8 F &!& %

$:

<, = % ? @ > ? @ ! ! !/ & -% 7 8 ? @ ? @ / %! !/ 8 & <, = %! /
$

: - ! <- ,"= && ? @ / 8 7 % ( !!&!! $% Hypergraphs

!&! ( ! ! !& /8

&!7 &!/ / -% 7 8 !/! ( ! ! / % DEFINITIONS

: 4 &! 7 !& / 8 & %!& ! 4 / ! ! ! / 7 &!!

:

L ! / ?½ ¾ @8

?½ ¾ @ ! ! / & ! & %! &8 %! & !!& & ! & &

FACT

: ?- . $ #"#",# <5&(0=@

¿?" "@

>

REMARKS

$):

&!& / ( $% ! ¿?" "@

© 2004 by CRC Press LLC

854

Chapter 8

ANALYTIC GRAPH THEORY

$+: L& + ! ! %! & 8 %! " %! ! 7!/ &8 ! ! &!/ ! %!

References

3 ?0 3@8 0W3

© 2004 by CRC Press LLC

Section 8.3

855

Ramsey Graph Theory

0 ?00@8 3*W
8 , ?0 @8 *W <,,= P 78 & / ( 8 0

, ?0,,@8

<-,= P 7 L - 8 !' ( /8 , ?0,@8 "W"+ <( = P 78 P ( 8 ' !8 E 8 ( %! ! / 8 " 8 0W" <,,= 5 & 8 ( 8

?0,,@8 0W0

<(*+= ( (8 4 !! && ! 8 + ?0*+@8 ",W" 0 <L(, = 8 ( S L 8 ( 8 ( - & 8 !' ( 8 % && ! / !' ( 8 " 0 ?0, @8 "*W+ <L(, = 8 ( S L 8 ( 8 ( - & 8 O & & & / ( 8 ?0, @8 *W+" <L( *= T8 ( S L 8 ( 8 ( - & 8 5!! /W / ( 8 * ?0 *@8 W

© 2004 by CRC Press LLC

856

Chapter 8

ANALYTIC GRAPH THEORY

<(0= ( 8 !' ( & !! /8 ?00@8 *0W+ <'*= ' $ 8 4 &!! ! / 8 ?0*@8 "+W",3 <

<

0= 8 4 !/ !!'! /! ( 8 ,*W,0 !

?U 47! @8 458 0 0 0= 8 4 % ?* *@8

?0 0@8 0,W0

< 0= 8 4& : O ( ?" +@ ?* +@ ? @8

* ?00@8 * < 3= 8 !&! 6 / ! / 8 ! ?33@ < -5 = 8 - -8 5 / 8 ( 7 8

*4 ?0 @8 ? & 7 N!! !! @8 ,,W +
8 3W 0 ! 8 & 4 58 00
8

?00@8 0W"

& & / ! 8 " ?0 3@8 0W33
© 2004 by CRC Press LLC

Section 8.3

Ramsey Graph Theory

857

<( ,= ( . P (T8 ! ( 8 ' ! ? E! @8 !/ 2!7 ! 8 0 , <(03= ( . 8 N . (&!8 S - & 8 4 8 S E! a 8 003 <U+ = S 7 S U&$ 8 / !& &! %! ( 1 8 " ?0+ @8 *W,* <**= ( % 4 5 8 !! ! &!& /8 , ?0**@8 W, <( = ! ( 8 O ( ? @ ? 0@8 ?0 @8 ,W* <-5! = L - # 5! 8 !' ( P: !' (

/8 ,W ! " ! ! $ " 2 ? T8 . 48 -'8 4 $T' 8 @8 N!$T 8 0 <- ,"= L - !8 !' ( /8 P: ( !!&! /8 6E " ?0,"@8 +W, <- M0*= - 8 U M $%8 !' ( 8 0 0 ?00*@8 +W," <- ,= ( - 8 !/ ( / %! 7 / 8 & ?0 ,@8 W" <- 0= - 8 ( / %! 97 7 !& 8 ?0 0@8 "*W"

<- 0= - 8 ( ? ¾ G @ ? G . @8 & * ?0 0@8 "3W*" <-#0 = U ( -/ M 5 #/8 4 % & &!& ( 8 ?00 @8 ",W*3 <M+*= S M; !&8 &! &! / &!& /8 ?0+*@8 *,*W* " <M++= S M; !&8 !& / ( 1 8 " 2! 7 ! E 8 ?0++@ <M +"= M 8 O ( ?-/!@8 * ?0+"@8 3"W" <M!0*= S - M!8 ( ? @ /! / 8 4 , ?00*@8 ,W3, <M (0 = U M $%8 - S T 8 P (T 8 & ( 8 ?00 @8 ,W"3"

© 2004 by CRC Press LLC

858

Chapter 8

ANALYTIC GRAPH THEORY

<.0"= .&'$8 O ( !! /8 ?00"@ <.00= .&'$8 ?. . .@ ,"W ,

?" G +?@@8

,* ?000@8

<50"= S 5&$ 8 !! ! 8 " 8 2!7 ! -%!!8 00" <5&(0= N Q 5&M ('!'%$!8 9 &!& ( / ! & 8 3"W3 ! " $ ! 0 OQ4108 L&!&8 00 <5&(0*= N Q 5&M ('!'%$!8 ?" *@ > *8 ?00*@8 30W

0

<5&(0,= N Q 5&M ('!'%$!8 / &!/ ! !! ( 8 +0 ?00,@8 0W30 <5&#0= N Q 5&M M 5 #/8 7 ( ? @8

+ ?00@8 00W3* < 0+= S X X!8 ( 8 * ! $ 00 ?( . 8 5 & 8 . .7 ' @8 5 8 ?00+@8 W"3 < (, = S X X! P (T 8 & &!!& /8 ?0, @8 0*W33 <!3= P !$!78 & & & / ( 8 ! < , = Q 8 ( / 8 +W " ! 8 ?. E N ! $ ( S E! @8 4& !& 8 0,

<(3= (!'!'%$!8 ( 8 Q !& 7 ?33@ <(M = (!'!'%$! Q . M 8 & /! ( / ! / !8 ?0 @8 *0W, <(3= L ( 8 O /!&8 " %A, 3 ?03@8 +"W + <(,= P (T8 ! ! / / !' ( 8 !8 2!78 8 0, <(0+= P (T ( 8 4/ !! &!6 !7!!8 +W0 ! $ " : ?( . S X X! @8 !/ P /8 00+ <(,= P (8 O ( S 4 N T 8 8 * ?0,@8 0"W3" <(,= P (8 O ( S 4 N T 8 8 * ?0,@8 3*W3

© 2004 by CRC Press LLC

Section 8.3

859

Ramsey Graph Theory

<(= ( S 8 !' ( & ! !! /8 ! <(U0= (%! U U/8 O ! ( / %! 97 / 8 ?00@8 W < = S N 8 4 ! & !/ /8 "+ ?0 @8 W , <, = Q 8 & & ( 8

< = 4 8 4 ( 8 ?0 @8 *30W*,

?0, @8 "0W**

© 2004 by CRC Press LLC

860

8.4

Chapter 8

ANALYTIC GRAPH THEORY

THE PROBABILISTIC METHOD "

#

" L! 5 5 " 4 ! " .7' .& . "" (T ! "* N ! Q!!! ( &

Introduction

!!!& & 7 !!!& ! / !&8 ! ! 7 Æ&! / / &!9

8.4.1 The First Moment Method DEFINITION

) ! 7 %!& !Æ& !! 7 !/ &&!7

: * !77 9!/ 7! 8 $% / % & 7 ?! 8 9 @ & 7 6 ! ! & !& & EXAMPLES

:

7 ! ! %! 7 !& Z ´ ½µ -! 8 & & & -! !& Z ! ! & /8 & !/ ! ! ! %! !! ´ ½µ8 & ! Z ´ ½µ ! !! ! Z

:

: 4 > ? 0 : & > @ ! *( ! ! !! ! 7 ! % & & & 0 > 0 > & / ! ! % / ! & -% 7 8 <+= % ! 0 &

© 2004 by CRC Press LLC

Section 8.4

861

The Probabilistic Method

½

! & ! !! % & / %!& & !

:

<0"= 4 ! ! 7 3 ! 7 3 7 !& ? @8 ! 7 > ?3 @ 3 & / %!& F! 3 ! & % 8 ! 8 7 ! 3 Q 9! ! %! 3 I % ! 7 !8 ! & %!& !9

? @

?@

! / . @ 7 !& %!& %!& ?3 @ ! ! ?@ ! & 7 @ 8 9 7 ! & %!

3 L / ! /!7 ¾? @? G +?@@ ! ! & !/ ! 8 !& ' $ 7 ! ? @ ! 7 !& ! %! 3 8 ? @ & 3 ? 5 <5,0=@ O !$ ! /!7 ! $ 4 & <433= Ramsey Numbers DEFINITIONS

: 4 / ! / &!/ ! %!& 7 / ! & !

: ? @ ! ! / & ? @8 7 / &!/ & / &! !

Q !!/ &! 7 ? @ 7 !Æ&8 ? @ ! $% & * 4 % 7 &! FACT

O ! % 7 !/ % % ! ! !!!& 7 ! % !!!/ & % ! !

:

<",= L Æ&! / 7 8

? @ ? +?@@

¾

?@

REMARKS

$:

L& ! 6! O % % ! ! !/ ! ?@8 & 9 / &!/ %! &!& &

7 7 !Æ& & !&! &!/ %! /!7 ! % & %:

, - 8 0

© 2004 by CRC Press LLC

862

Chapter 8

ANALYTIC GRAPH THEORY

, ,# :## :* *## # -# 8 ,# :: , , 8 %!& !

%! &!& ! 7 + & &!/ %! & !&! &&!/ ! ! & !!!&

$:

&&! 7 && 7! @ %!& & &!&

! &!/ 4 ! &&! % & 7 ?@ @ & &

??@ @ ?@ @

%

$:

??@ > 3@ 3

L& ! !& ! !Æ& &! 8 8 $ 8 (&!8 & <(03= ! (T < (T 03=

8.4.2 Alterations O ! !!& &!6 && ! !! % &!& /! DEFINITION

: ! 9 / F & !8 ! ! % ! EXAMPLES

% &F & / %! / /! 7 &!& 8 ! 8 ! &! & 7 / %! & & / ) & & %! ?)@ & ! ! 8 ! F !

: <*0= 7 ! ! / B ! / %! /! &!& B L !!!& 8 / %! 7 <= > ! %!& & ¾ ! / && %! !! 68 %! & &!& 6 > '?@ Æ&! / % & / 7 !& & / %! 7 !& 8 /! ! !' B 4 1 / %! &7!& 6!

: N !/ & & / 7 ?@ & % ? @ ? +?@@ ¾

: 4 !!8 !!& ! % N &$ b? ½¿@ N &$1 % !9 !7 & 7 ! > b? ? @½¾@ ! % ($! !!7 <(33=

© 2004 by CRC Press LLC

Section 8.4

8.4.3

863

The Probabilistic Method

The Lovász Local Lemma

4 9 8 ! ! L& FACTS

: ?.7'@ , 7 ! .& . : !7 & &! 7

! ! ! 8 % %! 7 ! ! ! !! & %!& ! && ?!@ ??! @ 6 & > ?!!@ ! / C D !9 ?C G @6 ? 7 "C6 @8 ?? ! @ 3 ! ! ! !! & !

$

: & ! !/ !7 % ? @ E & /! & 7 !& &!6 / & &! !/ !7

?@ ? @ ? +?@@

%!& ?@ EXAMPLES

: .

> ? 0 : & > @ / ! %!& 7 / & / ! & / ? G @ ! & E & 7 ! ! 9 / 0 ! & !7 7 ( > ! & > 8 % 9 D %! 7 ( & 7 ! ! ! 7 %!& F& ! ! / D & / &!/8 6 > C > & 7 + &!/8 ! 8 7 / ! &

): ! && ! &!/ - % 7 /

> ? 0 @ & ! % & A 6 ! ! & & & A & &!/ ! / ! ! & ! A !! &!/ ! ! / ! !! & ! ! & %!/ ! &

7 ! !&! & ! $ 5 ( <5( 3= 7 %!/ ! ?!@ A B ?!!@ & A + ! & ! !/ ! ! & & ! A & & &! 7 ! !#8 %!& & ! & / & ! ! !! ! !!7 !! 7 %! && &!/ ! &!&&

+: - ! ! !&!

> ? 0 @8 > 8 !7! . C ! / C % & % !! ! 8 > !' 8 ! ! !' %!& &! & & !

& & & % ! ! ! %! !!7 !! 7 9 / 7 !# %! ! &

© 2004 by CRC Press LLC

864

Chapter 8

ANALYTIC GRAPH THEORY

REMARKS

$:

! !9&! & & 8 7 % 7 E ! ! ! !/ % $ ! & ( & & <0= 4 8 L! ' ( <4L( 0*= % ! & ! & & &!/ -! & &

$:

4 &! & ! &&!7 ! ! ! /! 9!/ F & ! !/ %! $/

N &$
8.4.4

The Rödl Nibble

! & !/ ! $ ! % 9 ( <(T *= Æ &F & -! ! !!/ & & % &!& !/ DEFINITION

: &! & ! 6! F & ! ! ! & ! FACT

:

<(T *= . 1 ? B@ !! !' ! <= %!& &! 7 B <= & 8 % 7

1 ? %@ > ? G +?@@ + + ! % / !' / / %! ( ! / & < ! 0= EXAMPLES

: S <S0+= ! % &!& !/ / ! '? @ ! ! ! & &! 7 8 ! & 7! & 7 ! &!& & & ! 48 & % %! !!7 !! 7 &!/ /!

!

:

¾ M! <M!0*= ! % ? @ > b? @ % ? @ ! !! & 7 ( N &!/ & / &! ! ¾ ( !/ N & ! &!&! %! '? @ 7 ! 4F!8 M8 ' ! <4M' 3=

© 2004 by CRC Press LLC

Section 8.4

865

The Probabilistic Method

: M <M0+= ! 7

+ /

! C G +?C@ - % & / / 8 !/ ! &

& /

: 5 ( <5( 0 = ! %

! / ! C G '?@ &!& ! !! & & / 7 !& / /

7 ! !&! cF& / c7 &

8.4.5 Bounds on Tails of Distributions !!!& %! 7 % !! ! !! / 7!! 7! FACTS

%!/ % ! 6!! %! ! !!!& &!!&

: ? 4' - H!/ ! 6! R /8 <433=@: .

@ > @ ?5 5 5 @ 7! 8 %! 5 5 5 ! &/!/ 7 7! 5 &/ 7 @ 3 % 7

??@ ?@ @ @

?"@

: % & 3 4 8 & & 4 ! & ! %! !! 6 L & &! ! ! ! 4 8 % % ! !! d 3 &! ! S <S03= 7 d %!& ! ! 6! % d ? ??! 4 @@ ?*@

% ! & LM ! 6! L!8 M !!
: 6! ?"@ % L! ' 8 8 M$!8
! % % !/ / / %! ! / % !/ 7 ! % !/ ! !&! / 4 % !/!/ ! ! 7 7 !H % !/ / C?@ / ! !! %!& % !/!/ ! O
© 2004 by CRC Press LLC

866

Chapter 8

ANALYTIC GRAPH THEORY

& 7 ! & %! !! 6 > % ?*@ !! % & % / & & REMARKS

$: ! ! 6! / <0+= & ! & 6! ?"@

$: ! & ! F & F & / <43"=8 <5( 3=8
References <4FM' 3= 5 4F!8 S M8 ' !8 4 ( 8 0 ?0 3@8 *"W+3 <4L( 0*= 5 4 8 4 5 L! ' 8 N 4 ( 8 5!& -! & & 8 $ ?00*@8 (3 <40= 48 4 /!!& 7 ! & 8 4 ?00@8 +,W,0 <433= 4 S & 8 " 8 & !!8 E! &! & 8 333
, ?333@8 W, <",= 8 $ /8 $ * ?0",@8 0W0" <*0= 8 !! 8 ?0*0@8 "W <+= 8 O &!! 8 6 ?0+@8 *W3 <0= S & 8 .! & ! 7 8 3 ?00@8 *W*"
© 2004 by CRC Press LLC

Section 8.4

867

The Probabilistic Method

/8 " ?33@8 3W, <(03= ( 8 N (&!8 S & 8 4 8 E! 8 003 <S03= S8 ! !! / 7!!8 4 ?003@8 W3 <SY.(33= S8 . Y &'$8 4 (&! $!8 4

8 E! 8 333

<S0+= 4 S8 4 !& &!& !/ /8 Q54 & !& 8 00+ <M!0*= S M!8 ( ? @ /! ¾ / 8 4 , ?00*@8 ,W3, <M0+= S M8 4 !& / ! &!/8 , ?00+@8 W*0 <5( 0 = 5 5 N 4 ( 8 L /!!& & & 8 " 5F $ ?00 @8 *"W*0 <5( 0 = 5 5 N 4 ( 8 4 &!& 8 ! ?00 @8 "W 3 <5( 3= 5 5 N 4 ( 8 !/ 8 33

" 8

<5,0= S 58 8 -8 ( !8 E!8 0,0 < (03= S X! P (T8 $ 4 8 !/ P /8 003 < ! 0= ! / S & 8 4 !& 7! &&! ! /8 * ?0 0@8 "W" <(33= S ($! 4 !!78 7 /! / % &!/8 4 + ?333@8 "W <( *= P (T8 O &$!/ &7 !/ 8 + ?0 *@8 +0W, <3= 5 ( 7!8 4 ?G>@ !! /! !!!/ / !/ % /!!& 7 ! .7' & 8 !

" $ # ! 0 % / ,8 33

<0"= S & 8 " 8 & !!8 45

!&!8 00" <0+= 5 /8 & ! ! ! !& ! 6!! ! & & 8 " 1 ?0 ?00+@8 ,W3*

© 2004 by CRC Press LLC

868

Chapter 8

ANALYTIC GRAPH THEORY

GLOSSARY FOR CHAPTER 8

9 0 # 8 : ! /

.

:

&

8 & ?&

@

% & &

% 8 * 8 :

: /

8

7

% 7 !& & ! & %

& !/

F& &! &

,# 8 8 : 0

&&

!

!

? @ W && ! ( : % %! @ ! / &!/ ! & &!& * & & ! & ?

:#- ' - ? 4

4 4 4 = 4 :#- ?# ' - ? 4 < + 9 [ > ?

?

? [

8

@

>

*

<

%* *#"

/

: & &! ?

W /

:

G @ &!6 !/ &

: /

%!

-#-0 - -#'

8

%

!/ & /

- #- :

6 & &

?@ 9 ! /

!

/

G

¾ Ô% # ½

4 W 7! : 6 & & 38
ZG

: & &!

8

@ >

W 7! :

@ >

&

4 +?@

*?@

/

>

?@

/

& ! 7 7

&!/

! /

&8 7 & &

7 !&

W /: / / & &

W ! /: ! & /K % !H

9!!

-#'

W ! /: & /8 & ! ! !&!K

% !H 9!!

-#' * - ?

@ W /

:

7 !& ! / &!6

?@ )? @ G )?@

W /: / !

F& 7 !&

%!

-:@ 8 : - - # # ' #

: & & /

%!

&& !7 F!!/

0 ?@ ?@

: !! 7! !H

! / K &!6 %!!

*

%-

- 8 :

: & & / & & %! % 7 !& !

!

- @ 8 : ::

:

/

%- - # 8 8 : :: # @#

)

7

/

7

W /: / / %! 7

7 8 ! /

) /

: & !/ /

W /

8

: !

W /

:

0 ?@ ?@ ! ! / !7!

7 !& 8 $ 7 /

© 2004 by CRC Press LLC

869

Chapter 8 Glossary

# @- @ 0 -# ¾

: ! 0 ?@

W /

%

: !H & 0 ?@

W /

?

?@

@ W ! /

!

: 7 /

&! % 7 ! / / ! &! ! /

@ 8 : %0 - F %0 8 : 8# # :: ) ) :# 0

W ! /

!

:

/ %!

! / %! / ! / W /

7 !&

: & &! !F! &! /

&7

: !/ & 7 7!

!! K &!6 %!! W %

! /: / / !!&

W /: / ! & & W /:

/ / 7!/

! %! 7 !&

: ! /8 !/ : ?@ ? @ &

0 ?@ !! ? @ ?@ 0 ? @K / /8 ! / !/ : 0 ?@ 0 ? @ & ! ! / 0 ?@8 ? @ ? @ ! ? @ 0 ? @

:8 : % #0 %:8 :$ * : 7 !&

8

8

: / % / !'

!

!!& &

! &

# - # 8 8 : :: # : - * # : 6 " 6- 6 # 0 8 : 8 : :: -## 8 : : --#- 8 : : ## :, 0 -- :

?

& %!& !

@: ! / & &8 %! &

& %!

&

! / &

!/ /

?

@ W /

:

/ ! 7 !&

! !

W 7 !& ! / : %!!

:

F& 7 !&

!/ %!

&

7

: /

6 & / !

&&!

: ! ! !& !/ & !/

: /

1 1 !

!& 6 & &&

!&!7 5 &! $!

: -!! / 7!/ & & ! / !

7 !&

W /: 8 / ! !/ %!&

/ & /% 7 7 /

: /

!

&"

>

* * *

© 2004 by CRC Press LLC

%!

?@ >

?& " @ 0 ?@ !

870

Chapter 8

:* *###- :

ANALYTIC GRAPH THEORY

7!/ ! & F & %!/ ! !

%! !!7 !!

$2 #**: %!!

!' &&!7 & ! K &!6

$ 8 : ! ? @: / & ? @ $ #:#-# ? @: !! &!& &! /

! / &!/ ( /

$ * / &!/ & & 8

?½ ¾

@:

%

&

>

!!7 ! /

!

*#: # & ? @:

? @

&

&!& & /

7 !& !! /

? @

!

&

- #- W ?½ ¾ @ % & ! & / 8 -

-8 ? @: 7 !& ! / 8

? @

&

& / !& & & ! & &

# 8 : ( ? @ %! > ? @: !!7 ! / & 8 # -8

8

/

!& &

& / &!/ ?( N @ ! ( !& &

$ %F # : # ? @:

$ %# F # : # ? @: @8

8 :½:

! N

%!

? @

! / & !9!

!!& !! /

?½ ¾

%!

? @

W ?½ ¾

@:

/

/

@

7

! ?

K !! &! !

! &! !/ / && ! %! !!

*# # ? 6@:

&

% ?½ ¾

!! & % ! !

@ / 7 <= >

8

!

!

! / & 9!

!!& !! /

C$ D # # 8 :

!! & % ! !

? @ / 7 <= >

½ ¾

&

7

!

K !! &! !

! &! !/ / && ! %! !!

6 > 6?@ 8

&K &

#0 ? @:

! !! & ! ? @ /

7 <= >

K

¾

!

%! &

/ K &

% 7 !! & / !

- .(

6?@

>

: 0 -# W /!/ !! / ? 6@8 /

!

;:

&!

´µ , ´µ >8

?@

? 6@

> 38 ? 6@ 7 ; ´µ G > ? %! A , ´µ

&

;

!

7 B AB ! &/ @

: 0 -# W /!/ ! / ? @8 /

!

;:

&!

´µ , ´µ >8

?@

? 6@

7 B AB ! &/ @

© 2004 by CRC Press LLC

> 38 ? 6@ 7 ; ´µ G > ? %! A , ´µ

&

;

!

871

Chapter 8 Glossary

#" $ *: ?½ ¾ 8

#-:

/ /

&

@

!! !' 8 ! 8 / 8 /

7 !& &

%!

? @

? @

' 0 --: & & & & 0: / ! %!& & & ! : !! / ½ #- * - 8 :: / % ! !

! !& /

! ! ! /

0 -# W / ;:

6 ?

@ 7

;

!

6

>

6 6

/!/

+? @8

?

?

;

? %! A 7 B AB ! &/ @ ( /!/ 9!! ! %!

8 : ?@: @

6

&

&

! /

?@ & > +?6@ ? @8

@8 &!

@

!

7 !& % & ?

6 !

* ?@: &!&

/

%: :*:

, : --#- 8 ::

© 2004 by CRC Press LLC

7 !&

9!/ / ! /

/ 7!/ / ! &!9 !

&!& &

!

!

/ 7!/ & & ! / !

Chapter

9

GRAPHICAL MEASUREMENT 9.1

9.2

DISTANCE IN GRAPHS

DOMINATION IN GRAPHS

9.3

TOLERANCE GRAPHS

9.4

BANDWIDTH

! " "

!# $% GLOSSARY

© 2004 by CRC Press LLC

Section 9.1

9.1

873

Distance in Graphs

DISTANCE IN GRAPHS

Introduction ! " ! # $ $ %

& ' $ $ $' ' & $ $' ( ) ' * $ !

"

+,$ -.#

9.1.1 Standard Distance in Graphs ($ $ /$ ' 0

* & 0

$ ' 0 Distance and Eccentricity

' *

$

* 1 2 * 1 DEFINITIONS

3 3

4 * 1 & "#

3 4 * & " # "$ # % ( % " # %

* 1 &

"# 5 1 " # Ü¾Î

EXAMPLE

3 6 * 1 4 $ * 1 * 1 "# 5

© 2004 by CRC Press LLC

874

Chapter 9

Figure 9.1.1

GRAPHICAL MEASUREMENT

FACTS

3 4 * & " # ' %!

3 0 0 * % " & " "# # % &#& & 0 $

3 " # - "# " # 5 - '

5

" # 5 " # "# + " # " # 7 " #

. "# + .

Radius and Diameter

! $ 1 $

* DEFINITION

3

$ ' *

& "#& 1 $ ' & "# REMARK

3

/$

1

¾

" #

EXAMPLE

3

6 * 1 4 $ '

"# 5 "# 5

Figure 9.1.2

© 2004 by CRC Press LLC

Section 9.1

875

Distance in Graphs

FACTS

3

4 * ' * & "# "# "# '& 8 9 $ "# & '& % $& 0 1 & /$

"# "# "# '

3 +:;. 4 * ' * & 1 "# 5 "# 5 4$ & $ "$ * # $ 7 Center and Periphery

$ * $ DEFINITIONS

3 ( * 1 "# 5 "#< * 1 "# 5 "#

3

$ $ ' * & "#& $ $ ' *

& "#

EXAMPLE

3

6 * 1 4 $ ' "# 5 "# 5 ' 4 $

Figure 9.1.3

FACTS

3 3 3

+,$ =. 6* ' " # +> . * '

)

+?@ . * '

* 1 ¾ 4$ &

* 1 ' " # 5 " #

!

3 +, '=. ( * " # ' ' * ' * 1 ' * 1 '

© 2004 by CRC Press LLC

876

Chapter 9

GRAPHICAL MEASUREMENT

Self-Centered Graphs DEFINITIONS

"# 5 !3 (

$ $ $ 3 (

EXAMPLE

3 * 4 $ %

¾ % &

Figure 9.1.4

"# $ $# $

FACTS

%3

+,$; . A $ 1 % & 2 & $ '

" #" # " 7 7 @ # B

5 5 & 5

&

3 4

% $ &

& 2 &

3

+A C -. 4 * % & 1 % 1

' $ "# $ $ "# $ $ $ $

3

+A C -. 4 * ' 0 $ D& 1 % $% $ D

9.1.2 Geodetic Parameters Geodetic Sets and Geodetic Numbers

4 * ' & 1 *

* ' * 1

*

© 2004 by CRC Press LLC

$

Section 9.1

877

Distance in Graphs

DEFINITIONS

%3 ( * 1 % * 1 $ 5 " & * 1 # &

3 4 * & + . & & * ' % < "#& + . * ' + . 5 + . ¾

Ë

3 ( * + . 5 "# ( $ ' ' $ & "#

3 ( 1 % $ $ $ " # $ REMARK

3

6 * 1 $ $ * ' B $& %* 1

* '

EXAMPLES

3

5 4 $ % & $ " # 5 B & 5 $ & "# 5

Figure 9.1.5

'

3

5 $ 4 $ @ $ $ ' " # & $ $ u

v

C4 :

G: x

Figure 9.1.6

© 2004 by CRC Press LLC

u

v

x

w

w

( )

878

Chapter 9

GRAPHICAL MEASUREMENT

FACTS

3 3

+A . $ >%

+EF-. B & "# 7 & $

3 +,$G$. A * "# "# 5

' 5 " # "# 5 ' 5 "½ ¾ #7 & ' 7 7 7 5

3

+EF-. 4 * '

* & & & 1 "# 5 "# 5 "# 5

3 +EF-. ( * $ $ '

* ' * 1 ' * 1 '

!

3 +EF-. ( * $ $ ' ' Convex Sets and Hull Sets

* ' * ' * ' * DEFINITIONS

3

( * + . 5

+ . * 1

3 4 * & +. 5 & +. 5 +.&

+ . 5 + + .. 4 & /$ * + . 5 + . & " # + . & & * 1 $ & + . & "#& * '

"# 5 1 " #

3 A * B + . 5 "#& ( $ $ ' ' $ $ "#

##

3 ( 1 $ $ $ " # $ $

EXAMPLES

3 B 4 $ ; & 5 5 +. 5 5 +. 5 & * 1 < * 1 4$ & * 1 $

© 2004 by CRC Press LLC

Section 9.1

879

Distance in Graphs

u

u

x

G:

w

x

w

v

S1 = {u, v, w}

v

S2 = {u, v, w, x}

Figure 9.1.7 *

!3 B 4 $ =& 5 + . 5 ¾ + . 5 "#& $ B & $ $ "# 5 4$ & " # 5 s

t x

G:

u

z

w

y

Figure 9.1.8 ( ## REMARK

3 6* ' $ * $ $ "# "# * ' FACTS

%3 +> =. A $ "# 5 5 5 -& 5 5 & 5 7 &3 +6* =. B & "# 7 3 +6* =. B % & "# 3 +EF--. 4 * ' & 1 % $ "# 5 "# 5 3 +EF--. 4 * ' * & "# 5 " ¾ # 3 +EF--. 4 & "# 5 '

5 "½ ¾ # 7 & & * & 7 7 7 5 3 +EF--. ( * $ $ $ ' * '

© 2004 by CRC Press LLC

880

Chapter 9

GRAPHICAL MEASUREMENT

9.1.3 Total Distance and Medians of Graphs Total Distance of a Vertex DEFINITION

3

"

# * 1

0 '

" # 5

" #

¾ ´µ

ALTERNATIVE TERMINOLOGY3 * 1

* 1

EXAMPLE

%

3 6 * 1 4 $ * 1 $&

" # 5

¾ ´µ

" # 5 - 7 7 7 7 7 7 7 7 7 7 @ 7 ; 5 ; 2 u

0 2

G:

4 3

6

5

7

1 1

4 2

Figure 9.1.9

#

FACT

3

+6?;@. A 2

"# " #" 7 #

"#

$ ¾ The Median of a Connected Graph

' $ H

$ DEFINITIONS

!

3 ( * 1 *

%

3 *

$

! "# $

© 2004 by CRC Press LLC

$ '

Section 9.1

881

Distance in Graphs

EXAMPLE

&

3 6 * 1 4 $ - & * 4 $ - 38 37

29 28

G: 37

26

26

u

v

42

34

52

M(G) : u

vv

29 38

Figure 9.1.10

(

FACTS

3 !3

+=-. 6* ' " # +=. * '

)

Centers and Medians of a Connected Graph

* DEFINITION

&3

4 $ &

" # 5 " # 3

" # " #

FACTS

%

3 += . 4 * ' * & 1 $ "# & ! "# & " "# ! "## 5

&3

+> . 4 * '

& & &

$ $ & 1 $ "#

& ! "# & "# ! "#

9.1.4 Steiner Distance in Graphs 2

*

' * $ 6$

)& * & ) Steiner Radius and Steiner Diameter

( 1

© 2004 by CRC Press LLC

882

Chapter 9

GRAPHICAL MEASUREMENT

DEFINITIONS

: * 2 ' & " 5 * & "" # 5 " #

3 4 ' " * & "" # " $ 2 "$ # $ " > '& $ $

&

" 3 A 4 & * 1 & "#& 1 $ % *

3 & "#& & "#& * '

"# 5 "# "# 5 1 "# ¾ ´µ

¾ ´µ

EXAMPLES

3

A 5 4 $ " # 5 *

2 &

4 $ u

u z

v

y

w

z

v

T:

G:

w x

x

Figure 9.1.11

( "

3 6 * 1 4 $ % ' ¿"# 5 ¿"# 5 @ 6 5

5

G: 5

5 4 6

Figure 9.1.12

© 2004 by CRC Press LLC

5

5

6

$

Section 9.1

883

Distance in Graphs

FACTS

3 +E: F= . A 4 * '

& " # 5 " # 3

+E: F= . 4 * '

" #

* '

&

" #

4 $ + : -.

3

+ : -. B &

"# "# "#

" # "#

"#

3

+ : . 4 * ' * '

"#

&

7 "#

Steiner Centers

$ DEFINITION

3 4 & * 1 "# 5 "# $ $ ' % *

FACTS

3 +: -. A 6* ' " # %

3 +: -. A

" # %

' %*

9.1.5 Distance in Digraphs

$ 0 * 1

DEFINITIONS

3 A * # B # % & " # %

#

© 2004 by CRC Press LLC

884

Chapter 9

GRAPHICAL MEASUREMENT

3 ( # " # # % % * ' * # Radius and Diameter in Strong Digraphs

0 '& $& $

$ "

0 # DEFINITION

3 "# # * 1 # $ ' * # & "##& 1 $ ' & "## EXAMPLES

3

% # 4 $ ( % " # 5 : & % # B & % # ' * 1 "5 # #

- & # u

D: v

(

Figure 9.1.13

3

* # 4 $ ' %

: * "## 5 "## 5 & & $ "## "## 5

D:

2 4

4

3

Figure 9.1.14

2

FACT

3 +E? . 4 * ' * # "## 5 "## 5

© 2004 by CRC Press LLC

& 1

Section 9.1

885

Distance in Graphs

The Center of a Strong Digraph DEFINITION

!

3 "## # $ $ ' * "# 5 "## EXAMPLE

3

# 4 $ 4 $ & 5

u u

D:

4

2

4

C(D) : v

3

Figure 9.1.15

2 v

FACT

!

3 +E? . 4 * ' #& 1 " # # Strong Distance in Strong Digraphs

' ' 0 & 0 $ $ "

0 # DEFINITIONS

%

3 4 #& " # $ 2 $ #

&

3 "# * 1 # * 1 #

3

$ ' * # 1 $ '

"## "##

EXAMPLE

3 B # 4 $ @ & " # 5 & " # 5 & " # 5 * #

& "# # 5 @ "# # 5 - © 2004 by CRC Press LLC

886

Chapter 9

10

w

v

GRAPHICAL MEASUREMENT

10

10

D1 :

6

D2 :

x

u

6

6 10

10

y

Figure 9.1.16

10

"

FACTS

%3

+E6F %. * 1 $ "## "## "## * ' #

&3

+E6F %. 4 * ' & & 1 # "## 5 "## 5

3

+E6F %. B # $

" #

& "##

Strong Centers of Strong Digraphs DEFINITIONS

3 ( * 1 # "# 5 "# $ $ ' * #

"## #

3 ( # & #

"## 5 "##&

EXAMPLE

3 "## # 4 $ ; 10 10

10

D:

u

6

6

6

u w

v

w

SC(D) :

v 10

10 10

Figure 9.1.17

FACTS

3

+E6F % . 6* ' " #

© 2004 by CRC Press LLC

Section 9.1

887

Distance in Graphs

3 +E6F % . 4 * ' % $

& 1 0 ' ' '

References +, '=. ,

) '& * &

= " =#& @ I; +,$; . 4 ,$ ) '& % * $& " ; #& I +,$ -. 4 ,$ ) ' 4 '& & ( %C '&

-

+,$G$. 4 ,$ ) '& 4 '& A J G$ & 61 $ $ & +,$ =. 4 ,$ ) '& F & ? & : * & " =# ;I +E6F %. E& 6 & & F& & " #& I +E6F % . E& 6 & & F& :

& " #& I@ +EF--. E& 4 '& F& : $ $ & ; "---#& I= +EF-. E& 4 '& F& : $ & "--#& I@ +E? . E& A ?& & 3

& = " # = I +E: F= . E& : : & & , F$&

& " = #& I- +6?;@. E 6 & 6 ? )& 6 ' & & @ " ;@#& =I @ +6* =. 6* , & $ $ & ; " =#& ;I +A . 4 '& 6 A$)) & E $& $ & ! ! ; " #& = I +> =. 4 ' ? >

& E* 1 ' & " ! @ " =#& =I -

© 2004 by CRC Press LLC

888

Chapter 9

GRAPHICAL MEASUREMENT

+> . 4 ' F >& ' $

& # = " #& I + : -. ( & : : & E & : $ & " -#& I + : . ( & : : & E & : * 1 $ ' & $ $ ! ! " K (*& 4 L E$& A & 4 $#& B( $ " #& I- += . L & ( & % & J & $ " = # I= +?@ . E ?& $ & % ;- "=@ #& =I - +A C -. A

E C& : $ $ % & &!! % " -#& I@ +> . L >*' & : & ' & J & $ " # ;I-- +: -. : : & & " -#& =I ; +:;. ( :& 0 $ & " ;#& ;I; +=-. ? & ' & " =-#& = I +=. $2 2'M ) & E $ ' & " =#& I=

© 2004 by CRC Press LLC

!'

Section 9.2

9.2

889

Domination in Graphs

DOMINATION IN GRAPHS

B$ ' E ,$ >$ >$%$%' $ @ J 2 H E! $ ; E =

9.2.1 Introduction C * 8 9 " # *

' 1

0 B ' $ )&

& 2 & & $ 1 '& ' 1

) ' ' &

+ =& = . 1 $% * ' $* '

$ +( ;.& + @.& + A -. 4 * ' &

$ + =. * --

DEFINITIONS

3 ( 5 " $ # * 1

! * 1 ( * 1

3

% "# $ ' C $ % "#

EXAMPLE

3

5 & 5 ' 4 $ * & $ ' " % "#% # % "# 5

© 2004 by CRC Press LLC

890

Chapter 9

Figure 9.2.1

5

& 5 '

GRAPHICAL MEASUREMENT

+

Equivalent Definitions of a Dominating Set $ ' N 1 % ) C /$ * 0

DEFINITIONS

'(

3

* 1

" &

' * 1#

* 1

&

!

( "#&

#

3

'

Ë

¾ ( +. 5

$

' 3

$

-

¾ * *$ $

'

'"

#

&

) %

3

)%

3

A

*

( +. 5 ( "#

* ' * 1 & ( +.

"

3

* '

* 1

& " #

* 1

!

3

" %%

$

% "# 5

$

!

© 2004 by CRC Press LLC

% "# *

( )

-

'3

Section 9.2

Domination in Graphs

891

Applications of Domination

*

' 0 * $ ' % $ ' 4 & $ )& ' $ & $* ' & ' EXAMPLES

3

, +, ;. )

$ $ $* ' $ $

$* ' $

3

A $ +A @=. $ $ )&

& & * ' ' )

3

' B 0 * % * % " # * ! ' N % & * " #% * & & *

&

1 & LO & & +L;.

3

( '

* ' )

& $ % (

' /$

9.2.2 Minimality Conditions > & * ' $ * & * ' $ ' DEFINITION

%

3 ( $ EXAMPLE

3 4 4 $ & ¾ 5 '

$ 6' ) $

C $ : +:@. FACTS

3 ": H # +:@. A # 5 " $ # # ' #

3 +.3 1 * 1 # $ ( "## 5 < +.3 * 1 ! * 1 #

3

+:@. B 5 " $ # * 1 # & #

© 2004 by CRC Press LLC

892

Chapter 9

GRAPHICAL MEASUREMENT

3 +,E; . B * 1& 1 $ * * ' * 1 '

The Domination Chain

$ /$ ' DEFINITIONS

&3 3

( *

'

* !

&

* "#& 1 $ %

3 & "#& /$ $ ' 1 "( 1 $ #

3 C $ % "# ' & & D"#& 1 $ '

3 4 ' & ( " # 0 ¾ ( "# ( + . 5 ( " # ( *

* ' * 1 & ( +. ( + . 5

3

$ ' 1 $

3

1 $ ' $

& "#

& +"#

EXAMPLES

3

4 $ 1 N 2 3 @ ; = $& "# 5 * " # 5 @

Figure 9.2.2

!3

( + " # 5 * " # 5 @

4 $

N 2 3 & @ ; = @ ; = $& % " # 5 D" # 5 @ ">

& " # 5 +" # 5 @#

© 2004 by CRC Press LLC

Section 9.2

893

Domination in Graphs

4 $ 1 $ N 2 3 = @ = - $& " # 5 +" # 5 ">

& % " # 5 # %3

Figure 9.2.3 (

+ " # 5 % " # 5 +" # 5

FACTS

3 +, @. ( 1 '

3 +, @. 6* ' 1 3 +E ;=. ( ' $

3 +,E; . 6* ' 1 $

* ' 1 & * ' 1 $ & * /$ ' & 0 * ' E )' &

& ;=

!3 +E ;=. 4 * ' &

"# % "# "# * "# D"# +"# %3 +E4=. B & * "# 5 D"# 5 +"# REMARK

3 /$ ' & ) &

'< 1 ' -- *

$ * $ /$ /$

4

1 & E )' & 4*& '& $ +E4'$--. 2

% " # 5 " # *

$ $ 2 ' ' +' . $

$ $ ( % $ * 2 $

* + '-.

© 2004 by CRC Press LLC

894

Chapter 9

GRAPHICAL MEASUREMENT

9.2.3 Bounds on the Domination Number $ >% "

+?; . E + =.#& 0 $ REMARK

3

: * $' & % "# 6/$ ' $ ' P"# 5 & /$ ' $ $

' P"# 5 -& & 5

Bounds in Terms of Order and Minimum Degree

$ * $ * & * /$ 4 $ $ $ *

% "4 -# DEFINITION

3 5 Æ ' " #

* 1 ! * ' * 1 ' Æ & $& $ ' * 1 " #& * 1 ¼ ¼ FACTS

&3 3

+:@. B * 1& % "#

+4 ?L =& Q$=. B * 1& % "# 5 ' ' Æ '

3

% "# 5 2 ' +,E--. +J =.

3

B $

Æ "# & $ $ 4 $ * % %0 1 * 1 " $ 1 * # '& E + = . 0 8 9 4 $

Figure 9.2.4

3

( ## , ,)-

+ = . B Æ "# & & % "#

© 2004 by CRC Press LLC

Section 9.2

895

Domination in Graphs

¾ -Minimal Graphs

E + = . $ ' $

/$ 0 0 ' $ ' 2 *%% & " 0 = # DEFINITIONS

!3

( %* 1 '

3

%

Æ "# & "

# & "

# % "# %3 4 & ' ! * 1 ' %* 1 " 7 # ' - "

4 $ "## &3 ( $ ! ' '

' * & ' & * 1 B $& ' * & ' ' #" # " #

"

4 $ " # " ##

Figure 9.2.5

.

3 ( ' & ) ' - "

4 $ @#

Figure 9.2.6

( $/0 $/)0

3 ' %"# $ %! * $ & ' %" # $ * 1

$ "

61 - #

A ' ! $ ' . ' %"# ' %" # $ ' . ! ) * N $ : *

3

6

© 2004 by CRC Press LLC

896

Chapter 9

GRAPHICAL MEASUREMENT

3 ( * 1 ) * 1 * 1 ' %"# $ $ $

) "

61 - #

A 5 #" ;# #" # & 4 $ ;

Figure 9.2.7

1

FACTS

3 + = . B & & Æ "# 5 & % "# 5 & ' * 1 & % "#% B $& % 4$ & ! " # 5 " # 7 % & 5 & $ * 1

3

+ = . ( % ' & #" #

3 + = . B = Æ "# & % "#

!3

+ @. B Æ "# & % "# = * & $ "

61 #

EXAMPLES

&

3 ( '

' %"# $ ' %" # $ 4 $ = ) * ' ) * & $ * ) *

Figure 9.2.8

© 2004 by CRC Press LLC

( #

Section 9.2

897

Domination in Graphs

3 % $ 5 = " 4 $ # * $ 5 =

Figure 9.2.9 + $#

3 ' %

' $

'

$ 4 =

/$ ' A ¼ ' 4 * 1 ¼ & " ! # ' % $ 4 $

' ' * A $ ' $ "

4 $ -# 6

' $

%

Figure 9.2.10 ( #

More Bounds Involving Minimum Degree FACTS

%3 +E=& E -. 4 ' $

Æ & % "#

Æ

Æ

Æ7

&3 +( & (;& ;. 4 ' $

Æ & % "#

7 "Æ 7 # Æ7

3 +(;& ;. 4 $

Æ & % " #

© 2004 by CRC Press LLC

Æ 7

Æ

/

898

Chapter 9

GRAPHICAL MEASUREMENT

3 +C =. A 5 "¾ ¾ ¾ 7 ¾ ¾ # * ' & 7 % "# 7 Bounds in Terms of Size and Degree

$ FACTS

3 3

& % "# " 7 # + ;. B Æ "# & % "# " 7 # /$ ' ' ' " # B

DEFINITION

3

( 3 " # "

# "

#

¿ % 0

Æ "# & & % "# 0 &

EXAMPLE

( 1 ¿ % 4 $

3

Figure 9.2.11

( 2 + %"# 5 " 7 #

REMARK

3

( 2 ¿ % $ +

.

Bounds in terms of Order and Maximum Degree FACTS

3

+, ;& C( ; . 4 '

7P

© 2004 by CRC Press LLC

1 $

P&

% "# P

Section 9.2

899

Domination in Graphs

3

+4J -. 4 ' * 1& $

Æ 1 $

P&

% "#

7 "Æ #

P

Æ

/$ 4 @ 3 +;. 4 * 1 $

Æ &

% "#

" 7 Æ #

Bounds in Terms of Order and Size

J 2 +J @. $ 2 * * $%

B * $Æ

' ' & $

* 0 DEFINITION

3 ( $ $ /$ * * FACTS

!3 +J @. B 2 $ % & " % # " % 7 # 4$ & 1 $ 2 ' % 7 * & * $ * & % * %3

$ 4 =

* $ $ 4 <

& $ P"# 5 % "#

&3

+ . B 2 $ % P"# % & " % # " % 7 #

3

+, @&J @. 4 ' 2 &

% "# 7 7 4$ & % "# 5 ' * ' Bounds in Terms of Packing DEFINITIONS

3 ( 5 " $ # * & & & " #

!

3 1 $ ' )

B 1"#

© 2004 by CRC Press LLC

900

Chapter 9

GRAPHICAL MEASUREMENT

REMARK

3

> ) & * ( +. 5 & ) $ * $ % "#

& ( + .

FACTS

3 3

4 ' & 1"# % "# + ;. 4

& 1" # 5 % " #

9.2.4 Nordhaus-Gaddum-Type Results B @ +>@. ' >$ $ B ' * $ $ $ $ $ $ *

* * 4 0 $ $ DEFINITION

%3 4 & * 1 "# " # * !

' ' /$ ! FACTS

3

+?;. 4 ' &

" # " #

3

% "# 7 % "# 7 &

% "#% "#

+Q$=. A & % "#% " # 5 ' 3 & ! $ ' Æ ' &

3

+?( . B * & % "# 7 % "# " 7 #

$ *

A % !

3 + . A 5 & " # % " # 7 % " # 7 % " # 7 & " # 1 $ *$ $ % " #% " #% " # ; 7 R" # REMARK

3

4 4 ;& 1 & & $ 1 $ $ '

; 7 & ; 7 &

© 2004 by CRC Press LLC

Section 9.2

901

Domination in Graphs

9.2.5 Domination in Planar Graphs >% * 1 $

"

+?; .# $ $ $ B '

$ * ' $ /$ ,$ $ "

+4 .# FACTS

!3 + @. $ $ * ' & $

%3 + -. 6* ' $

1 4 $ & 4 $ $ /$ $

&3 + -. 6* ' $ $

3 + -. 6* ' $

3 + -. 6* ' $Æ

' $ @& $ 3 + -. 4

$ & 0 ' ' %

$

3 + -. 4

$ & 1 $ $

EXAMPLES

3 4 $ & $ ' *' 'N + @.& $ 4 =

Figure 9.2.12 ( #

© 2004 by CRC Press LLC

)

902

Chapter 9

GRAPHICAL MEASUREMENT

3

$ 4 ' 4 $ B ' ' $ ' *

4$ & ' ! *

& $ $ $

/$

Figure 9.2.13

( # + )

REMARKS

3 S 4 =& *' 'N + @. *

* $ $ B $&

$ $ ' : & ' * & & * $ $ $

3 C 4 0 ' ' $ @& $ )

9.2.6 Vizing’s Conjecture : $* ' ** $ B @ J 2 +J @. $ $ $ $ $ 4 * ' N ! $ & $ ** CONJECTURE

! +J @=.3

4 '

& % "#% " # % " #

DEFINITION

&

3 ( % "# 5 "#

$ $ $ $ + . $

© 2004 by CRC Press LLC

Section 9.2

903

Domination in Graphs

FACTS

3

+, ; . B $ % " # 5

% "#& ' & % " #% " # % " #

3

' 4 $

' & J 2 H E% ! $ $

" ' * ' ? L +?L =@.# ' " ' * ' +6 .#

$ $ E) $ +E$--. 3 +E$--. 4 '

& % "#% " # % " #

REMARK

!3

4 $* ' $ J 2 H ! $ &

+ .& + .& H E ; + = .

9.2.7 Domination Critical Graphs B $ * ' $ 4 & '

$ ' $

+,> $=. +C( ; . , & E & $ +,E $==. $

* 1 $ ' * 1 * & '

$ ' "

+$,=.# DEFINITION

3 ( * ' $"#& %" 7 # 5 % "# B % "# 5 & % $ %% % % "# 5 % " 7 # 5

$ "# FACTS

!3

%3 &3 3

% "#

%% % "* $$'#

'

%% % ' +$ -. ( %% % ' 5 2 ,

2 * , ' %% % 2 , +$,=. (

$ %

3 3 3

+$,=. %% %

+$,=. 6* ' %% % * %

+C -. 6* ' %% % 1 * %

© 2004 by CRC Press LLC

904

Chapter 9

GRAPHICAL MEASUREMENT

REMARKS

%3

' %% % %% % *

2 4 0 & $ $ %% % 1 ) $ ' %% %

&

3 B * ' %% % 1 *

3

4 $* ' $&

$ H E @

+ = .

3 $ $ * 1 & *

$

4 $* '&

E + =.

3

> 1 $ N

% 0 * $ & & $ $ 4 1 & $ ' ( ' $ ' ' ' + -. * 1 & $ J & 2

$ J

9.2.8 Domination Parameters ' ' ' C 0 ' ' ' + . 2 0 DEFINITION

3 4 * ' &

' + . 0 '

% " 3 #

$ $ $

EXAMPLE

3

1

+ . " +E ;;.# + . * " +E =-.# + . " +C; .# + . " /$ +L =.&+L E==.# + . " % + =.# @ + . ' " ' +A C ;;.#

© 2004 by CRC Press LLC

Section 9.2

905

Domination in Graphs

REMARKS

3 ,' 0 & % "#

% " 3 # ' ' C 1

& 1 * & * * % 3 $ 2 * 0 ' * ' N

3 (& ' 0 ' 4 1 & /$ * 1 $ % +4 ?=.

References +( . > ( ? & ! & ? C ' & +(;. J B ($*& 6 1 ' $ '

* & "$ # ! ' ($ "@# " ;#& I= +,E--. Q$ , & 6 ? E )' & C ' &

& F & 61 /$

** & ) @ "---#& I- +, ; . ( , ) A 4 & 1 ' $ E $ & &! % !'& "# " ; #& I= +,> =. ,$ & 4 '& ? >

& E A $N & & ; " =#& I@ +, @. E , & # ! " $ & A& @#& -I +, ;. E , & & >%& ( & ; +,E; . , , M 6 ? E )' & % %

& & $ & " ; #& I +,E $==. E , & F E & $& J 1 % & = " ==#& ;I; +E=. K E K '& : * 1% $ % & - " =#& @;I=- +E -. K E K '& ( % $ & * " -#& -I-@-

© 2004 by CRC Press LLC

906

Chapter 9

GRAPHICAL MEASUREMENT

+E$--. C 6 E) $ & ( /$ ' J 2 H ! $ & +!) ; "---#& & > & " # +E =-. 6 ? E )' & &

& & - " =-#& I +E4'$--. 6 ? E )' & : 4*& E '& ? $ & ( % 2 "% #%

& "# "---#& ;;I +E4=. 6 E )' & : 4*& E '& ( & E $ ' & $ & " =#& I= +E ;;. 6 ? E )'

& ' & ; " ;;#& ;I@ +E ;=. 6 ? E )' &

& ? &

' ' & &!! " ;=#& @%@= +6 . 6%F E

)& $ $ & " #& I; + -. O & C & ( & BB& + '-. O & C & ( & E '& ( % 2

/$ $ & +4 . 4 & & L 'N& A * %

1 $

& ! @ " #& I +4 ?=. ? 4 4 ) ? & % & B K (* (? )& & ! ! "L2& B =#& =& =I-- C ' +4 ?L =. ? 4 4 )& ? & A 4 L & ? & : * $ & @ " =#& =;I +4J -. 4 A J)& 6 $ & =- " -#& I +?; . ' ?& * !, # )!& C 4

& 4 " ; # + . C & ( & E & >$%$% ' $& @ " #& I + -. C ( & & - "--#& I + . 4 ' C ' & E ' BJ3 & -! - " # ; I

© 2004 by CRC Press LLC

Section 9.2

907

Domination in Graphs

+ . , 4 & : J 2 H ! $ & = " =;I @

#&

+ . , 4 & : J 2 H ! $ % $ & " #& -I@ + ;. C ' & 3 (

* *

& " ;#& I; + =. C ' &

& ? & .! # ) & )) & > K)& = + = . C ' &

& ? & , ' & )) & > K)& = + =. C ' ? & % & " =# I-@ + -. C ' ( & E $ 3 0 & + A -.

E A)& & & *$ = > & > K)& - + @. ( & 3 ( $* ' * ' " E ? #& @ " @#& I; + . ( & ( $

%

$ 1

2 & " #& I@ +?L =@. ? A 4 L & : $ BB3

& - " =@#& ;I-@ +?;. 4 ? E '& $ ' >$%$ $ H H$ & % ; " ;#& ;=I;- +?( . ? ? ($$& & * ) " #& ;;I= +L;. ? LO & & ? & : *

% & ( " ;#& I- +L =. A A L & ! & & > S *& =

!

+L E==. A A L , E22 & )& ! @ " ==# @;I; +A C ;;. A A )%4 ? 6 C : $

& &!! - " ;;# I-

© 2004 by CRC Press LLC

908

Chapter 9

GRAPHICAL MEASUREMENT

+A @=. E A A $& * ! & % & > K)& @= + @. *' L 'N& $ & " @#& I + = . C E$ , & $

& " = #& ; I;@ + ;. ( ? C &

) * $

& / @ " ;#& I +' . E '& J * ' $

& "# " #& @I;; +>@. 6 ( >$ ? C $& : ' & ! @ " @#& ;I;;

0 +;. E '& $ H H$ & + % 10 ; " ;#& -;I; +Q$=. E ' > Q$& % & @ " =#& I +:@. : : & # & * & B& @#& -@I

! = "( &

+J =. , A J)& E 2 /$ %

* $ & " =# I@ + @. , (

& & $

& " @#& ;;I +C; . 6 )$ , C )& $ & " ; #& @-;I@ + . A ( & 1 $ $ * $ & =; " #& @I; + ;. A ( & ,$ $

& " ;#& I +$ -. $ & E & =@ "

-#& I@

+$,=. $ , & & & " =#& @I;@ +J @. J J 2 & $ & ( !2

" @#& -I

+J @. J J 2 & ( $ 1 ' $ & ! @ " @#& ; I;

© 2004 by CRC Press LLC

Section 9.2

Domination in Graphs

909

+J @=. J J 2 & $* '& - "@# " @=#& ;I +C( ; . , C ) , ( '& & ! 3 " ; #& ;-%; +C( ; . , C )& , ( '& 6 )$& *

' & B $ & ( & B 3 & " ; # +C =. L C & $ * ' & % 4!!5& @ " =#& I +C -. 6 C! )&

% & " -#& -I

© 2004 by CRC Press LLC

910

9.3

Chapter 9

GRAPHICAL MEASUREMENT

TOLERANCE GRAPHS ! " " B

Introduction $ * 1 * ! 2 $Æ

' &

& 1

B & * * N ' $ * $

(

( $ $$' ' * '& $ ' !$1

9.3.1 Intersection Graphs ( ' ' * & * ! ' * ' Ordinary Intersection Graphs DEFINITIONS

3 A 5 ' & T"#& * * 1 ! ' 5 / 5 ( 1 ' $ 5 T"#& ' ' ' ' "# 5 $& $ "# ' 5 C 5 T"#&

3 & "#& $ % ' $ ' $

3 ( ' ' 5 $ $ * ' $ " # $ " # 4 $ $ $ 1 /$ * 5 ' 3"#

6 & /$ /$ 1 $ %

© 2004 by CRC Press LLC

Section 9.3

911

Tolerance Graphs

EXAMPLE

3

$ 5 5 & 5 & 5 & 5 & 5 5 T"# 4 $

Figure 9.3.1

( 1

FACTS

3 3 3 3

" 2 ) H # +. 6* '

& "# 5 3"# +LC;=. 3"# >% +6@@. 4 ' 5 "#& "# +6@@. 4 * '

REMARK

3

' $ $ +

.

Interval Graphs

B * 0 $

%

' +;. $ ' +, . ' $

* $ +4 =.& +=-.& + .& + =. :' )$ $ DEFINITIONS

3 ( '

*

3 ( * * *

3 ( * * ' 3 (

* $ & ' * & $

!

3 ( $ $ % 2"# 0 " # " # 2"# " # 2"# * "#

© 2004 by CRC Press LLC

912

Chapter 9

GRAPHICAL MEASUREMENT

EXAMPLE

3

4 4 $ & *

Figure 9.3.2 " #

# #

FACTS

3 +A ,@. ( * ' $ $

3

+ @. ( * ' $ $ *

3 +@ . A /$ * 3 "# * " # $ * " # * $ $ Chordal Graphs DEFINITIONS

%

3 (

$ $

'

&

3 ( ' $

3 3

( * 1

"% ' # 0 &

$ $

( * & * 1 $ $ * 1 $

&

REMARK

3

$ : $ +,A . + .& 1

FACTS

!3 %3

+,$;& ;& C;=. ( ' $

+4$@& ;-. ( '

© 2004 by CRC Press LLC

Section 9.3

913

Tolerance Graphs

Competition Graphs DEFINITION

3 A # * 1% % 2 4 2& ' 4" # 5 " # 2& 5 4" # 4" # " T"# ( 1 # $ 5 T"# ' $% * # EXAMPLE

3

4 $ #

Figure 9.3.3

( #

REMARK

3 E ' $ ' E +E;=. % $ '

' B 1& # $$' ' +;=. $Æ

$ * & $ % ' $ * /$

1 * ' $

"

A$= & L # $ * %

2

+ . ' FACTS

&

3 + $,=& A$=. ( '

' * 1% "# 5 $ /$ * & & 0 5 /

3

+ $,=. ( ' '

3"# "#

3 + $,=. ( '

* 1 "# 5 & /$ * $

5 /

3

+$ =. ( "# ' /$ * % & " & 1 & 5 & $ #

© 2004 by CRC Press LLC

914

Chapter 9

GRAPHICAL MEASUREMENT

9.3.2 Tolerance O$ & +=. +=.& $ 2 * B +? $ . 1 N ' $

, ) & $ 0 DEFINITIONS

3 A 5 ' $ 0 & 6 % * & %*$ $ $ & * $ (& 7 % * & %*$ & ' $ 0 * $ 7 ' 7& 6& H * 1% "# 5 $ "# ' 5 / 6" # 7" #

3

7 1 & 7& 6& * & $ 7% 7& 6& H (

H& 0

p-Intersection Graphs

' ' 7% 2 & 7 $

6 $

DEFINITIONS

3 4 & ' 5 $ 0 0 * "# 5 $ "# ' 5 / ( 1 ' %

3 ( ' ' $ "# $ & * ' $ & 5 ½ $ $ & $ /$ *

!3 $ ' $ % %3 ( %

' $% * FACTS

3

+? . 6* ' % & ' *

3 +BL . ( % '

' % /$ * ' "#

3

4 * ' & % $ ' % /$ *

© 2004 by CRC Press LLC

/$ $

Section 9.3

3 !3

915

Tolerance Graphs

+L . % $ /$ "#7

+BL . ¾ % 5 %3 +? . %

'

'

REMARKS

3

E ' 5 & % % /$ * ' /$ * & * '

3

% $ Æ $& * ?L C & 6 ;.#

5 "

+EC &

3

,' ' ' * $

&

$ % 4 1 & '

* ' & % ' $

' + .

3 4 $ ' ' 7

Tolerance and Intervals

$ +=& =.

C 5

' *

DEFINITIONS

&

3 ( 7% ' 5 * 6" # 5 & * & 7" # 5 " #

% * " # ' & : ' 7% * & & 7" # 5 1" #& & 7" # 5 7 &

3

( & $ $ & ' '

3 ( % *

'

EXAMPLE

3 E ' * ' * % * & $ *

* 4 $ % * & $ *

© 2004 by CRC Press LLC

916

Chapter 9

Figure 9.3.4

GRAPHICAL MEASUREMENT

$# #

FACTS

&3

+=. ( * ' % * 5 / ' $ % 5 /

3 3

+=. % * )' +=. % *

Some Special 7-Tolerance Interval Graphs DEFINITIONS

3 ( 7 *

7% * $ %

3 ( 7 7% * $

* $ * '

$ 7 # * ' & 0 -& 7" &# 5 7"& # 5 3 ( # 7 7% * * ' ( $ 7

3

FACTS

3

+,4 BA . 1 % * % $ *

3

+? . ( $% * ' $% $ *

3 3

+? . (

( 7% *

+?-. 4 * ' &

( 7% *

REMARKS

!3

$ $ 7% * 7% $

* $ * $ 7 % %

%3

4 7% & /$ ' $

7 5 * & ' ( & $ $ $ $&

© 2004 by CRC Press LLC

Section 9.3

917

Tolerance Graphs

1 $ ( B

& +=.&

% * EXAMPLE

3

4 $ % *

Figure 9.3.5

( $# #

Tolerance and Subtrees

4 ) ;& $ $

* $

U

REMARKS

&

3 B $

$$

$* $ ' & $ $ +, .

3 C

0

&

$ +?$--& ?$-- . DEFINITIONS

3 ( " # % ' $

& 1 $

1 $

!

3 ( " #% * ' & * B & $

' ' ! * &

FACTS

3 +?$--. B " #% & " #% 7 @

!3

+?$--. 4 & 1 " #% ' ) 1 " #%

%3

+?$--. 6* ' 1 " #% * ' 7 "#

© 2004 by CRC Press LLC

918

Chapter 9

GRAPHICAL MEASUREMENT

Tolerance and Competition graphs DEFINITIONS

%

3 A 7 ' $ % * % * ( 7 '

7% ' $% *

&

3 A 7 * 5 " # %$ ' % * ( 7$ ' $ $ "# ' 7" # FACTS

&3

+, J . ( 7% '

1 7%% /$ * "#

3

+, J . "# 6* ' % " # 1% " # $% $ $% %

3 +,EJ --. B 7" # 5 & 7%

' ; 5 REMARKS

3

B $ & * - 8 "

+, J @.#

3 B +,EJ --.& $ % N $ 7& % 7% 2

3

' % $ +(AA$ .& $ 0 %

3 $ * $ 7% $

+-.& 0 ) * 1 $ * ' B * ' $

' ' 3 *& $

& $

References +(AA$ . E ( ( & A A '& ? A$ & ( L L 2& > % 7% & -- " #& ;I-; +, . , 2 & : ' 0 $ $ & ! - " #& @-;I@-

© 2004 by CRC Press LLC

Section 9.3

Tolerance Graphs

919

+, . 6 ,

) & > $

& ! " #& I@ +,4 BA . L ,& E 4 $& B) A ? A '& $ & ! @- " #& I; +,A . ( ,V & J , A ? & !, '& B( ( & B(& & (& +,EJ --. E , & ? E J '& , $ N & "---#& I; +, J . E , & 4 J '& & 3 ! ! ; " #& I +, J @. E , & 4 J '& %7% % & ! @@ " @#& -I-= +,$;. ,$ & ( $ & -I

" ;#&

+EC . E$ , C & % $ % $ * /$ & " #& I@ +E;=. ? 6 E & . 6 & S * ' & & >?& ;= + $,=. $ E , & ( 2 & ! @ " =#& I; +6 ;. > 6& B $ & & ; " ;#& I +6@@. 6W& ( C A M& ' & = " @@#& -@I +4 =. E 4 $& * '! 1 * '! & C '& > K)& = +4$@. 4$) : ( & B * & / " @#& =I= +;. 4 * & $

1 ' & & @ " ;#& ;I@ + @. E ( ? N& ( 2 ' * & @ " @#& I= +=-. E $ & ! # & ( &

& =-

© 2004 by CRC Press LLC

920

Chapter 9

GRAPHICAL MEASUREMENT

+?-. E $ & 6 ? ( > )& ( 7% & $ & -- +=. E $ E A & ( 2 * & " =#& I +=. E $ & E A C & & ! " =#& ;I;- +-. E $ ( > )& ! & E S * ' & & -- +$ =. $ & E & * " =#& =I V ( * & * " ;#& +;. !M& S @ +BL . B)& %L L & ( L

& 4 4 & % & * " #& I= +? . ? & : % /$ * $ & * " #& I +?L C . ? & ( 6 LM 2' , C & % $ $ %

& " #& @I@ +? . ? 4 & $% * ' $% $ * & *# @ " #& I= +? $ . ? & 4 $ & ( $ & B $ ! "K (* & & 6# C ' B

& > K)& * " #& ;-I; +? . ? & 4 6 & $ & " #& ;I;; +?$--. 6 ? $ & E

& "---#& ;I +?$-- . 6 ? $ & '

& "---#& I +L . % L & $ * & B 7 ($ ) 8 "? & & 6#& + * .& >%& ( & & I@ +L . % L & ( L

& 4 4 & % & 3 ! ! ; " #& @;I;=

© 2004 by CRC Press LLC

Section 9.3

Tolerance Graphs

921

+LC;=. A L$& A ? ) ' E L C& E* ' /$ ) ' O & " ;=#& I +A ,@. E A )) ) ) ? E ,& 0 ' * & . " @#& I@ +A$= . ? A$ & 4 & & % ' & B ! # &! ! ! "4 & 6#& & > K)& = & I +A$=. ? A$ ? '

& ( 2 $ & ! @ " =#& I

+. 6 "2 !%# 2 ) & $ $1 M M H & . " #& -I-; + . ( L

4 & * & B( ( & B(& & (& + =. , ) > & & & , & = +@ . 4 & B N & B # 5 "4 '& 6#& ( & > K)& @ & I@ +;=. 4 & 4 & & 1 ' & B ! # "K (* X A )& 6# +3 * @.& & > K)& @=& ;;I - +;-. ? & $ & ! ! " ;-#& ;I@- +C;=. ? C & $

& " ;=#& @I@;

© 2004 by CRC Press LLC

922

9.4

Chapter 9

GRAPHICAL MEASUREMENT

BANDWIDTH !# $% 4$ 6 ' $ ,$ , : , E , B : B* @ E

Introduction +@. $ & * 1 % ' $ B / ! * & P 0

/ $ *$ P * B $ ) ' 8( & ' * & 3 $ * % $ P 2 9 ' ' $ & 0 )

+EE =. +AC .& 1 *

& * % * $* ' 4$ $ $ +E==. + . ( $ $ 0

9.4.1 Fundamentals The Bandwidth Concept EXAMPLE

3 4 $ N * 1

% ' $

9¿

! '

Figure 9.4.1 3 )

© 2004 by CRC Press LLC

9¿

+ #)# 4

Section 9.4

923

Bandwidth

Figure 9.4.2

( #)# 9¿ 4

4 4 $ & ! ' 1

* * 1 ' & $ $ $ 1 4 $ ' $ "8$ 9# $ $ $ & ) * 4 9¿ $ $ $ 4 $ '

$ 2 DEFINITIONS

( " * #

3 3

(

! ' 3

A ' $ ' & , "#& * ' , "# 5 ' " # ' "# 3 $

3 3

, "# 5 , "# 3 '

$

( $ ' $ , "# 5 , "# " & $

* , "##

Applications

$ ' +AC . $ $* '& & EFFICIENT MATRIX STORAGE

$ $ ! ' 1 * $ * & ' , "# *

B , "# & 0 * VLSI LAYOUT

$ JAB $ % $ $ $ $ ( 0 ' * ' * $

$ , "# 1 $

$ $

© 2004 by CRC Press LLC

924

Chapter 9

GRAPHICAL MEASUREMENT

INTERCONNECTION NETWORKS

(

)

'

*

) $ '

' 3 "# " #& $ ' ' " # & ) $ '

' " # ' " # B $ ) &

' " # ' " # B 5 & ' $ , "#

)

'

)

' %%

BINARY CONSTRAINT SATISFACTION PROBLEM (

** *

*$ &

&

* *$ $ 0

* *

&

* * * '

B

, "#

' *

&

' $ *

Algorithms DEFINITIONS

3

'

'

$

$ 8K69

, "#

8>:9

3

4 * 01 *

& $ 8K69 , "#

$ ' 8>:9 3

(

1 $

$

'

' & $ *

!

3

(

%&

* 8K69 8>:9

* ' ' & $

' $ '

$ ' Æ $

FACTS

3

+;@.

>%

3

+?L;=.

>%

1 $

3

+=-.

*

%

© 2004 by CRC Press LLC

* ' ' 01

Section 9.4

925

Bandwidth

REMARKS

3

% $ * % 01

$

3

B *

4 * & ' $ ) ' ' $ $ *

1 $

& $& >% & ' '& * '

& $ 4

3 ' 1 *

* & % 1 /$ * & * *

+AC . 4$ $* ' +6*; & ; & =. C ' $ * $ 1 '& 1 *$ $ $ & ' $ $ 2 ! ' 1 * 0

3

( ' 0 * 1

+(@.& N * $ 0 * $ +@=. +E$ @ . ) $ ;-H& * + ;@. ' * * $ +EE =.

3 ) 1 1 0 $ +: -& L @& ;& K =& ,LJ --& 4 --& $-& L-& E-& L-.

9.4.2 Elementary Results The Bandwidth of Some Common Families of Graphs DEFINITIONS

%

3 ½ ¾ * 1 2 * & & * ! '

'

&3

9 * * & %

% ' /$ & * ! '

N 1 ' FACTS

3 3 3

, " #

5 &

* *

, " #

5 &

' * *

, " #

5 &

© 2004 by CRC Press LLC

* *

926

Chapter 9

GRAPHICAL MEASUREMENT

3 +6 ; . A * , "½ ¾ # 5 "½ ¾ # " 7 # $& , "½ ¾ # 5 7 +E;-.

!3

+@@. , "9 # 5

EXAMPLE

3

4 $ $ 9 4 $ % $

Figure 9.4.3

5 + )

A Few Basic Relations

%%% %(%% * Æ "# Æ "#& * ' * $ $ Æ "# P"# FACTS

%3 +E L;. ,"# Æ "# &3 B $ & ," # ,"# 3 B & ,"# 5 ,"# ,"# , " #

3

+E L;. B & , "#

On the Bandwidth of Trees DEFINITIONS

3 ( " # 5 ' ' 5 0 * 1

(

' ' 0 $ $ *

3

* * * 1 ' & * * *

© 2004 by CRC Press LLC

Section 9.4

927

Bandwidth

FACTS

3

+E L;. 4 '

& , " #

" # *

3

+CK & (L @. A

" #

6/$ ' '

$ * * , " #

3

+ . A %'

, " # 5 " #"" ##

3 +E==. B

0 , " #

'

&

EXAMPLE

3

4 $

$

& , " # 5

/$ ' $

Figure 9.4.4

5 + ) +

Alternative Interpretations of Bandwidth

* : * +A --. DEFINITIONS

3

& & * * 1

* '

3 ' ' ' "# 5 7 ' "# * 1

' 0

FACTS

3 +EE =. 4 ' 1 ! & *$ " / # E 0 ' $ $ ! $ 1 $ / & ) * " / # 2 & 2 H /$ * ' 2 ' ! ' $ 1 ! ' 1

© 2004 by CRC Press LLC

928

Chapter 9

GRAPHICAL MEASUREMENT

!3

+EE =. ' $ *

%3

, "# 5 , "# ' $ $ $

9.4.3 Bounds on Bandwidth Two General Bounds FACTS

&3

+@@. 4 & : $ $ , "# 1 : 3 5

3

+E=-%. 4 & P $ 1 ' ½¾

, 1 3 5 Subdivisions, Mergers, Contractions, and Edge Additions DEFINITION

3 & & ' ' ' $

B 5 & FACTS

3

+E:=@. B ' $ * &

, " # ", "# # $

3

+E:=@. 4 ' * "#&

, "# , " # , "# $

3

+E=-%& E:=@. 4 ' $ "#&

, "# , " # +, "# . $

3

+CC K . A , "# 5

" "## 7 , " 7 # " "## 5 " "# #

© 2004 by CRC Press LLC

1 $ *$

"# 7

7 "# @

"# @

Section 9.4

929

Bandwidth

EXAMPLE

3

4 $ $ 4 ' * 7 * & $

Figure 9.4.5

( + ) + )# +

REMARK

3

4 * $ /$ ' ' 6V3

, " 7 # , "# & 7 ' ' 0 +E=-%. Nordhaus-Gaddum Types of Bounds DEFINITIONS

3 & '

& * 1

3 ( ' $ %* 1 * ' $ %* 1 0 '

FACTS

3 3

& "# , "# 7 , "# +E6E=. * & $ , "# 7 , "# "# & "# ' !3 +E6E=. * & $ "#& "# , "# 7 , "# %3 +4$C -. A ' "# 5 1 ,"# 7 ,"# 3 %* 1 " 7 # ' "# 7 ;" # +E6E=. 4 '

Other Bounds FACTS

&3 3 3

+E L;. A , " # , "#

& , "# "# ' +E=-%. A $ " # * ' * , "# 5 "# ' * ' * 1& & +E=-%. 4

© 2004 by CRC Press LLC

930

Chapter 9

GRAPHICAL MEASUREMENT

9.4.4 On the Bandwidth of Combinations of Graphs *

' ' & $ Cartesian Product DEFINITION

!

3 & & " # 5 "# " # " #" # $ " # ' " # 5 $ " # "

# 5 $ "# FACTS

3

+E;& E L;. 4 &

, " # " #, "# "#, " #

3 3

+E;. 4 & 1 & , " # 5 B & , " # 5

EXAMPLE

3

4 $ @ $ & ' 4 & @

Figure 9.4.6

( ) + )

Sum of Two Graphs DEFINITION

%3 & 7 & " 7 # 5 "# " # $ " 7 # 5 $ "# $ " # 3 "# " #

FACTS

3 +A C C . A $ "# " # , "# 5 "# , " 7 # 5 "# 7 " #

© 2004 by CRC Press LLC

Section 9.4

931

Bandwidth

3 +A C C . A $ "# " # , "# 0 "# "#7 " # , " 7 # , "#7 " # 1 , " # 7 "# " # 7 "#

!3 %3

+A C C . 4 & , " 7 # 5 7 +A C C . 4 ' &

, " 7 # 5

7 @

5

5 5

EXAMPLE

3

4 $ ; $ 7 & ' 4 &

Figure 9.4.7

( ) + ) 7

Corona and Composition DEFINITIONS

&

3 & Æ & $ ' "#

& * 1 B "# ' & * ' " #

3 & " #& "" ## 5 "# " # " #" # $ "" ## ' " # $ "# "

# 5 $ " # FACTS

&3

+E=-% . 4 & , " Æ # , "#" " # 7 #& $

3

+E=-% . 4 & , "" ## ", "#7# " #& $

© 2004 by CRC Press LLC

932

Chapter 9

GRAPHICAL MEASUREMENT

Strong Product and Tensor Product DEFINITIONS

3 & "# #& ""# # 5 "# " # " #" # $ ""# # ' $ "# $ " # 5 $ " # 5 $ "# 3 & "# #& ""# # 5 "# " # " #" # $ ""# # ' $ "# $ " # FACTS

3 " # " #

& , " " # # 5 7 7 7 B & , " " ## 5 7 B & , " " # # 5 7

3

+AC . "# B

+AC ;% .

7 , " " # # 5 7 7 7

* & *

9.4.5 Bandwidth and Its Relationship to Other Invariants ' $ * *

$ * * $' $* ' +, $=& , $ .

$$ $ & , 5 , "#& Æ 5 Æ "#& Æ 5 Æ "#

5

"#& $

5

$ "#&

Vertex Degree DEFINITION

3

& $$'

*

FACTS

3 +E;-. B

/$ & , 1 "/ # / 5 '

, Æ 3 +EE =. B

& , "Æ # © 2004 by CRC Press LLC

Section 9.4

933

Bandwidth

EXAMPLE

9¿ $ 4 $ = , 1 * ¿ & Æ 5 & , 5 ""# # 5

3 4 % ' $

Figure 9.4.8 5 + ) +

Æ 5 , 5

Number of Vertices and Edges for Arbitrary Graphs FACTS

3 + $,= . , "$ 7 #

3 +, $=. ,

" # =$

& , & $ " # " # $ & $

!3 +AC ;%. B

%3 + $,= . B , &

$ " #+" , #. &3 + $,= . B , & $ " # + " #. 3 +(A 6 . A , 5 " <# - 5 < 5 * % & " <# & $ & < " , # & " "<# < " , # $ $ * ,

Number of Vertices and Edges for Graphs with no

FACTS

3 +E=. A " , # 1 $ $ %* 1 * , * , " , # ,

2 & & , & " 7 #, " 7 #, $ , 7 " 7 #, 7 " 7 #, & $

3 +,E $4 J --. A

© 2004 by CRC Press LLC

934

Chapter 9

GRAPHICAL MEASUREMENT

3

+,E $4 J --. A 2 & 5 " 7 #, 7 =, = 01 $ - = 5 & , " 7 7 #" =# $ , " #& $

Radius and Diameter DEFINITIONS

3 & "#& $ $ * 1 * ' * 1

3 & "#& 1 $

' * FACTS

3 3

+EE =. 4 ' & , Æ "Æ #

+E;-& E L;. 4 ' & " # "# ,

"#

3 +E = . 4 ' & , 1 " "¼# # "¼ # 1 % $ ) * $ ¼ * * REMARK

3 '

* $ 4 @ 4 $ % $ * 5 & 5 & , 5 &

* $ $

Figure 9.4.9

5 + ) + 5 5 , 5

Vertex and Edge Chromatic Number DEFINITIONS

3 "* 1# & >"#& $ $ $ ' 3 "# ' &

& ' " # 5 ' "#

!3

& >¼ "#& $%

$ $ ' 3 $ "# ' &

* 1& ' " # 5 ' " #

© 2004 by CRC Press LLC

Section 9.4

935

Bandwidth

FACTS

!3 %3

& , >"# +, $=. 4 ' & , > "#

+E L;. 4 '

¼

Vertex Independence and Vertex Cover Numbers DEFINITIONS

%

3 & "#& ' * $

&

3 & ="#& % ' * $ * ' *

FACTS

&3

+E;-& E L;. 4 '

&

"# , "#

3

+ ;@. 4 '

& , ="#"#

Girth, Vertex Arboricity, and Thickness DEFINITIONS

3

& "#& 2 $ '

3 & ; "#& $ $ $ "# $ * $

$ ' $

3 & $ $

&"#& $

FACTS

3 3

& , " "# #" ; "# # 7 . B &

+, $ . B +, $

, +" "# # " "#. "# 7

3

+, $=. 4 '

© 2004 by CRC Press LLC

& &"# 1 ", #

936

Chapter 9

GRAPHICAL MEASUREMENT

9.4.6 Related Concepts $' * *

' Bandsize

, 2 $

O' +AC

.

DEFINITION

3 A ' $ ' & "#&

$ N $ ' ' * ' "# 5 "# 3 ' $ FACT

3

+6 C = . 4 ' & , "# "#

Edgesum (Bandwidth Sum)

$ 0 +@. 6 $ $ + ;-& B;& B;@& EE =& E==& KC & K$$ & K$$ @& AC . DEFINITION

' $ ' "# 5

¾3 A ' " # ' "# * ' "# 5 "# 3 '

$

FACTS

3 3

4 % ' $ 9& "9 # 5 " # A ) & $ >%

Cyclic Bandwidth

E' $ +A & A ;& L

& AE-.

DEFINITION

3 A ' $ ' , "# 5 1 ' " # ' "# 3 $ 5 * ' , "# 5 , "# 3 ' $ Edge-Bandwidth

6 % $ * $ +? $C

.

DEFINITIONS

3 (

'

© 2004 by CRC Press LLC

! $ "#

Section 9.4

937

Bandwidth

!3

A ' %$ 1 ' " # ' " # 3 ! * ' , ¼ "# 5 ,¼ "# 3 ' $

' , "# 5 ¼

%

3 -"# $ "-"## 5 $ "# * -"# ! ' ! FACTS

!3 %3

4 ' & , ¼ "# 5 , "-"## 4 ' & , "# , ¼ "# & & , ¼ "# , "#

EXAMPLE

!

3 4 $ - % $ $ -"# > % $

$ * 1 -"# B Æ $

, "-"## 5

Figure 9.4.10 , "-"## 5 , ¼ "# 5

Profile

&+ $ +A K$ & AC

.

DEFINITIONS

&3 A ' $ & * 1 "# ' "# 5 1 ¾ "' "# ' "## ( +. 3 A ' $ ' ' "# 5 ¾ "# ' & "#& * ' "# 5 "# 3 '

$ FACTS

&3 3 3

"# 5 "# 5

B & " # 5 7 " #

© 2004 by CRC Press LLC

938

Chapter 9

GRAPHICAL MEASUREMENT

Cutwidth

$ $ +$$=& A =& E=& $=& K=& E==. DEFINITION

3 A ' $ ' & "# 5 1 $ "# 3 ' "# 5 ' "# & "# 5 & "# 3 ' $ FACTS

3 3 3 3

& "# 5 & "# 5 & "# 5 ¾ & "# 5

3 & " # 5 " #" # 7

Topological Bandwidth

+E=-%& $=& E==. DEFINITION

3 & , "#& * ' , "# 5 , " # 3 0 FACTS

!3 %3 !&3 !3

4 ' & , "# & "# 4 '

& , " # & " # , " # 7 , " # 7 B & & & , "# 5 & "#

, " # 5 5 & "#

Additive Bandwidth

$ 9 * ! ' % 1 $ * & $ ' , "9 # 5 & $ * & "$ ! ' 1 $ # ( % * & 2 $ +,$ .&

' #, ! ' 1 & * 9 & '

/$ '

© 2004 by CRC Press LLC

Section 9.4

Bandwidth

939

DEFINITION

3 A ' $ ' , "# 5 ' " # 7 ' "# " 7 # 3 $ & , "#& * ' , "# 5 , "# 3 ' $ REMARKS

!3

1 ' " # 7 ' "# " 7 # & ! ' 1& $ " 2 # & $ ** * 8 * 9

%

3 ' * *

*

* &

* >% &

* * ' '

*

&3

, "9# 5 5 , "9 # 5 &

* 0 ' $ ' B & : & * ' " ' 4 4 =

#

3

B

$ * & *$ , "#

*

& * *

* %'

* Æ $& ' $ * C *

$ B & * +,EJ C K . ( * + & EE & J, & ,EJ @& E @& $, ;. FACTS

!3 !3 !3

+,$ . , " # 5 +,$ . B , "# & , "# , "# +,$,EJ . , " # 5 , " # 5

References +(A 6 . K (* & ? A $& ? E& 6V& : $ 2 * & &!! # * # ! @ " #& I +(@. (' C & ( $ 1 ' 0$ & ! = " @#& @I; +(L @. L (& ( L )& * &

* & - " @#& -I-@

© 2004 by CRC Press LLC

940

Chapter 9

GRAPHICAL MEASUREMENT

+,$,EJ . 6 , $Y M& $ 2& E , & E& ? & J '& : * & ! # ! ! = " #& I +,$ . 6 , $Y M & $ 2& ? & * & == " #& I +,LJ --. ( ,$& L! *& * & J & % 0 1% $ * 1% & ! "---#& I +,E $4 J --. E , & ? E & $& ? 4

& J '& ( 1 & ! # "---#& ;=I= +,EJ @. E , & ? E & & J '& : * %'

& = " @#& - I +,EJ C K . E , & ? E & J '& ? C & ? K & E * & +, $=. E , $& (

* & " =#& ;I-; +, $ . E , $& (

* I$ B& " #& I +E @. E& 4

* & " @#& I; +E-. ( E& 4 $ & & : % & ) ! ; "--#& I +E;-. J E*M & ( ) '& !' ! ! - " ;-#& - I +E;. ? E*M*M& : $ & " ;#& I +E=-%. ? E*M *M& : & & E : 2 & S * ' C & : " =-# +E=-% . F E & & & $ S * '& ( & E " =-# +E=. 4 L E$& : $

& * ! ! @ " =#& @=I;;

© 2004 by CRC Press LLC

Section 9.4

Bandwidth

941

+E==. 4 L E$& A & ! & ( % A &

& E( " ==#& I@= +EE =. F E & ? E*M *M& ( L '& > 6 & I $* '& ! # @ " =#& I +E L;. ? E*M *M & ( L '& > 6 & L & 3 $& & E$

& S * ' C : & A& : " ;# +E6E=. F E & 6V & 4 L E$& A & : & ! # & E& 6& C '& > K) " =#& I +E:=@. ? E*M*M ? :M '& & @ " =@#& I- +E = . 4 L E$ '$& $ & ; " = #& I +E=. 4 L E$ C & ?& %

& & ( & & : " =#& ;I - +E$ @ . 6 E$ ? L

& $ ' % & 9: ! # # " @ #& ;I; + ;@. ( L '& I $& ; " ;@#& ;I== + $,= . $ E , & : 2 * & ;@ " = #& I + $, ;. $ E , & B* ** * & ! # ! ! " ;#& ;;I= +6 ; . 6 & $ & ! ! # " ; # +6 C = . 6V& & C ) & , * $ 2 & ! # " = #& ;I +6*; . E 6* & (

/$ $ 1 0 * & * ! ! # ! + " ; #& =;I= +4 --. S 4 & (1 * *$ & ) ! # @- "---#& -I

© 2004 by CRC Press LLC

942

Chapter 9

GRAPHICAL MEASUREMENT

+4$C -. F 4V $ , C & ' ' & ; "--#& @I; +?L;=. '& A & ?& 6 L$& E 1%

' $ 2 & * ! ! " ;=#& ;;I + ;@. > 6 & C & ?& L ) ' & ( $ 0 1& * ! ! ! " ;@#& @I- +: -. E ) ? :M'& > ;@ " -#& ;;I==

$ &

+$-. ( $& B* 1

& ! # ! - "--#& I@ +$$=. 6 $ B $ $& B* ' % 2 $ & & 6 6

E$

& > S * '& 6*& BA " =# +@. A & : $ * & ! # * " @#& I +@@. A & : $ & ) ! # ! " @@#& =I + . 4 C ) & * $ & H & & S * ' E 4 " # +EE . 4 C ) & E& ? E & * $ & - " #& I@- +L . 4 '& E L & ( ) & 6* ' '

& &!! # * # ! ; " #& =I= + ;. ? 4 ) & (1 % 2

& # - " ;#& @;I - + ; . 6 & : $ 0 $ $ '& $ " ; #& I@ +B;. ( B) Z[& $ *

& ' ! " ;#& I +B;@. ( B) Z[& $ *

& ! 4 ) " ;@#& - I

© 2004 by CRC Press LLC

Section 9.4

Bandwidth

943

+? $C . ? & $ ' & ( & , C & 6 % & * ! " #& -;I@ +L @. L & & & /$ & * ! " @#& -I@ +L-. L) , & , H& ! " ;#& ;I +L-. L A & (1 ' 1 '$

* & * ! "--#& I +AE-. E , A& C E $& C E& E 2

/$ ' & "--#& =I= +AC . K A L C & , $ ' & - " #& I= +AC ;%. K A L C & $ & $& 0 & " ;#& I +AC ;% . K A L C & : $ ' & ! ; " ;#& I +AC . K A L C & ( $* ' * & $& 0 & ! # " #& ;I +A =. A $ & S $ 1 ' % $

& * ! ! " =#& I +A . K A & ' & ! ) ; " #& =I== +A ;. K A & $ ' & " ;#& I- +A --. K A & : 2 & 1% "---#& I@ +A C C . ? A $& L C & ? 4 C& , $ & = " #& ; I= +A K$ . K Q A ? ? K$& $ 0 )& ! ; " #& @I@@ +$=. 4 ) & E $& B $ $& & * ! ! @ " =#& =I + . F & '$& ! # & ? L "6 #& % & > K) " #& @I

© 2004 by CRC Press LLC

944

Chapter 9

GRAPHICAL MEASUREMENT

+;@. E $& >% 2 % & @ " ;@#& @I;- +@=. & 1 2 & 9; ! # # & , ' & & > ? ' " @=#& =I +=-. ? , 1 & ' % 2 % ' & * ! ! " =-#& @I@ + ;-. ( Z[* & $ *

& <= ! ; " ;-#& @I; +=. C 4 '& ( $ 1 0 & ! # ! ! % " =#& I@ + . A & , %'

& " #& -I +J, . J E , & : *

& - " #& I@- +CC K . ? 4 C& , C & , K& 1 $ $ & ! # - " #& =;I - +CK . ? 4 C , K& : $

$

&

! >+ ! ? " #& I

+K=. K)) & ( ' % $

& ! # # " =#& -I == +K =. ? K& ( % & 1@# ! # ! " =#& I= +KC . , K ? 4 C& : $ & ! >+ ! ? " #& @ I;= +K$$ . ? K$ G $& ( $ $ & ! # ! = " #& I +K$$ @. ? K$ G $& $ $ & ! " @#& @I=

© 2004 by CRC Press LLC

945

Chapter 9 Glossary

GLOSSARY FOR CHAPTER 9 , "# I 3 , "# 5 , "# '

) +

$

, "# ) ' I 3 , "# 5 ' " # 7 ' "# " 7 # $ "#

) +

# ## I '

*

3

$

' $

%* 1

0 '

3 $

7 & "

# 5

7 $

%* 1

* '

& 0 -&

7 & "

# 5

# #3

* $ & ' &

$

3 $ $

3

' 3 $ N ' < "# ) 2 I 3 "# 5 "# 3 ' $ ) + , "# I $ ' 3

) 2 I $ $

'

, "# 5 ' "

#

' " #

$ "#

) + )#3 8K69 , "# 8>:9 * * ) + ) I 3 $ ' $ , "# 5 , "# ) +$ )#3 01 & 8K69 , "# 8>:9 *

I 3 " # 5 "# " # " #" # $ " # ' " # 5 $ " # "

# 5 $ "# ) +

, "# I 3 , "# 5 , "# 3 '

I 3 $ $ &

$3

$ $

'

$

' *

% *

# I 3 * 1 ' /$ $ &

$3

* 1

% '

% $

# 3 $ '

>"# I 3 $ $ ' 3 "# ' & & ' " # 5 ' "#

) $

© 2004 by CRC Press LLC

946

Chapter 9

# # I

GRAPHICAL MEASUREMENT

* 3 *

* '

# # I 3

$ *

* '

*

# ) I * 1

< ( +.

# ) I

< ( + .

3 ( "# & ( "#

3 ¾ ( +.& ( +.

%

3 ' $ *

$3 % ' $ * & 7$# 3 7% ' &

$% *

5 " $ #3 "# 5 * ' ' ! < *%&%

# ) ' I $ ' 3 ' " # 5 7 ' " #

* 1

# $ 3

* * * 1 ' & * * *

# $ ½ ¾ 3 * 1% 2 * & & * ! ' '

# I !

" # I 3 "" ## 5 "# " # #" # $ "" ## ' " # $ "# "

# 5 $ " #

I 3 5

"

## I 3 * 1 *

3 * *

I

3 ' "# * 1 ! * ' * 1 '

+ I $

5 ' "#

'

3 & "# 5 1 $ "# 3 ' "#

3 & "# 5 & "# 3 ' $

# ) + , "# I $ ' 3 , "# 5 1 ' " # ' "# 3 $ & 5

# ) + , "# I 3 , "# 5 , "# 3 ' $ 3 $ ! ' ' ' * & ' & * 1 )# 3 * 1% % "# $ % $ $ $ + . $

+ I

© 2004 by CRC Press LLC

947

Chapter 9 Glossary 6 I

3

*

& $$'

3 1 $

' * < "# & $3 1 $ % ' * 3

I

I

* 3

*

I

$ 3

$

* 1

$ * 1 $

¾ '

I

&

"

#

3

$

' 3

-

$

3 $ * ' * 1 & I 3 $ + . & #6 I 3 $ + . /$ & # I 3 $ + .

I

! * 1

' & #3 $ & I

+ .

& # I * 1

3

$

$ $

3

$ * ' * 1

!

' ' $ $ $ & $% 3 $ ) I 3 $ ' & #6 I 3 $ ' /$ & I 3 $ ' & # I 3 $ ' ' & I 3 $ ' 1

# 3

"( 1 $

#

3 $ ' & # I 3 $ ' & I 3 1 $ '

I * 1 3 * 1 * 1 & $3 1 $ % * & I

* 1

© 2004 by CRC Press LLC

948

Chapter 9

$) + I %$

3 ,¼ "# 5 1 ' " # ' " # 3

¼ ¼ $) + I 3 , "# 5 , "# 3 ' $ ) > "# I 3 $ $

$ ' 3 $ "# ' & * 1& ' " # 5 ' " # #6 I 3 ' 5 $ $ * ' $ " # $ " # "6 &

'

GRAPHICAL MEASUREMENT

!

/$ /$

' $ "# $ & 5 ½

$ $ & $ /$ * & 7$$ I 3 ' $ $ "# ' 7" # I 3 * * $ ) ' I 3 ! ' 3 $ "# $ "# "# I ' $ ' 3 ' " # ' "# "# 5 &

$ I 3

1 $ #

'

$ & * '

¾

"# I 3 "# 5 "# 3 '

3

$

*

) 3 $ ' 3 * * * 1%

< "# ## ) I 3 $ ' $

I

3

2 $ '

## 3 * * 1 $ * 1% ) I

%

% 3 * * &

' /$ & * ! '

N

1 '

I * 3 * * ! I ' * 1%

!

5

'

3 5 / 5

*

$3 7% 7 $ /$ & 7$# I ' 5 3 * ' 5 / 6" # 7" #& 6 %*$ $ $ 6" # 5 & 7 % * & %*$ & &

' $ 0 * $

3

) I

$ &

2

$ I 3 2 $

© 2004 by CRC Press LLC

%

949

Chapter 9 Glossary # 3 ' *

7$# 3 7% * 7 7% ' *

& (

* ' ( $ &

7$# 3

7% * 7" # 5 1" # 7% * 7" # 5 " #

& $# 3 & $# 3

& 3 * $ * * '

& $# 3

7%

*

7" # 5 7

& 3 * $ *

) I $

& I

3

( +. ( + . 5

< - # -"# I

$ ' 1 %

1 $ ' $

I * 3

73

3

* $ * ' * 1

' ! * 1

3

&

%* 1

* 1% /$

* ! ' !

# 7 3 ! ) * # 7 I ' "# $ 3 %! * $ # 7 I ' " # $ 3 * 1

$

$ 3

&

"

#

0

3 " #

% "# 0

Æ "# &

"

#

3 * 1 $ *

) 3 $ $

' *

I * 3 ' ' ' $ $ # 3 %* 1 % '

3 " # Æ "# & "

# & "

# % "#

## ) 3 $ $ * 1 $ $ '

89$ # I 3 * 8K69 8>:9

* ' ' & $

'

) I * 1

( "#

3

* !

<

3 ¾ ( "#& ( "# < ( " # $ I * 1 3 * $ " #

) I * 1

© 2004 by CRC Press LLC

950

Chapter 9

GRAPHICAL MEASUREMENT

3 1 $ ' ) 7 I 3 * $ * # I 3 * $ & & * 1 $ $ * 1 $ 7 ) I

# 3 * 1 ' /$ %

3 $ $

' *

# # # 3 1 $

$

' '%

& $ *

3

:# "# I $ ' 3 "# 5 ¾ "# :# "# I 3 "# 5 "# 3 ' $ :# + " # I * 1 "# $ ' 3 "# 5 1 ¾ "' "# ' "## ( +. ) I 3 ! ' 3 "# "#

"# I 3 $ $ * 1 * ' * 1 < /$ * '& $ + I

* * 1%

* '

' *

% ' *

: I 3 ' 0 $ $ &

$3

$

*

# $ 3 # # 3 * 1 $ $ " I * 3 $ 2 $

I 3 1 $ ' *

I * 1 3 * 1 * 1

I 3

$ '

*

/" 0 I

* 3 $

2 $ *

I 3 $ $

'

*

# I 3 * 1 ' $

3 '' '

"# # I 3 ""# # 5 "# " # " #" # $ ""# # ' $ "# $ " # 5 $ " # 5 $ "#

© 2004 by CRC Press LLC

Section 9.4

951

Domination in Graphs

# 3

* '

*

# # $ 3

3

) I

'

5 0

* 1

) 3 ' $

7 I 3 " 7 # 5 "# " # $ " 7 # 5 $ "# $ " # 3 "# " # " # # I 3 "" # # 5 "# " # " #" # $ "" # # ' $ "# $ " # 7 & "# I 3 $ $ $

# # ) + , "# I 3 , "# 5 , " # 3 0

# 3 $ * 1 *

I $

3

$

' *

$" # 3 ' $" # 3 ) ' )

"#

; "# I 3

-

$ $ $

$ * $ $ '

$

)

="# I 3

' *

$ * ' *

)

"# I 3

'

* $

) I

3

1 $ ' 1

+7# #3 & $ $ & '

© 2004 by CRC Press LLC

'

Chapter

10

GRAPHS IN COMPUTER SCIENCE 10.1

SEARCHING

10.2

DYNAMIC GRAPH ALGORITHMS

! ! " # $ %& ! " ! $ %& ! 10.3

DRAWINGS OF GRAPHS

10.4

ALGORITHMS ON RECURSIVELY CONSTRUCTED GRAPHS

'& ! $ (

( ( ) '* & ! $# & ) $ & ! $# & GLOSSARY

© 2004 by CRC Press LLC

10.1

SEARCHING

!" # $ % & '( ) ' ) *

Introduction ' ) ( + )" " , )-. .

/ ) ")0

1 2 3 . " ")0 ") 0 4

() 3 ) ) ) (#+, # Æ 3 )" #+, # " " 5 ) 0 . 5 . 63 ) ' ) ) 0 ) 0 " ) ) ( / ) 78#. 9* . %. :*8;. . <88=1 ' ) /( Ü &1 " ) 5 ) >" . )

Æ

0 / , ) )-1

" " . ")0 ? / 1 . ) 0" ) " ) . . /1 !" . "Æ " . ) . ) + ) . 3

")

> + " . ) ( 0 0 !" 5 + . ( " 0

10.1.1 Breadth-First Search ) /00 1 4 ) ( + @ >5 A ). 3 3 ") / 1 " " . 0 "

© 2004 by CRC Press LLC

DEFINITIONS

B ' 4 ") 2

") . "

( "

B ) ( )

B B

( . / 1.

? ' ) ( . . 0 ) ) (

. . / 1 )( )") "

/ 1

EXAMPLES

B

" ) ( a e

s

b

s

g

f

c

Figure 10.1.1

d

a

b

c

e

d

f g

B + # 7$ #&. C= " 3 3 ) " 3 ) . "

& 3). 3 3 D ( 0 ) 3 ) 3 3 30 0

""

B . ) ) )" -. ) 4 ) ) " 2E . 3 > 3 ) ) 3 "

B

) / & 0 ( ) 3 ) 3 > 3 ) ) ) ) #

B

+ )" ))" 35 . )

0 ) 0

3 )

B "@@ 5 ) 9A , "@@- 78#=. 3 0 ( ' / 1 ( 3 ) ) <

5 ) 3

© 2004 by CRC Press LLC

Ordered Trees DEFINITIONS

B

+ . ( ( . ) 3 )

/. " ( 1

.

B F( ( / 1 ?

B ' 3 ( + 3 " . / ) 4 1

B F( ( ) ( .

. 3

B + . ) ( 0 ) + . + ( Æ . >

. 0" > / 1 (

Æ . ( Æ

FACTS

B B

' 0 4

' 0 ) 03 ' ) + " 0 4 + ) . 4 ) . .

Algorithm 10.1.1:

! B " 1B 0 4 )

?

? / 1. (

)

¼ ? )5 ? ? G " ·½ ·½ ? G ! ( , - ! / 1 ! Ë )5 ( ·½ ? H

© 2004 by CRC Press LLC

B ) 0 )) " ) / H 1 ) "" !"" 0 . 0" 3 0 )"

B

+ 3 ) 5 ) / H 1 0 5

) / . / 1 )1 (

B

I 0 4 /. " 1 " 0)

B B

) 0 " 0 0 4 ) (

>5 A ) )" ) 3 " + @ 0 4 9 5 0 4 ) . " '

"" )) ) ) / H 1 7;&. 9* =

10.1.2 Depth-First Search /00 1 3 8 "

( )@ 79";. 8= ") 4

3 0 : > 7:&. &= > ) Æ ) / Ü #1 4 ) ( " DEFINITIONS

"B B

3

+ " ? / 1. /00 1 ) ( "0 . . 0 )

/ 1

2

/

1

/ 1

.

2

" 3 05

B + " .

4 . ) 2 ( 0 (

B 9 ? / 1 0 3 ( 0 ) ( ' ) . 2 2 / 1 0

4 B

'

'

'

> 3 "

"

/ 1

© 2004 by CRC Press LLC

B 9 0 . 3 3

") ) ( '

( 0 (

4 4 3

0 B ' > J ) / . ) ")0 3 ")0 1 EXAMPLES

B

" " 4 " + 3 4 . 0 ) 4 3 ) / #1 " 0 0 / &1 1

1

2

3

2

3 5

4

6

5

4

6

7

7

Figure 10.1.2

B " " 4 . 05

1

3

1

2

3

2

4 3 5

6

4

5

6

Figure 10.1.3

FACTS

B

' ( " 4 ) ' ( 4 "0 " 0 0 ) ' ) 4 " 3

© 2004 by CRC Press LLC

Algorithm 10.1.2:

! B " 1B 4 )

)5 / 1

? / 1. (

# / 1 ( ! / 1 / 1

! 0 )5 ( /1 (

B

" " . . " / 1 ' ) " ). / H 1

"B + / 1 . / 1

) (

"

(

B

" " " 0 ' ) 0 4 . / 1 )" < K )) " 0 3 0 ) / 5 / 1 " 1 / 5 / 1 " 1

B " 0 4 . / 1 )" 0

0 < K

" 0 + 3 0 / 1 3 0 3 / 1 I( " ) + ) / 1 05 / 1

B

' ) 0 ( " " 4 B " 3 ? + ( . " / 1 3 4 ) .

B

' ) " ) / ( 0" . ( ) , - " " 3 "1 REMARKS

$B

< 3 " ). 0 " 4 4 )

$B

< 3 0 ) (

). 0 4 © 2004 by CRC Press LLC

Discovery Order DEFINITIONS

B ")0 )

B +

")0 ) )

0 4

FACTS

B

$ ) 0 4 " ) ( 3 ")0. 3 ) "

B

+ . ( ")0 " . 3 4 . 3 0 !" 5 3 ( ) ( B 9 ( 3 H ) 0 )) " / 1 / . 1 ) REMARKS

$B

3 4 ) ) ) 4 "" L 0

" . 0 4 ) 35 0 ) " 3 / 1

$B

$ ) 0 @ 3 " " " 3 4 ) 3 35 3 . 3 ) )

$B

' ) () *)5 3 " 3 " 3 ) )") Æ Æ B (" / 1 / 1.

4 ( 0) 4 " 0 " ) Æ A 0 )" 0 0 4

$B

M 3 ) 0 0 3 4 # 3 ) !" " 3 4 & 3 0 3 4

10.1.3 Topological Order ") +

>" 3 ) )) . Æ ) ) ") L I )

© 2004 by CRC Press LLC

DEFINITIONS

B B

' . .

' ( 3 . ( 3 "

B '

( ) 3 ")0 ")0 EXAMPLES

B " " 5 ' 0 + ) ")0 + 4"

0 a b c

d f

Figure 10.1.4

e

"B

' 3 0 3 33 . " ")0 " ) " " ) 3 / 1

B

!" " ) B " N !

" " O. 3 3 ) N O 0 . " "P " ")0 ")0 B !" " 3 3 ")0

B ' )0 " 3 . 3 05 05 )5

B

' ) /

") " 31 4 0 " 3 " 4A ) ) ) ) 4 " ) ) ) ")0

B

)" . < " . " "

B

+ . 3 ) ' 3 3 ) " ) ))

") 0 . 3

© 2004 by CRC Press LLC

FACTS

B 2 ) " ) 5 0 0 () )( ) 4 / 1 ( )" 0 " / 51.

3 " 0 ( 0 /1

B ' 3 ")0 0

")0 . " ( " ( B ' ")0 " " ")0 3 3 ")0 .

3 ")0 " " . " 3 ")0 "B ) " ")0 0 ")0

5

3 ")0. "

B 0 )) Æ 0 4 3 3 4 . 4 ( 0) 4 5 $ " . 3 3 4 " 5 3 ) Algorithm 10.1.3: % &#'( )* +, -

! B ? / 1 1B ")0

B (

")0 7 =

" B 3 " 5

7 = ? . 0 )

Q

)5 ) Æ . 3 " ( .

0

0 3

B ' 3 )) ' ) " ) " 7 = " B

7 = ?

0 0

")0

Algorithm 10.1.4: % &#'( )+ +, -

! B ? / 1 1B ")0

? R

B (

")0 7 =

7 = ? ! 7 = ? / 1 # / 1 ! / 1 ! 7= ? /1 3 5 ) 7 = ? R 0 3 ) ! ( ! (

© 2004 by CRC Press LLC

B ' ) " ) +

/ 1 ) (

EXAMPLE

B " " 3 ) ")0 " a

a

a

a

a

a I[a]=1

b

b

b

b

e

I[b]=3

c

c

I[e]=2

d

I[c]=5

I[d]=4

f I[f]=6

Figure 10.1.5 . # ! #'( ' FACTS

B 9 4 ) /4 #1

B >A ) 7&0. 9* = 0 ' ) >A ) " . " # 3 " 2 ( 0 0 ) 4 ")0 "0 4 ")0 ) & ")0 " a,6

b,4

c,2

e,5

d,3

f,1

Figure 10.1.6 % / #'( (/ '

B ' ) ")0 ) 7C&= 35 0 " ) ) !"" " . 3 ( + 0 ) ) 35 ' ) . )5 3

B ) ) ) ")0 3. / 1 )

0

© 2004 by CRC Press LLC

/ 1

B

' () " " 7= < 4 ) 7 = " 7 = 0 )" . " "

7 = ? )( 7 = H 7= B / 1

+ ) " "

) ) 7 = "

0 4

( "

9 " " " I )

B

) ) 0 " " ) . ) ( .

"

B $ ; " 3 !" ) )) 0 " 0) ) )) 0 ) " 79* = EXAMPLE

B

" & " 3 ) 4 2 0 3 . 0 3 7 = " 7 = ". 3 R " ) " 5 4

1 5

4 0

1 2

2 2

Figure 10.1.7

+ '

10.1.4 Connectivity Properties 4 ) " 3 ) " + 3

. () " ) " > 7&=

3 0 ) 7%=. 3 ) 4 ) ) 4 Strong Components of a Directed Graph

+ .

? / 1

DEFINITIONS

"B

3

© 2004 by CRC Press LLC

.

B ' ? / 1

3 .

B + . 3 0 " 0 05 . 4 B 3 Q ) . . 4 !"

B

. ( 1

/

B ' " > 0 ( ) " 0 "). 3 / 1 0 FACTS

B

9 0 ' ) ( 3 )

B B

3 3 )

' ")0 ") 5 3 . 3 3 ")0 . ") / 1 / / 11 " ")0 "!" L 0 ")0 L 0 ( .

B

' 0 ) + ) 0 . " )

B

' 5

(

+ A A

B

M >" 3 ) 4 + 5

B

I( 3 ) 4 ) 4 ) Algorithm 10.1.5:

0'

! B ? / 1 1B )

" B 3 " 5 "

5 B )5 Q ) Q B 9 5 ' ) . Æ 3 0 3 "

0

( .

B

) 3 )) 4 ) 7%= 5 ) ' ) "

© 2004 by CRC Press LLC

5

5 0"

B ) 5 A 0 ")0 0 ) ' )

"B 4 ) ) ) " > 7&=

+ )" " ! / 1 ( ! / 1 3

")0 ( / 1 A 0 ) 0 / )1 3 0 05

/ ! / 1 !" ) ")0 ( 0 1 3 ! / 1 ? , - )

B ' ) ) ) " 7 ; = C >" /""0 R 79* =1 + 4 . 3 0 4 )5 L 4

3 3 . 3 0 0 3 EXAMPLES

B " ; 3 . ) . 2 ) 1

1

2

3

3 {2,4,5,6}

4

5

6

Figure 10.1.8 ' !

' ) ) :) ) # " 3 "

B " 8 " ; / 0 " ) J ) " ; " 1 2 ( 0 0 ")0 3 0 3 " 1(1,1)

3(2,2)

2(3,3)

4(4,3)

5(5,3)

6(6,4)

Figure 10.1.9 . # ! ' '

© 2004 by CRC Press LLC

"B " 3 ")0 0 0

")0 . 4 ")0 . ")0 . > ) ( 3 3 ( ")0 " 05 " 0" ) ")0 3 ")0 " > ) ( + " " ( 05 " a b d e c

a 0 0 0 0 0

b 1 0 1 0 0

d 1 0 0 1 0

e 1 1 0 0 0

c 1 1 1 1 0

Figure 10.1.10 ( 1 # 2 / ' .

B 2() 3 3 " "

) 5 %"

) . ) ( . 4 " . ) ( # . 3 / 1 > ) ( 2() ". / 1 0 " " 05 " " ) (

4 %"

) ) ( 7:#8=

B " " ;

1

03 " (" )

1

1

1

1

1 {1} an SC

2

{2,4,5}

{2,4,5}

{2,4,5}

{2,4,5,6} {2,4,5,6} an SC

4

5

6

6

3 {3} an SC

Figure 10.1.11 . # ! ' '

B " 03 3 ")

"

.

4 .

B ' $5 /@1 00 REMARK

$B ) ) ) (05

+ 0 0 3 3 ) . 0" )5 3

)) !" )

© 2004 by CRC Press LLC

a

b

c {a}

d

{b,d,e}

{c}

e

%# ' 0

Figure 10.1.12

Bridges and Cutpoints of an Undirected Graph

+

? / 1 "

DEFINITIONS

B ' (

"

B B

'

'

)( ) "0 "

' ' 0 A ' 0

B 9 $ 0 0 ) $ 2!" " "0 )( ) . 3 3 "

0

B "B B

( . ' "

" !"

' " ( (

"0 3

TWO EXAMPLES

B " 3 3 0 . # " . & 0 ) + " 0 " "

:3. " 0 0 1

2

11

12

14

Figure 10.1.13

© 2004 by CRC Press LLC

3

5

4

13

15

7

6

8

9

10

( #

B + ))" 35 /. +1 0 . 5A " 0 ))" . . )

+ 35 " . A " 0 ))" FACTS

B ' ) + 0 ) 0

B ' ( A

( 0 + A

B >" 3 ) 4 0 0 + ) )) ) ' ) . ) ) < ( Algorithm 10.1.6:

! B " ? / 1 1B 0 ) 0 " B 3 " " B B )5 . )5 0 /

1

B ' ) ) ) 4 " 0 ) " 7%=

B ) ) : > " 0 ) 7&= 0 3 / 1

3 '

") ")0 ( 4 3 ")0 4

! / 1 ? ) B ) 05

: > / 1 (

)

0

( /3 )" 0 ( 1R

! /1 ? R / 1 ( ! /1 . 3

/ 1

" ) @

. ! / 1 ? ) ! /1 B B / 1 05

93 )" 0) "

MORE EXAMPLES

B " 03 " (" ) "

© 2004 by CRC Press LLC

1

1

{1,...,10}

{1,...,10}

{1,...,10}

{1,...,10} {1,...,10} a BC

2

2

{11,12,13}

{11,12,13}

{11,12,13} {11,12,13} a BC, (11,2) a bridge

3

3

4

4

{5,6,7}

{5,...,10}

14

15

{14} a BC, (14,12) a bridge

{15} a BC, (15,13) a bridge

8

Figure 10.1.14 . # ! ( '

B " 03 " $(( 3 % ' "

0

7*8= + @ . )5 0 .

Figure 10.1.15 /

B 4 5 3 % ' 7C&8= " !" ) )" 0 " # 3 3 " !" ) L ) 3 I 0 ) 3 " !" ) 0 " Æ 4 " !" ) 3 ( 7%C =

Figure 10.1.16 6 # 7# ! '

"B 8 /3 % '

0 ) 7< =

10.1.5 DFS as a Proof Technique + 0 3 " ) . 4 0 " ) ) /+A 3 ))0 ) P1 "0 ()

© 2004 by CRC Press LLC

DEFINITIONS

B B

'

0 "

3 / 0 !"

"

1 ' ) (

3

B ' ) ( " 0

3

B ' ) (

" !" " EXAMPLES

B

*00 A ) 0 " 0 ) /'

) #1 ) ) /' ) 1 < ) (" 0

. 3 ( " ) 3

(. " )5 (" ) ) ) (" ) ( " / )1 ) ) " " &. 3 3 3 4 (" " " 3 4" 2 3

Figure 10.1.17

! ! ! $(( 3 % '

B

) @ *00 A ) 0 7 ;= 0 0 + 0 " ' ) . 3 5 " 0 " ,

5-. . ( 3 "

B

C@ A ) 0 0 7%&8= < " 0 )

B 0 3 " !" ) ( 3 3 ). ") / 1 ) < 0 ( ( + 05 ) . 3 ) 3 " ; 03 ' ) ) ) " !" 7%C =

B ! " # 7*= ") :) . . ) " 0 B 4 ) :)

© 2004 by CRC Press LLC

Figure 10.1.18

! ! ! 4 5 3 % '

10.1.6 More Graph Properties 0 4 3 0 : >

Æ ) "0 0 " ) + " ) 0 0 Æ ) 4 ) Planarity Testing

4 ) ) ) 3 " : > 0 " 3 . " ". 4 DEFINITIONS

B 9 0 0 3 ) "0 3 B / 1 '

/ 1 0

> )

/ 1 ) ) ) " > )

/ 1.

B

3 ) % /% 1 / 1 " /

" 1 " %

/ 1 . EXAMPLE

B " 8 03 3 /1 3 ) ) % % . % 3 0 % % FACTS

B

< A ) /2() 1. 0 0 )

© 2004 by CRC Press LLC

a

S1

b c

d C

e S2

f S3

g

S4

S5

h

Figure 10.1.19 9 '

B " 0 : > 3 ) '" 7'"# =B 0 /1

%

) % R

/01 ) 0 3 ) " 3 ) ) )

0 /1 /01 ' " ) ) 72&8=

B : "" ) : > 7:&= 3 ) 3 2 0 )

' 4 ) " '

.

" 05

) "B

)R / 1 ) ) % ) 0 ( 3 % 3 ) / 4. )" > . " "0 . " 05 > 3 > 3 ( 1 )

) % " . 5 % % 0 )0 "0

/ / 1 05 > 3

" 1

"B ' ")0 " )

!

" / #1 " " " + , 3 - " ' 4 ) . 0"

© 2004 by CRC Press LLC

Triconnectivity

: > 3 3 4 ) ) 7:&0= 9 5 ) 0 ) DEFINITIONS

B ' "

) 3 3 3

B 3 0 ) ) " 4

EXAMPLE

B

& '

+ " 8 0 B

"R R R R

FACTS

B

3 @ 9 0 0 3 9 " 0 " 0 0 0 0 )) ) $. " " 0 0 /1 ) )

% /% 1 / 1 ? " /% 1R

/01 " 3 ) . ) ( 0 / )0 ,- )1

B ) @ " : >A ) ) 3

)

B

' " " 3 /4 1 Ear Decomposition and -numbering DEFINITIONS

"

B ' " ) ¼ ) ½ " . > " / 1 . . / 1 / 1 / . 0" ). " < 1

B 9 / 1 0 0 ' ")0 ) ")0 . ")0 . ( 0 ")0 0 3 ")0 0

© 2004 by CRC Press LLC

EXAMPLE

B

" 3 ) ) ")0 ")0 / ) 1 2

P1

3 4

P4 7

P0

P6 8 9 10

1

5

P5 6

P2 14

11 12

P3

15 13

Figure 10.1.20

' #'( ! (

FACTS

B < 7< 0= " 0 )

B

' ) 0 ! " 0 " 4 )

0 ) /#, 72&#=. " ) , ) - " 1

B

' ) 3 / 1 0 " ")0 ) 72&#=

REMARKS

$B

")0 0 ) ) 9). 2 0") 792#&= + " )0 0 ( $ 3 )0 ( ( (. " )0

$

B 2 ) 4 ' )

0 " Æ )" 3 ")0

R ) 0 4 2Æ ) 0 0 ) 7*8= Reducibility DEFINITIONS

B ' $ 3 " ( (. . (

© 2004 by CRC Press LLC

B ' 63 0 ) (

!" 3 B / 1 (

(

0

? (. / 1

/ 1

B < 4 0) "" ) ) / ( 1 ' " ) 0) ) " + ) ) " 0 $ 0 " ) $ ' )

/

) $

1. 3 3

)

/ 1 )" ) "

!"

) ( ,

EXAMPLE

B

" 3 " 0 63 + . 63 0 "0 @ " 0 B 63 " 0 "0 ( " /3 ( ) 0" 3 1 > 0 ( > ) ( . ". . " " r

a

b

Figure 10.1.21

c

:

# ( ;

REMARK

$"

B 3 ) "" )" ) ' ) 3 " A " 0 63 $ ) ) @ /. ) )) "0(

. 0 . 4 "

4 . 1 0 " 0 7' D;#= FACTS

) " 0 ) > 7&= 63 )" " 0 + " !" " ( ( 63 . 3 4 / 1 ? B ) 3 05 /

© 2004 by CRC Press LLC

1

B 7&= ' 63 " 0 /1

B

7&= '

") 63 ( 0 ")0 9 0 ( / 1 3 05 " /1 + 3 /1 ( . 3 " 0 3

B

; 4 !" "0 " 0 /1 0 )" ) 0 053 . + . " 0 Æ # "

"B

) 0 ) ( ' ) " ) ( ) " 0 , /4 1

B

0 53 ) ) 0 3 " )

" > 3 )

) 0) ) /*/ 11 : * '5)A " 3 3 7&. 9* =

B %03 > 7%;= 3 ) 0

) 0 ) D ) )5 " 0 ) " )

B

" ")0 + )) 7&= 0 " EXAMPLE

B

" 3 4 63 3 0 0 ")0 ' ) ( ( ;

" 0 )) ;. . 1 2

6

3 4

5

7 8

Figure 10.1.22

#'(

Two Directed Spanning Trees DEFINITIONS

B

+ 63 . ( ( (

© 2004 by CRC Press LLC

B 3 ( 2 / 1 63 ( " / 1 EXAMPLE

"

B + " /( 1 0 " 3 > ( / ( " ( "1 FACTS

3 ( (

4

/1 ? B ) 4 05 /"

? 1 I

) 3 &

B ' 63 3 > ( ( ? ( 3 > ( / 4 1 % " #. 3 ) ) @ ) + 72&=

B B

' 63 > ( 0

> ) ) 4 > ( ( 7&= $ 0 63 ) 4 ( 3

0 ")0 )) ) 0) 0 0 " ) 3 > ( / #1

B

+ ) . / 1 0 / 1 /1 >A ) 4 0 " !"

) " 0 ) )" )

B

) ) ) 5 5 . " 0 ) ' ) "" 7%;= " ) Dominators DEFINITIONS

B

+ 63 3 ( (. (

(

(

?

B . /1. ( " ) )

B

3 ( ( ( ? ( / 1

"B B

)

(

3 ( ( ? (. . /1. 4 0 /1 ? ) B )

© 2004 by CRC Press LLC

EXAMPLE

B

" 3 ) " I ( ) 0" # & ; ( ) ) ( ( 3B /&1 ? . /;1 ? & ' " ( )) ) & ) ) &

1

2

3

4

5

6

7 8

' ! # "

Figure 10.1.23

FACTS

B 0 ) " 93 $5 79$#8=B 2 ( ( ( " !" )) ) >" 4 ) ' ( ) )

"B

B

9" > 79&8= Æ ) 4 ) + 4) ) > 7&0= (

.

/ 1

3 ) #

FACTS ABOUT SEMIDOMINATORS

) ) " " 0" ( 3 0 9" >B

B 5 ( ? ( 9 0 ( 3 ) )") " /1 ) ) /1 . (" /1 / 1?

B

/ 1 / 1

/1 ?

3

/ 1

) ) 0 )" 0 " 4 B / 1 ? ) B / 1 /1 B ) )

/I ) 3

B

!

#1

) 9" > 79&8= )" ) ) "

& 053

/ . 1 )" )) ) " & 3

B

) ) /*/ 11 ' )) ) ) 7':9 88=

© 2004 by CRC Press LLC

10.1.7 Approximation Algorithms ) "0 3 4 " "

Æ" /I 1 0) (). 4 0

"0 3 3

0 ")0 I / "0 :) 1 4 0 " ( ) ) " Æ" 0) : 4 "0 ( 3

0 ")0 . 3 ) " " 4 (

) ) ( ) )

" 7C 8&= DEFINITIONS

B

) @ 0) 5 4 ) 0 "

' * ) ) ) "

4 " @ ) * 79* = 0) . @ " ")0

B

0

"

0

"0 3 ) )")

0 ")0

B ' " +

) 3 3 + )5 )" ' + % + "0 R

") 0 + 0

"0 2 ' + % 3

0 ")0 ) 3 , ) 0

"0 -. 3 " !" . , )

2- ALGORITHM

'( ) ) ) 2 " 4 ' (

) 0 ) ' ) # 3 3 C " F 5 7C F 8= 3 4 0 ) 0 , - " 3 ) 4 ' ) # 0 ) Algorithm 10.1.7:

' 0 < . '

! B 0

" ? / 1 1B . ( ) ) 2

?

" (B 3 " 0 0 0 0 0 ) 0 / 1 Q

© 2004 by CRC Press LLC

EXAMPLE

B " 03 ) (" ) :) . ) 2 ) 3 4 3 ) .

) ( + ' " 0) . 3 3 (

' . )A " B ()

r

r

Figure 10.1.24 ' 0 . ' '

FACTS

% ( ) ) !" 3 0" @ )")

" < @ ) " 3 0"

B & 2 " ) ( )" B & ' ) & . 2 7C F 8= 0 ) ' 2

3 ") . 3 0"

B ' ) & ( ) 3 0" ) " + 2. / 93 "1 / 93 "1 " B F) F 7FF= ( ) )

) 2 ) 0 4 0

> . 4 + " ' & B ' 2 ) ) "0 3 ( F) F ) ( ) ) ) 0 "0 0

© 2004 by CRC Press LLC

"B S . * Q F> 7S*F= " $ 93 " ) )

) 2 F 7F = " $ 93 " ) ( ) ) "0 3 B 93 " ( + 2B + ' ) &

+ 2 + 7C F 8= 0 0 ) ,-A +A &&

B %03 7%= ) ( ) ) 2 )" + " 0 7C F 8= 2. 93 ". )

B C " * 7C *8#= 4 ( ) ) ) + 2 )" + 0

" 2 0 ) 7C F 8= 0 4) 93 " %03 7%= ) 3 # ( )

References 7' D;#= ' F ' . * . S D). . ' < . 8;#

7':9 88= ' ". :. < 9" . $ ". ) ). ; / 8881. &M 7'"# = 9 '" F . )0 . / 8# 1. &M 7 ;= * . *00 A ) ) ( )" . ;& / 8;1. & #M& 8

78#= %

. . :. 88# 79* = : ). 2 9 . * 9 * . . . 2 . $%3 : . 72&= S 2) . 2 > 0 . 8 M8# 0 * *" . ' )

. I3 O5 / 8&1 72&8= 2.

.

. )"

. ).

72&#= 2 * 2 >. )" ")0 . / 8. 8M

8&8

7;&= $ 9 ) * 2 >. 0 " ) 35 ) @ ) . / 8;&1. 8#M#

© 2004 by CRC Press LLC

7%&8= : I %03. ' ) 3 03

. ! # / 8&81. M 7%= : I %03. 0 4 0 )

. " & /1. &M 7%= : I %03. ' ) ( ) )

"0 )" . #$ % ! /1. ;M8 7%= : I %03. ) 0" 4 ) +

"0 )" . #& % ! /1. #M#8 7%;= : I %03 * 2 >. ' ) ) > " . / 8;1. 8M 7%C = : I %03. : C * 2 >. D !" )( )") ) ) . / 1. 8M ; 7%= $ % * )

. '(. S < Q . 7:#8= :.

!

. ' < . * $'.

8#8

7:&= S : * 2 >. 2Æ ) ) " . # / 8&1. &M&; 7:&0= S 2 : * 2 >. )

. / 8&1. M ; 7:&= S : * >. 2Æ . 8M#; 7:*8;= 2 :3 @. *> 5.

. ). 88;

/ 8&1.

. )"

7S*F= * S . * F>. ' ( )

) ) )") . #& % ! /1 &M& 7C 8&= C ". '( ) ) 4 "0 . ( ) % . 0 : 0"). < "0 . 88& 7C *8#= C " * . +) ( ) ) "

) 0) . / 88#1. M 7C F 8= C " D F 5 . ( ) . / 881. M 7C= S C 0. ) 3 )B ' ) . $* /1. #M &

© 2004 by CRC Press LLC

7C&= 2 C" . . 2 . ' < . 8&

+ # %

T 7C8= ' C@ . 4 3 +. %, -. . + 8 / 881. &M8 792#&= ' 9). 2 + 0"). ' ) . M . 0 * . % . 8#& 79&8= 9" * 2 >. ' ) 4 ) 63 . " / 8&81. M 79$#8= 2 93 < $5. 0> ) @ . M 79"B ;;= 2 9" .

/0 0 .

7$ #&= $ ). ) 3 0).

/ 8#81.

;;

/ 8#&1. #M#&

7*8= F *) . ) 3 0 . . 0 S : * . $ C" ). 88 7*= 9 *U . 2 5)0 @.

" , & /

81. 8M

7*8= : 2 *00 . ' ) . 3 0) Æ . # / 881. ; M; 7= * 3 5.

11 2 . ' < .

7 ; = $ . ' ) 63

. 3 & / 8; 1. #&M& 78#= + 3. ,3 )@-. 88#

#

; $ ) * .

7&= * 2 >. 4 ) . / 8&1. #M # 7&= * 2 >. 63 " 0 . M#

7&0= * 2 >. ) . / 8&1. #M;8

8 / 8&1.

7&= * 2 >. 2Æ 0" " ). / 8&1. M 7&#= * 2 >. 2 > 4 . / 8. & M ;

© 2004 by CRC Press LLC

#

7B ;8= % . 9 0U) 0 . )- ;&

/ ;81.

7FF= F) ' F. ( ) ) )")

"0 . #M& ( 4,% . 0 C S C ". 9" I )" 8 . F. 7F = ' F. '( ) ) )") "0 )

3 0". #* % ! / 1. &M# 7<88= $ ' <

. ! ' < . 888

11. 2 .

7< = : < . I

0 . / 81. 8M#

7< 0= : < . " . / 81. M #;

© 2004 by CRC Press LLC

10.2

DYNAMIC GRAPH ALGORITHMS

!

! " # $ % & %

Introduction

& ! ! ' ! ( )' *+ & !' ! ' !' ! , !' ! & ( ! ( ! ! ! !' ( ! !

( . DEFINITIONS

/ 0 ! ! !

( ! ' /

0 ! ! !

REMARKS

/

& ! ' ( ) ( !' ' ( ! ' (

( (

/

! ! ! Æ !' ! ( ' ! ! Æ ! 1

© 2004 by CRC Press LLC

986

Chapter 10

GRAPHS IN COMPUTER SCIENCE

DEFINITIONS

2 ! ! (

/ 0 !

!

/ /

0 ! (

/ 0 ! ' ' (

0 ! (

REMARKS

/

& . ( ) ( ( ! " " ! ' ( ( !

/

& ( ) ( ( ( " " !' ( ( ! !

/

& ! !' ! !' 3

PART 1: DYNAMIC PROBLEMS ON UNDIRECTED GRAPHS ! ! ! Æ ! ! ! ! . !' 4 ! ) ! ! ' !

( ! Æ EXAMPLES

/

! ! ! ! !' !' ! !

/

0 ! !' !' ( ( ! ' ( (

REMARKS

/ ! ! 1

/ 0 ! (

! 5 6 ( ! . ! !

© 2004 by CRC Press LLC

Section 10.2

987

Dynamic Graph Algorithms

& ' . ( ! ! ! ! ! 5Ü6' ( ! ! ( ! / 5Ü6 ! 5Ü6

10.2.1 General Techniques for Undirected Graphs ! ! ' ( !

! 0 ' ! ' ! ! ' ! ! ! ) ! ! ! ! ! ! EXAMPLES

/

/ ( ! !' ( ( ) ) ( ( ! 7' ( ( !' Ü Ü ( ( ! ! !

/

8 / 7' ( ' 90: ;<' 0: ' => 2 !' ) ( 7 ! & ( (' ( . 3 ! ! / ! ' ? ' @7' ( ! !/ !' . ' 1 & ' ( ! ! ( 4 ! ' ( (

Topology Trees

! A ) 9A=#> ! DEFINITIONS

/

/ /

Ì '

! Ì

!

© 2004 by CRC Press LLC

988

Chapter 10

GRAPHS IN COMPUTER SCIENCE

/ 0 Ì / ! Ì 7 ¼ ) ! ! 7 ! 5 ! 6 ' ! ! ' !

! ( , ASSUMPTION

& A ) 9A=#'A;<>' ( 7 7 ! / ( ! ' 9:$;> DEFINITION

/ 0 7 7 ! / 56 ? 7 ! 56 ? 7 ! 56 @ ( , 0 7 ! ' ! ( ' ! A ! REMARKS

/

!

' 3

/

B 56' - ! ! ! ' ( ( ' ( ! ! , ! APPROACH

2 ) ( ( ! A ' ! ( ' ½ ¾ ' ( '

( ( ! 7

& ' ! 0 ' ( ) ( ( ! , ! ' ( , ! & ' ( ( 56 & ( ( ' : (' ! ! ' ( ) ( , ' ! 56

© 2004 by CRC Press LLC

Section 10.2

Dynamic Graph Algorithms

Figure 10.2.1

989

Ì

56 56 . !

!

! !

? ! Ì ! ( ! ,

& ' 5 ! ! 6 7 / ! ! '

! 7 ! 3 (

© 2004 by CRC Press LLC

990

Chapter 10

GRAPHS IN COMPUTER SCIENCE

FACTS

!/

! ' 5 ! 6 5 9A=#'A;<>6

( ) 5 ! 6 ! !

!/

5A ) C 6 9A=#> ! !

5 ! 6 ET Trees

? :1 ! D ! 9:D ;;> ( ) ( ( ( ! ( ! ) (

!' ! ) ! 3 ' ( ! ) 7 ( !/

5 6/ ( 5 6/

5 6/ ) ( ! E ) ( !' )

DEFINITIONS

/ 0 7 () !

! ! ( ! ( () ! 7 E ' ' ? ! 5 A ! 6

/ 0 # " # 5 ! 3 6 ? @ ' ? ' ( 5 A ! 6

Figure 10.2.2

© 2004 by CRC Press LLC

" " "

Section 10.2

991

Dynamic Graph Algorithms

REMARK

/

0 ! 7 Ì ? 5 ) 76' ? 56

APPROACH

& ) ' ( ( ? 56E ? 56 3 ! 5 ! 6

# $

5 6 5 ! 6 . ! ? ! ) !

%& '( $ '

? ( / 5 6 ( ( ! ) 5 6 ( 5 ! 6 ( ( 5 6 5 6 5 ! 6 7 / ? ! 5 6 5 6 ' 9:D ;;> FACT

!

/ B 5 ! 6 ! ? 5 9:D ;;>6 Top Trees

0 90: ;<> Æ

' ' !' 7 ( ! ( ) A ) C ! ' ! ' ( ) ! !/ 7 DEFINITIONS

/

9A=#'A;<>' ' ( ( ' ' !

/ ( 7

/

0

7

/

!

'

( ' ( !

! 5 ! 6

2 A ! 7

© 2004 by CRC Press LLC

992

Chapter 10

GRAPHS IN COMPUTER SCIENCE

APPROACH

! Ì ) ! (

Figure 10.2.3

"

' & ) ( ( ! ,

% ' ! ! ! 4

( ! ( ! . ( - !

7 ! ' ! 4 !

( !' 7 ! ! ! 4

© 2004 by CRC Press LLC

Section 10.2

Dynamic Graph Algorithms

993

FACT

!/ 90: ;<> A ( ! 5 ! 6

! ! ( 5 ! 6 . 5 ! 6 REMARKS

/ ! 7 0

7 )

' 7 ( ! ! ( ( 90: ;<'0: ': >

/ ! 1

' (

! & ! '

, Clustering

! 9A=#> ! ! ' REMARKS

/ ' . ' ! ( !

/ 0 . ! # " 9A;<>' ( ! ! ! ' ( ! ! ! ! EXAMPLE

/ 2 F

! !

! 9A=#> + G 5 6 ! ( ! ! ! ! 7 ' , ( 0 ! ! A ' ! 1 ! ' $%" ' ! ( ! 9A=#>

FACTS

!/ A ! ! !

5¾ ¿6 5 9&;'%;#>6

' 5½ ¾6

2 ! - ! 5' ' 9A=#'A;<>6

!/ 5A ) C 6 9A=#> ! ! ! !

© 2004 by CRC Press LLC

5½ ¾6 ' (

994

Chapter 10

GRAPHS IN COMPUTER SCIENCE

REMARKS

/

9A=#'A;<> A ) C ! 2 ' 5½ ¾6 -! 9A=#'A;<>

/ ! -' (' ( ) Æ ) 7 Sparsification

. ! ? ' 9?&@ ;<> ) 7 5( ! ) ( 6' ! 1 ! ! & -- ( ! !' ! ! ! ! ' ( ' 5 6

! ( ! 5 566' '

! ( A ' 5 6 G 5½ ¾6' ( ! 5½ ¾6 0 ) . DEFINITIONS

/

A ! ! ' ! ¼ ¼

/ /

0 !

56 !

0 56 ' ' 56 56 2- ! ( F ( ( A ' (- APPROACH

+ ! ( ! 2 ! 56 ! !

1 . . ! ' ! ! ! ( ! ! .

( . ? 5 !566 ! ( 56 ! ' ! ( ! NOTATION

& 7' ! 75 !¾ 6' !56 ' REMARKS

7 ( .

/ . ( !

) ( 0 ! ! .

© 2004 by CRC Press LLC

Section 10.2

Dynamic Graph Algorithms

995

3 ! & . 5 H 6' ! 56

/ & ' . ! A ( )' # . ' ! ! ! ! . 52 9?&@ ;<> . 6 5 6 5 6 FACTS

! / 9?&@ ;<> + ( ( . .

5 6 (- ' ( ! 5 6 ( ( 5 6

! ( ! ' ( ! 5 5 5666 H 5 566' ( 5 566 ! / 9?&@ ;<> + ( . 5 6 ' ( (- ' (

( 5 6 5 6 5 5 5666 H 5 566 ' ( 5 566

REMARKS

/ B ' . . 5A <6 1 ! ' ( ( Æ .' ( 5A =6 7 ! ! ' ( ( Æ # .

/

. ( ! ' ! ! ' !- ' 7- 0 7 ' ! ' 5½ ¾ 6 9A=#'A;<> 5½ ¾6 9?&@ ;<>

/

. ( ) ! ! ' ( ) (

! ' ! ! / ! ! . Æ ! ! Randomization

! . ( ! Æ ! ( :1 ! D ! 9:D ;;>E 7 ! ( 1 APPROACH

2 ) ( 1 ( )' ) ! 7 & ! G 5 6 ' ! ! " ' ! % "

© 2004 by CRC Press LLC

996

Chapter 10

GRAPHS IN COMPUTER SCIENCE

' ! !

? 5? 6

' ( ( ! (

) ) 0 ) :1 ! D ! ( !/ ( ' ! ! - ! ' . ) ! ' * ) ) 1 ( ! 2 ! !

5 ! 6 ! , ! G 5 6 ! ( !- 5 ' ! 6' ( ! - 5 ' 6 A ' ! ! . ! (

REMARKS

/ @ !/ ' ! ! ! ' ! - ! !

/ ! 5 !¿ 6/ ! ' !' ( 3 ! 5 !¾ 6 : (' ! - ! ( , ' ) . ! ! & (

! !' ( ) 7 - ! , 0 ! !' ( ) 7 ' ( ( ( ! (

/ ! ! , ' 7 ) 7 ' ( 1 ! ! ' (' ! ! ' ! ' ! ! ' ( / !

& ! ! ' !

) . & ! ' & ! ' !

( ! ! H H FACT

!/ 5:1 ! D !C 6 9:D ;;> + ! ( ¼ !' , ! 0 ! 5 !¿ 6 7 1 ' I5¼ 6 5 ! 6

© 2004 by CRC Press LLC

Section 10.2

Dynamic Graph Algorithms

997

10.2.2 Connectivity 2 ( ! ! ! ! 9: >/ ! ( 5 ! ! ! 6 ( - ! ( ! ! 5 !¾6 1 + ) 1 ! 9:D ;;>' ! 9: > ! ! ! ! FACTS

! / + ! ' ! 2 ' ( ½ ¾ - ! ( ½ ¾ 2 ! " & ( '

! ' !E ( ' ( !/ !' ! ! 56 ' ! ' - ! / ' - ! !/ G ¼ ½ ¾ ' ( 7 !

!/ & ' ! E ! '

! ! ( !

' ( ! ! INVARIANTS

+# ,-/ 7 ! ! ! ( !

+# ,-/ REMARKS

/ & 56 ( + 5 6 - ! 5 6' ( 5 7 ! 6 + ! 56 56' 56 5 6 ! '

´µ ! 5 6' ´µ' ' ´µ / & 56 7 !¾ FACTS

!/ 2 ( ! ' ! & !¾ !

© 2004 by CRC Press LLC

998

Chapter 10

GRAPHS IN COMPUTER SCIENCE

!/ 2 ! G 5 6 56 ' ! ) -

! !

' 56' ! 56 :' 5 656

( 56 2 ( )

!/ 5 6 . ! ! ' & 7 ' ( ( / ' ( E ( ' G ' ( 5 6 ! / ! ' - ! ! ( + ! 5 6 ' ( !' ' 5 6 ( 56 ' ! H ! @ - ! . / ! ' ! ' (

! / 2 ?-'

'

' ! 5 ! 6

!/ 9: > 0 ! ( ! ! 5 ! 6 1 ( ! 5 ! ! ! 6 ( - ! REMARKS

/ & ! ! ! ' ?- . ! ! 7' ! 1 ' ' ' . ! ! - ! ! ! ?- ( / 9: >

/ ! 1 ! !' 5 ! 6

. @ ' ! ! 5 ! 6' ( ! 5 ! 6 ' 1 ' ' 5 ! 6 2 ! ' ! ) ! 5 ! 6 5 ! 6E ' 5 ! 6 ' 5 ! 6 1 ?- G ( ( 5 ! 6 ( - 0 ( 9: >' 5 ! ! ! 6 ! J5 ! 6- ?-

10.2.3 Minimum Spanning Trees 0 ( ! ! . ! ( ! !

© 2004 by CRC Press LLC

Section 10.2

Dynamic Graph Algorithms

999

! 9: > 0 ! 9:D ;<> ) - ! Decremental Minimum Spanning Tree APPROACH

& ! ! ' ! ' ( ! ! ( !' ' ?- Ü ! ( ! - ! ? ! ( 0 5 ! 6 ' ! 2 ( ! & ' ! ( ! !

/

! ( ! ! . ( ! ( ! ' 9: >'

! INVARIANT

+# ,-/ ?

- ! 7 ( !

! !

FACTS

! / & 56 ' ! !'

! ! 7 + ½ ¾ ( ! ( ' G ! ' 7 ( ! ! ! & '

5½ 6 5¾ 6' ¾ ! G 5 ½

¾6 5 ½ ¾6 ) & 56' ¾ ' ! 5¾ 6 5½ 6 ' ! - ! ! ( 5½ 6 5¾ 6'

!/ 9: > 7 - ! ! 1 ! ( ! ! 5 !¾ 6 Fully Dynamic Minimum Spanning Tree

! ! ! 1 :1 ! D ! 9:D ;<> ( ) ( FACTS

!/ 5:D ;<': 6 ( - !

! ' ' 1 ! ( ! I56 5 5 66' ( - ! 7 - ! ! ! ( ! ( !' ' !'

© 2004 by CRC Press LLC

1000

Chapter 10

! H

¾

GRAPHS IN COMPUTER SCIENCE

5 6

9:D ;<> 9: > A A A ( ! 5 6 G 5 ! 6 :' ! A A ' ( ! !/ 9: > 7 - ! ! ' ! ( ' ! ( !' ! 5 ! 6 1 ! PART 2: DYNAMIC PROBLEMS ON DIRECTED GRAPHS

& ( ( ! & ' ( ( / ' ! ( ) 1 !' Æ ' ' ' ! ! ' . ' ' F ( ' ! 2 . ! ! ! ! 5 "6' ( ( 5 # $' 6 & . ' ! ! , ! ) ( ! ! 5 - 6 9&'D ;;> & ' ( ( ! , !' ! ( ! ' - 9&>

10.2.4 General Techniques for Directed Graphs

& ( ! 2 . ! ' ( Æ ' ( ! )

! ! ! Path Problems and Kleene Closures

! 7 7 ! 5 9 +% > 6 B H '

© 2004 by CRC Press LLC

Section 10.2

1001

Dynamic Graph Algorithms

( - ! ' " G " 9 #> G 9 $ > H ! 9$ #> ' % G %9 #> G 9 #> ! 9 #> 2 ! !

7

! 7 ! 7

! ! 9 #> 5 #6

FACTS

!/

+ G 5 6 ! " 56 5F7 6 & & B , 7 ' B , 7 " 56 D & H B !/

& G

!/

&

+ G 5 6 ( ! ! ( ! -! & & ( ! 7 & 9 #> ( ! ! 5 #6 ' 7 D & !/

& G

&

7 ( ( (-) ( ! D & 7 &

.) ) 0 & '

!' 5 ! 6 ( - ' ( 5 6 ! ( !

!/

! & G & H & ' ( & G & & G & ¾

# )

0 ' 9<>' ! & 5 6 ( -

! /

' ( 7 & ' ! ' %' " 1 5 )( 6' &

' ' ' ' 1 ' & . ! ( ! /

G 5 H !% " 6 G !% G % " ' G % H % "!%

!' ! ! & ! & ! ( ! Uniform Paths

! & 9&> & ' ( 1 ( ! !' (

© 2004 by CRC Press LLC

1002

Chapter 10

GRAPHS IN COMPUTER SCIENCE

DEFINITIONS

/ /

0 ( ! (

0 ! ' ! ! ( .

/ 0 ( !

(

REMARKS

/ 0 . ' ( ! ! 5 6 ( . ) H H ) G 5( 6' ( ) ( 7 7 # !' ( ! ! 5 6' 5( 6 ( ! ( ! . /

& Æ 1 ! ! ( ! 5¾6 ' '

- -

FACTS

! /

5 & 9&>6 & ( ' * ' ' ' * ' ' ' !' ( ! / * * ' *

!/

5 & 9&>6 + ! , & ! 5$ 6 ( ' 1 5$¾ 6

REMARKS

/

7

! ! , 56 & '

! ! ( ! !

/ ) ! ' (

&'

! ! ! ( & ' 7 ! ' ! K ! ' 5K6 ! 5 ! 6 5 ! 6 ( ! 2 9&> ! ! 0 ! A ;' 5¾ ! 6 ! ! ( ! 5K6

© 2004 by CRC Press LLC

Section 10.2

Dynamic Graph Algorithms

1003

/ 0 9&>'

Æ A = ' ) ( ! Long Paths Property

& ( ) ! ' Æ ! ( ( ! . ! ! ! ) (!' ! ( . ! 9D=>' ! ! Æ ! 5 !' 9&'D ;;'L;'M(;=>6 FACT

! 5 ! 6 ' ( ' ' ' Æ ! ' + ! / 5 L)) 9L;>6 +

REMARK

/ 0 ( 9M(;=>'

! 9<;'+ <#> 0 9D ;;> Reachability Trees

0 ( ! . ) ;=' ( ? ( ( -. ! ! 9? =>E ) ! + ' :1 ! D ! 9:D ;;> ( ( ! D ! 9D ;;> ! 7 ( ! ! ! PROBLEM

& ( ! ' ! . 5BA 6 ! ! ! ! & '

7 ' ( ! BA )' ( ) ! G 5 6 7 ' ( ( ) 7 ( ! /

5 #6/ ! 5 #6 56/ 7 BA H ( )6

) 5

FACT

) ! 5)6 ( ! ! ! ( !

!/ 5D ! 9D ;;>6 ! BA

© 2004 by CRC Press LLC

1004

Chapter 10

GRAPHS IN COMPUTER SCIENCE

REMARKS

/

A ! BA ) ! ) ' ( 56 BA / 0 ( ( 9:D ;;'D ;;>'

7 BA ( ( ! ! & ' BA / ! ! ( !

' ) ! ,)C ! BA 9D ;;> Matrix Data Structures

2 ( 7 ) ! ! 0 ( 5 D 6' D ! & ! , ' ) 9&> PROBLEM

& B ' ( + B ( !' ' &½ & 2 ( , 7 ( ! ) / 5 N& &6/ - ( & ! - - ( 7 N& 5 N& &6/ - & ! - - 7 N& 5N& &6/ 1 & ! - 7 N& 56/ 2 ! ! / 5N& &6/ & ! - 7 N& : (' ! 3 REMARK

/

+ " ( ( ! ' , ( & ' ( , G " 8( ' , " ' ( ! ( ! () , / " 9 > F 4 & ' , 9 > F ( ( ( C '

) ! C !

© 2004 by CRC Press LLC

Section 10.2

Dynamic Graph Algorithms

1005

FACTS

!/ 5 & 9&>6 + ( ! B ! 0 ' ' ' 5¾6 1 ( !/ 5 & 9&>6 + ( !

H ! 0 ' ' ' 5% ¾6 1 ' ( %

7 3 ! (

10.2.5 Dynamic Transitive Closure

& ( ) ( ! ! ( !' ! 7 ( ! ) / 5 6/ ! 5 6 E 5 6/ ! 5 6 E 5 #6/ ( ! OP 7 7 # ' O P ( E FACTS

!/ 0 - ! !

' ! # 56 5 6' 56 ' ( ! ! !/ 0 - ( D

, 7 !' ! ! 9<> " 5 D 6 7 9 2 ;>' ) 5 6 ' ( - = 7 7

REMARKS

/ '

( ) ( ;;#' ( :1 ! D ! 9:D ;;> 1 ! ( - ! 5 ! 6 1 5Q ! 6' ( Q ! ! ! ! ( Q ! 56' 5 ! 6

/ D' ( 2 9D 2 ;$> ' ( ) J5 6 ' 5 6

© 2004 by CRC Press LLC

1006

Chapter 10

GRAPHS IN COMPUTER SCIENCE

/ D ! ! 9D ;;> ( ( 56 5¾¾6 ! ! 56

!E ! 1 ( - (

D ! 9D ;;>' ( 7

! ! ! ( 56 5 ! 6 1 ' ( ! 7 ! / ! 3 ( )' & 9&> ! 5 6 1 ! ! / 2 ! ! ! I5 6 ' 7 ( ( )' 56

/ & ( ! ' & 9&> ( ( ) 56 !- 7 / ! ! D ! ! 9D ;;>' 1 ! ( - ! 5 6 ( - 5 6 ( - / 8 9% M(> King’s /5¾ ! 6 Update Algorithm

D ! 9D ;;> . - ! !

Ü" 5% 6 ! Ü" 5 D 6 & 7 ! 5 ! 6 1 ' ! ! ! ! ( , & ! 7 ! ! !4 ! , ! APPROACH

! ! H / ' ! ' ! ( ! ! ! ! ' G FACTS

!/ ? ) ! ( 7 / - * 5 6 ! ! ( ! ' - ./ 5 6 ! ! ( ! 0 ! 5 #6 ( ./ 5 6 # * 5 6 ! / ! ./ 5 6 * 5 6 ( BA

" 5% 6

© 2004 by CRC Press LLC

Section 10.2

Dynamic Graph Algorithms

1007

! / ! 7 ' ! ./ 5 6 * 5 6 ' ! ' (

!

@' 4 7 ' ) ! D !C ! !/ 2 ! ' ./ 5 6 * 5 6 A ' 9D ;;>

Demetrescu and Italiano’s 5¾ 6 Update Algorithm

! & 9&> 7 " 5 7 6 " 5 D 6 & 7 ! 5¾6 1 ' ! ! 1 D !C ! ' ' !

7 ! ! ( , ) ( ( APPROACH

! D & , 7 & ! ( &½ &¾ NOTATION

+ ½ ! ! . & ' ¾ ! FACTS

! / B &½ &¾ . ! C "

5 D 6' ( ! ! 7 ½ &½ ' ( ! 7 ¾

&¾ ' &½ &¾ & ' !/ & ' ! & - ' ! ' " ' % 7 " 5 D 6' &½ ' &¾ - ½' ½ ' ½' '½ ¾ ' ¾' ¾' '¾' ! &½ &¾ ( ( ! = !

" 5 7 6/ 0 G H ! ¾ " ¾ G ½!'¾¾ "½ ½ G ½¾! ¾ G ½!'¾¾ ½ G "½¾ ¾ G '¾¾ "½ '½ G "½¾! 1 G % H "½¾! ( G % ' ½ G 0 ' '¾ G 1 D 1

© 2004 by CRC Press LLC

1008

Chapter 10

GRAPHS IN COMPUTER SCIENCE

½ ' 0' ½' ½' '½ ! ' ( ' ' ' 1 ( !/ ! 7

!/ & ' (

' ' ' (- 9&>

!/ A '

10.2.6 Dynamic Shortest Paths

& ( ) ( ! 5 0 6 ( ! ! ( !' ! 7 ( ! ) / 5 6/ ( ! ! 5 6 ( 5 G H ! 6E 5 #6/ 7 7 # ' H

( 7 E REMARKS

/ ) ! '

. ) # ! 9+ $<'$<'% $=> 0 ' ! 5' !' 9?=#'A@;='A@' %%;$'%%;$ '% =#>6' ! ( ( ! 0 / . ! ( -

! 0

' ( ) ! ( ! ( !

/ & ' 0 ' 90&@;> -

! ! ! ! ( ! " / 1 ! ! 5" ! 6 ! / :1 !' 9:D % ;<> ! ! 0

! ( ! ( !' ( ! 5 !5" 66 / A) % 9A%>' (

! ! !- ! ! ( ! 5 ! 6 1 ! / . ! ! ! ! ( ! ( D ! 9D ;;>'

( ! ! ! ( ! ( ! " / ! ! 5 " ! 6

© 2004 by CRC Press LLC

Section 10.2

Dynamic Graph Algorithms

1009

/ & 9&> ! . !

0 ! ! ( ( ! ! ! ( ! 3 ! & ' ! 5¾ ! 6 1 56 ( -

/ . !

! 9&> ! 7 ! ! ( ! 5 ! 6 1 ! ( -! ! ( ! ( ( )

/ 2 ! ! 7 ! I56 ' 7 7 ( ( )' 56

/ 8 - ! 0 !' 9B: >

' ( !

King’s 5 " ! 6 Update Algorithm

! D ! 9D ;;> ! " 5+ ! 6 " 5% 6 !

#' ! 1 ( ' ' 5 ( ! 6 ! 7' 5 ( !

6 ! ! ( , APPROACH

! )' !

! ! ! ) A ) ' ( ( ! !/ 7 ( ! ( ! "

9D ;;> FACTS

!/ )' ! D !

#' ! ) 4 ./ 5 6 * 5 6 ) 7 ./ 5 6 * 5 6 ( " 5% 6 & ' ) ( # )' ) ) H ) ./ 5 6 # * 5 6 ' ! 7 ! ./ 5 6 * 5 6' ( ! 1 5 )6

© 2004 by CRC Press LLC

1010

Chapter 10

GRAPHS IN COMPUTER SCIENCE

!/ ! )' ! 7 !

A / ' ! ' ' J55 ! 6)6 !' . ! ) ' ( ' ' ' ( ) ' O P' # ! ) ! ! ! ) . ! 7 ' ' , ( ' ' # 7

' . ! ( ' ! - - ! ' 7 ! ( ! ) ! 5¾ ' 6 G 55 ! 6)6

) G ! 5 ! 6 1 0 " 5+ ! 6' ' '

! / !

! 1 A ' 9D ;;> Demetrescu and Italiano’s 5 ! 6 Update Algorithm

& 9&> . - ! - ! . ! & ' ' ! , ' " 5 6 APPROACH

! A = ' ! ( FACTS

! / ! ( ! ' !

! ! " 5 6 ( ) ( & . !/ ' ( . '

0 ' ! - . ,)C ! 9 #;>' ( ( ( !

7 ( 7 ! ( 0 '

5$6 ( 5 A ;6 ! ! $ G 5 ! 6

! 5 ! 6 ' 5 ! 6 1 A ' 9&>

!/ !

© 2004 by CRC Press LLC

Section 10.2

Dynamic Graph Algorithms

1011

RESEARCH ISSUES

& ( ) ( ! ! ) ( ! ! ' ! ! ( & ' ( ! ( ! 1 ! 9: > ! & ! ( ( ! ( / ( ( - 5 6 '

( . 9?&@ ;<>

A !' ( ( ( - - ) I5¾6 ! - : (' , ( ( .7 ' ) ( A ' ( ! ' & 9&> ( ( ) 5¾6 !- 7 ! & ! ( ( 3 4 3 ! ! A ' ( Æ !- ! !R A ' ! ! F ( ! Further Information

% ! ! , '

! & ' ' & ( ' ' & ' ' " ' &( ' 2 ) ! ! ' & ( 5 86' ))) " 5A8 6 * & " + 5&0+ 6 1 7 ! & (,&( & 5 806' ) & 5? 06 Acknowledgments

( ) & ! ? & -;;;-"=$ 50+8-A6 & --### 58 &@6' & . % 5 , O0+&@2?B/ 0! & 2 P6

© 2004 by CRC Press LLC

1012

Chapter 10

GRAPHS IN COMPUTER SCIENCE

References

90: ;<> 0' S : ' D + !' ' 1 ! ' 5;;<6' +@ #$' <T= 90: > 0' S : ' ' ! ' ! "# $% "$ && 56' "$T#$ 90&@;> 0 ' A & ' 0 - ' @ ' & ! ! ' ' ( 5"6 5;;6' $#T$= 9B: > B(' % : ' ' & ! - ' ) * %# $% ( # $+,& 56' * U' 0 ! - ! ' +# - "56 5;<;6' T#

* (

9 2 ;> 2 !' 7 ! ' ' ( % # ; 5;;6 9 +% > : ' ? + ' % + % ' '

' ? ' & ' 9&> A & ' A / B) ! ! 5¾ 6 ' ( . /// %# 01

( # 0+ ,&& 56' =T=; 9&> A & ' A ( ! ( !' ( /// %# 0 ( # 0+ ,&. 2 3! 56' $T$< 9&> A & ' 0 ( ' )4 %# $% ( # $+,&) 5 56' #;T$$ 9 #;> ? 2 ,)' 0 ( ( !' 3 * 5;#;6' $;T< 9?&@ ;<> ? ' M ' A & ' 0 @ 1( !' . T 0 ! ! ! ' ' # * ' "" 5;;<6' $$;T$;$ 9?=#> ? : 1 ' ! !' +# - "; 5;=#6'
* (

9? => ? L ' 0 - ! ' ' #

* = 5;=6' T"

© 2004 by CRC Press LLC

Section 10.2

1013

Dynamic Graph Algorithms

9A%> S A) % ' !' ! ( ! !' ' ' ( /// %# 0 ( # 0+ ,&. 2 3! 56' T" 9A=#> @ A ) ' - ! ! ' * ' # " 5;=#6' <=T<;= 9A;<> @ A ) ' 0 -!- ! ' * ' # $56 5;;<6' "="T#= 9A@;=> A ! ' 0 - ' @ ' - ! ! ! ' 56 5;;=6' #T<" 9A@> A ! ' 0 - ' @ ' A ! ! ' ' ( " 56' #T= 9&;> M A & ' A- ! -! ' * ' # 5;;6' " : ? D' B )V' ;=

* ( % ( '

9:$;> A :' 6# $%' 0 -2' ;$; 9:D ;<> % :1 ! * D !' ! ! !' 1 5;;<6 #;"T$" 9:D ;;> % :1 ! * D !' % 1 ! ! ( ! ' ' # * "$5"6 5;;;6' #T#$ 9:D % ;<> % :1 !' D ' % ' ' A ! !' ' ( # % ##56 5;;<6' T 9: > S : ' D + ! ' - ! - ! ' ! ' -!'

' ' # * "=5"6 56' <T<$ 9D 2 ;$> D' % ( ' % : 2 ' 8 . ) ! ' 5"6 5;;=6' < * D !' A ! ! - !' &1 %# 0 ( # 0+ 5;;;6 9D ;;> * D ! !' 0 ! ! ' ).1 * %# $% ( # $+ 5;;;6' ";T";=

© 2004 by CRC Press LLC

1014

Chapter 10

GRAPHS IN COMPUTER SCIENCE

9+ $<> + ' 0 ( ) ' 7% - - &4 5;$<6' ;$T; 9+ <#> + + U1' 8 ! ' 5 * 1 5;<#6' =T; 9<> & ' ?Æ !' (

56 5;<6' #$T#= 9$<> S ' O 3 ! ! ! ! !P' ' +B -@-$' + B ' @( ) ' + ' D' ;$< 9%%;$> % ! %' 0 ! ! 1 ' ' ( 5;;$6' $ % ! %' 8 7 ! ' $ # #= 5;;$6' T<< 9%;#> %' A !' 5;;#6' #T #= 9% $=> * % ' ' 8 - # 1

* * % =5#6 5;$=6' $T" 9% =#> : % ' 0 1 - '

%# $ # ( # $ 94 3 .9

5;=#6' <;T=$

9 => % ? ,' 0 ' ' #

% " 5;=6' $T= 9L;> S L)) ' : !- - ! ' * ' # 56 5;;6' T# 9M(;=> M( )' 0 ( ! ! - 7 7 ! ' ( ) /// %# 0 ( # 0+ ,9 5;;=6' T;

© 2004 by CRC Press LLC

10.3

DRAWINGS OF GRAPHS

! " # $ % & ' ' ( ) ' * '

Introduction ' + + , . , , /, . , , , /,

- .+ ,

, + 0 / 1 , + + 2 3

+ + 4 +

10.3.1 Types of Graphs and Drawings , 5 , , +

, - - Types of Graphs

6 , 4 , + +, + ., / DEFINITIONS

3 3

# . / #

. /

3

#

© 2004 by CRC Press LLC

+

1016

3 3

Chapter 10

GRAPHS IN COMPUTER SCIENCE

+ # . / +

#

3 # . / 5, 7

3 #

- 8 , + 7 + -7

3 #

- 8 , + 7 . / + -7

3 # + , + .9 + + + /

3 #

+

3 # ./ + , # + +

3

#

./

Types of Drawings

1 , + . 4

/ + 4 4 + , 7 DEFINITIONS

3 3

, . 6 . //

, - . 6-

1

1 .//

3 1 , 0 + . 6 .//

3 # . 6 . //

3 # +

3 # . 6 .//

3 # + , , +

3

1

, . 6

© 2004 by CRC Press LLC

.//

Section 10.3

3 3

1017

Drawings of Graphs

./ # ./ 4 #

.,

/, +

3 1 , + . 6 .//

3 3

1 . /, #

+ 0

3 # , 3 #

-

+

3 # + 8 , + 0 , + +

3 # + + ./ ./ ./ ./, ./ ./ +

3 # - 0 + EXAMPLE

3

1 6 4 ¿ ¿

(a)

(b)

(c)

(d)

! " #"$

Figure 10.3.1

REMARKS

%3

+ : + ; + , Æ < = # , - ;,

, -

%3

1 + , +

© 2004 by CRC Press LLC

1018

Chapter 10

GRAPHS IN COMPUTER SCIENCE

10.3.2 Combinatorics of Geometric Graphs >)?, @)?A - * + -5 , , , + ,

-5 , : + , , , , 5, , , 1) 0 + % 0 + - , , + Æ # %

+ , + ., , >? , )?, B?, CC?$, CC, CC*9?(, C*, C)?A % 2 / 1 , : + -5

Delaunay Triangulations DEFINITIONS

3 # -

+

+ +

3 #

3

#

FACTS

&

3 >?A # - +

&3 &3

>? A # >)?A # -

-

REMARKS

%3

D >D?$A + 0 - % + +

%3

C *7 + D 1 , 0 D >C*A

© 2004 by CRC Press LLC

Section 10.3

1019

Drawings of Graphs

Minimum Weight Triangulations DEFINITIONS

3 # - +

3 0

-

3 # E E +

3

1

8 ,

FACTS

C 5

&3

1 5

F- . B >B&?A/ )+ + , + , + Æ # 5 *7 ' >'*?A, C >C(&A, G >G?A, 5 >**?&A, G 5 5 >G (A, # 0 >## ?$A, H >H A, 5 * >*?$A, C+ G 0 >CG?$, CG?(A

&3

>CC?$A #

. '#*/ + + 1 >CCA C C 3 , + 5 8 5 8 5 9 , , I >9IA 5

&

3 >CCA 4 0 . ,

/ 1 5 2 +

., , >G (, CG?$, *J&?A/, 5 2 -

© 2004 by CRC Press LLC

1020

Chapter 10

GRAPHS IN COMPUTER SCIENCE

Minimum Spanning Trees DEFINITIONS

3

#

, -

3

#

+

0

+ +

+

.

,

/

FACTS " + - +

&

3

>*)?A " +

4+

+

. '#*/

&

3

>*)?A F

&

3

>"9?$A 1 F- %

&

3

>C?!A F +

" - , +

Proximity Graphs 2 4 + 2

DEFINITIONS

3

#

+

3

3

-

/

.

, +

5

/

.

- +

© 2004 by CRC Press LLC

Section 10.3

Drawings of Graphs

1021

REMARK

%3 + 4

, + 5 + 6 , , + >B?A

FACTS

* , - + + , + % 7 + + 7 +

&3 1 5 , C , C >CC?$A, -

0 - +

&3 C ) >C)?A + +

&3 6 >CC?&A

: C , C, *7 , 9 >CC*9?(A , , *7 >*??A

Open Problems

'$ + 0

'$ C + + 1 "

- K 1 ,

'$ 4

6 ,

0 5

'$ 1+ #

K '$ 6 6 , 0

,

-

© 2004 by CRC Press LLC

1022

Chapter 10

GRAPHS IN COMPUTER SCIENCE

10.3.3 Properties of Drawings and Bounds 6 + , + L

4 + + ) 2 # + + # 4

9+ + , + ., % ,

+ / Properties of Drawings

1 , 5 5 + 6 , + 0 @ 1 , + + , 5 * + , 0 0 DEFINITIONS

3 3 3 3

-

+

3

3 , +

3 0 + + EXAMPLES

2 0 - 7+ 0 .,

/, 2 +

3

6 . M/ , + . / , ./ < 0 = 1 , , +

© 2004 by CRC Press LLC

Section 10.3

1023

Drawings of Graphs

(a)

(b)

(c)

(d)

Figure 10.3.2 (") * ++

!*" #"$

3 6 .M/, ./ , ./ 1 Bounds on the Area

0 + 1 , , 4 , + + <)C= <)C= < -= < - =, + 1 , 2 % @ , + 2 @ , FACTS

1 , 4

&3 C - + . M / &3 % . M (/ ; + ,

% - . /

&3 # + . (/, + & 3 N + 2

. ?M / 1, ./, - % . ?/

& 3 # % +

. / ) 2 Bounds on the Angular Resolution

0 + ; 4

© 2004 by CRC Press LLC

1024

Chapter 10

GRAPHS IN COMPUTER SCIENCE

( $$ , # " - #" $ C#)) @6 '#;)

'#91F I"

#'"#

)C

O. /

. /

)C

O. /

. /

-. /

O. /

. /

O. /

. /

!

+ )C

O. /

. ½· /

$

6

+ )C

O. /

. /

&

#DC

+ )C

O. /

. /

(

+ )C

O. /

. /

?

)C

O. /

. /

)C

O. /

. /

-. /

O. /

. /

-

O. /

O. ¾ /

. / . ¾ /

)C

O. /

!

)C

$

&

O. ¾ /

)C +

O. ¾ /

. ¾ / . ¾ /

O. ¾ /

(

-

O. /

. ¾ / . ¾ /

?

)C

O. /

-

)C

O. ¾ /

. / . ¾ /

O. ¾ /

. ¾ /

O. P /

© 2004 by CRC Press LLC

.. P /¾/

Section 10.3

1025

Drawings of Graphs

( $$ , #" " - # #" # #$ C#)) @6 '#;)

'#91F I" #FNC#' '")@CN1@F

-

O. ¾ /

-

O. /

-

O. /

. ¾ /

. /

.

¿

/

Bounds on the Number of Bends

0 + ) ! &

5 ,

( $$ . " # # #" " - #" #+ / "$ 0 & !$ C#)) @6 '#;) -

'#91F I"

@#C Q "F)

-

-

-

.

-

-

*#H Q "F)

P

P

P

P

/P P

P

P

Tradeoff Between Area and Aspect Ratio

-Æ + 0

, ; + , + 3 , + 5 , + ;, 5 1 , 5 + +

: 4 # + 2 + , FACTS

0 +

+ +

© 2004 by CRC Press LLC

1026

Chapter 10

GRAPHS IN COMPUTER SCIENCE

( $$ ( # # #" +## ! / " ! $ C#)) @6 '#;)

'#91F I"

#'"#

#)" '#1@

)C

. /

. /

)C

. /

. /

-. /

. /

. /

. /

. /

)C

. /

)C

. /

)C

. /

. /

-

. /

. /

-

, + +

. /

. /

>. / . /A . / ¾

1 , + <)C = < - ,= - 0 + - @

Æ

-4 + 6 + + REMARKS

%3 9 - + 0 , %

%

% 3 ' , + ; + ,

% 3 6 ,

, . / + + ., / ; + , , + 0 Tradeoff between Area and Angular Resolution

! 0 + + ; 4

© 2004 by CRC Press LLC

Section 10.3

1027

Drawings of Graphs

, N+ # ,

+

( $$ , # #" / " #" / # #1 +## ! $ C#)) @6 '#;)

'#91F I"

#'"#

#FNC#' '")@CN1@F

-

. /

O. ¾ /

-

. /

-

. /

-

./

. /

O. /

O. ¾ / O. / O. /

Open Problems

'$ % . / - . / 5 . / ' $ % + . /

' $ % . , , / . / 5 . / '$ O. ¾ / + . ¾ / - . /

' $ O. / + .

¿ / - . /

'$ + . / - . /

10.3.4 Complexity of Graph Drawing Problems $M ( 0

1 + 2 6 , , F- F , 6 .M/, Æ 9 Æ

+ -

© 2004 by CRC Press LLC

1028

Chapter 10

GRAPHS IN COMPUTER SCIENCE

( $$ (+ !+- / + /#"+ " + " " $ C#)) @6 '#;)

'@C"*

1*" @*C"H1I

0

F-

-

0

F-

F-

O. /

. /

O. P /

. P /

F-

O. /

. /

-

O. /

. /

5

F-

, 0 F- , +

Open Problems

'$

' . . //, + . /

'$

' 0 . . //, + . &/

'$

' 0 - . &/

10.3.5 Example of a Graph Drawing Algorithm 1 > ?&A ,

- , . / . &/

. $ 6 .// 3

2 3 +

,

4

#

© 2004 by CRC Press LLC

Section 10.3

1029

Drawings of Graphs

( $$ (+ !+- / + /#"+ " + " " $ C#)) @6 '#;)

'@C"*

-

F-

-

F-

-

+ +

F-

-

+ +

O. /

. /

- . / . /

O. /

. /

O. /

. /

- + . / . /

O. /

. /

- +

O. /

. /

-

-

O. /

. /

. / R . /

O. /

. /

-

F-

-

Q . /

O. /

.

-

Q . /

O. /

.

-

. / . /

O. /

F-

© 2004 by CRC Press LLC

. / , . / , . /

1*" @*C"H1I

/

/

. /

1030

Chapter 10

GRAPHS IN COMPUTER SCIENCE

( $$ (+ !+- / + /#"+ " + $ C#)) @6 '#;)

'@C"*

1*" @*C"H1I

"

F-

-

0

F-

-

0 L

F-

0 - S

F-

0 - S

O. /

. /,

0 +-

O. /

.

- . ¾/

O. /

. /

. /

O. /

. /

/

REMARKS

%3 5 : .6 . M/ /, : ;, + + ,

, +

% 3 6 : 5 5 +

C . /

" + ./ T : ,

. / : , . /

. / T . / P

" S ,

./ T

. /, , %

%3 F 5 3 . /,

+ 8 : . / +

, , ,

8

. /,

8 : . /

, , P ,

© 2004 by CRC Press LLC

Section 10.3

Drawings of Graphs

1031

%3 + : +

+ % + :

+

% .! /, ! 6

, + % .! P /

%3 1 + : " 5

, % : " ;, : : . ¾ / : - %3 + DC)1 ,

0 + 4

6 ./

10.3.6 Techniques for Drawing Graphs 1 % + + Planarization

0 + + Æ 0 . &/ 1 , 0 4 + 6 F- ;, 0 + >B*?$A , + # 0

-0 > (&A >((A .6 ./ / 1 + 0 Layering

#

3 # + , 0 1 4 + , + + . 4 + 4 / + 0 #7 + + 0

© 2004 by CRC Press LLC

1032

Chapter 10

GRAPHS IN COMPUTER SCIENCE

1

1

2

(a)

1

4

1

1

1

1

2 1

2

1 3

2 2

2 12

1

2

(c)

60 19

23

7

26

25

8

9

58

37 42

3

39 40

62

1

51

49

57 45

6 5 48

10

2

61

27 28

44

53 30

15

22

54

4

38

56

18 14

21

43

32

31

29 52

20

35

36

(d)

59

46 17

16

47

11

50

63

34

55

12

24

13

33

41

(b)

Figure 10.3.3 4+""" $ 5+#+ ! 6 7 6 - ! ! ! 8 6 +" ! # ! " !7 /! " - " . /$ ! ' " " / ++#+ #+ / "$ " . " " / "#!" " + " / " + / #!$

* + + + F- , ) >)( A + % >FD((, GFD?A 6 + BU *0 >B*?&A

© 2004 by CRC Press LLC

Section 10.3

Drawings of Graphs

1033

Physical Simulation

+ +

7 7 + ) 4 , + + 4 4 , ' + 2 % , + ., , + /

; + , >" (, G)(A ) + >;?$, ";, 6'? , G , C* A ' + 8 , >?!, C"?!, ?!A

10.3.7 Recent Research Trends + + + Compact 3D Drawings

+ 0 + 5 5 + 4 + + + 9 -0 - - , 7 FACTS

&3 5, , >?$A + + . / + -

& 3 , " , C '5 >"C'?&A + - - . ¿/ + , + &3 ) >)?&A -, -, - . / + . / + O. / + &3 6 #- , , V >?&A $. / +

&3 , , D >D?$A - . / . / + . / +

© 2004 by CRC Press LLC

1034

Chapter 10

GRAPHS IN COMPUTER SCIENCE

&

3 6 , C , 9 >69 A +

&3

7 +V, * , 9 >*9A . ¾ / +

- . / + -

&3

9 >9 A % + - . / +

& 3

# 7 +V 9 >9 A +

+ -8 - + , + -

& 3

, C , 9 >C9 , C9A

+ - . / + Graph Drawing Checkers

5 + + DEFINITIONS

3 # +

5 5 + 4 + + % 2 ., , >C?(, *)F)· ??A/ + < = 5 +3

3 5 @ , 5 5

3 3

!Æ + 5 Æ

3 " 5 4 2 : - 2

REMARKS

%3

5 + 1, 4, 4 # , , + + 6 , + 9 4

© 2004 by CRC Press LLC

Section 10.3

Drawings of Graphs

1035

%3 @ , 5 + , + Æ + 5 + + % 53 < K= < K= <

+ K= % 3 5 + * >*)F) ??A + >C?(A 1 + 5 + +

C >C?(A 5 + - +

Incremental Graph Drawing

1 + , , + , + N , 4 N , ) ,

%5 4 REMARKS

% 3 4 4 2

+ , + 1 , S >"C*)?!A, ,

, +, + 2 % + > A

+ + 2 S + + 2 , 0 , +

%3 )+ + 7 # 5 , , , >?!A 8

5 9 >9?&A8 4 + + 5 >?(A 5 , ) , >)?&A8 + G >G?&A, 9 >9?(A8 - , , B , 9 >B9A8 5 F 9 >F9A Experimentation

* + * , + ; + , + + 0 , + +

© 2004 by CRC Press LLC

1036

Chapter 10

GRAPHS IN COMPUTER SCIENCE

+ % + FACTS

&3

#

5 , # , >#A >A

&

3 4 -+ , ; >;?!A, + + +

&3

>C?&A + 0 ,!( , + , < - = # 0 >CA

&

3 ; + 5 G , G, *0 >GG* A > A

&

3 " % , , , 00 >A

&3

# + 4+ -

0 5 , ; , ' >;'?$A

&3

BU *0 >B*?&A - -

&3 # + + >DC A

Fixed Parameter Tractability

', 0 >6?&A Æ DEFINITION

3 # W 4 # , $%& , + W . ./ / , 0, 0, ,

FACTS

& 3

1 F- , + + ,

.

© 2004 by CRC Press LLC

Section 10.3

1037

Drawings of Graphs

+/ ; + , > A 4 % + 4 +

& 3 @ + F- + + 6 >6; , 6; , 9A8 0 ? 1 , % + 1 5, + , ? + + 6 , 0 . ./ ¾ / ./ > A , + 4 , 4 6

( $$ 9+ 0' " " + :-" + !$ C#)) @6 '#;)

F-;#' '@C"*

1*" @*C"H1I

-

- 3 +

. ./ P /

-

- 3 -

%-

3 +

%-+

%-

3

3

- %

-

. ./ ¾ /

. .% / / . .% / ¾/

. ./ ¾ /

10.3.8 Sources and Related Material 5 + >"??, G" , )A ) ) D ! " # .+ $!, ?(, & , !&, !, ?, &, (?/

© 2004 by CRC Press LLC

1038

Chapter 10

GRAPHS IN COMPUTER SCIENCE

REMARKS

% 3 ) + + >CC?!, ), ?!, ;**, B*?&, '+?, ) ??, ))D?!, ? , ?, ??, DC A ) + + $ .+ $, , ??$/, " % & $ .+ ?, -, ??(/, ' ( & " .+ $, ??!/, ' $ & $ .+ , , ???8 + , , 8 + $, , 8 + $, , / %3 ) 999 . / 999 . /

References >## ?$A @ # 0 , 6 # , )-9 , F G , * , ' , I-6 H, , $ . ??$/, ?M!? >*??A , # , ; *7 , ' :

4 -, !?M$( . S??/ . B G +/, + & , ) -D , ??? > A ) ) , , 9 , C , ' , C D , - ,

$ ./, !M? >;'?$A 6 , * ; , ' , # - 0 , &$M(& . S?!/ . 6 B /, + &, ) -D , ??$ >G?&A * G , # -Æ , &M! " ) # . ")# S?&/, , + (, ) -D , ??& >CC?$A , 9 C , C , 0 , $ . ??$/, (M . , ' / > ?!A 6 B , , $M& . S?/, + (?, ) -D , ??! > A ) ' , 2 + ,

./, &M& >A ) ' , # , ?M . /, + ?(, ) -D ,

© 2004 by CRC Press LLC

Section 10.3

1039

Drawings of Graphs

>9?&A N 9 , # , $M& . S?&/ . /, + !, ) -D , ??& >9?(A N 9 , , (?M?( # 1 ) # . 1)## S?(/, + !, ) -D , ??( >?!A ' 6 , , ' , 1 , 3 , - , - ,

. ??!/, ?&M >"C'?&A ' 6 , " , C, 6 '5, - , & . ??&/, ??M( >?!A 1 6 0 # , Æ3

, M ! !" #$%& . '

1 /, + (?, ) -D , ??! >B9A * , ) , B B , ) G 9 6 ,

! ./, M >?$A * 5, * , ' , + , ?M( # #* ) , ??$ >)?&A # ) , -, -, - &. ¾ / + , !M$ . S?$/ . ) F /, + ?, ) -D , ??& >H A )-9 I-6 H, @ -5 , $ . /, !?M& >A , 9 , * , * 00 , , '" () ./, $!M ?( >"??A , " , ' , 1 , -; , ??? >6?&A ' * ' 6 , " *

,

), ) ,

??&

>6;· A D 7 +V, * 6 , * ; , * G, C , * , F F , ' , 6 ' , ) , ) 9, ' 9 , @ 0 , ((M?? " ) , # . , ")# /, + $ , ) -D , >6;· A D 7 +V, * 6 , * ; , * G, C , * , F F , ' , 6 ' , * ) , ) 9, ' 9 , # 4 - 0 , M !

© 2004 by CRC Press LLC

1040

Chapter 10

GRAPHS IN COMPUTER SCIENCE

. / . *0, * BU , ) C /, + $!, ) -D , >C?&A , # , C , # , ' , " , 6 D , C D , #

,

& . ??&/, M! >CA , # , C , # , ' , " , 6 D , C D , 3 # ,

./, $M $( >;?$A ' + ; , , ! . ??$/, M >? A * , ' 0 , . ??/, (M(&

"

>?A * , ,

! . ??/, !&!M$

>C?(A C , N 5 <

=, . S?(/ . ); 9/, , ) -D , ??( >CC?!A , 9 C , C , 3 +, (M? . S?/ . ' 1 /, + (?, ) -D , ??! >C?(A @ + , C , 6 , ' , 5 + + ,

. ??(/, (&M( >C9 A " , C , )G 9 , - , @ . S/, >C9A " , C , )G 9 , 5- - , , , ), N+ C , >*?$A * 5 * ; * , # . K/

, M " +, , ??$ >*9A D 7 +V, * , ' 9 , - , M! . , S/ . * ) G +/, + !(, ) -D , >)A B 0, B , * ) , # + , - ./, M!$

© 2004 by CRC Press LLC

Section 10.3

1041

Drawings of Graphs

>)?A * 9 ), - 0 , (&M? " .

, ?? >D?$A C D , # , ? . ??$/, ?M!?

>9A D 7 +V ) 9, # Æ 4

- 0 , . /, !( ./, ) -D , (M ? >9 A D 7 +V ' 9 , - - , 9 5 ) . 9 S/, , ) -D , >9A D 7 +V ' 9 , F , ' '--, ) ), N+ , @ , >" (A " , # ,

.

?(/, ?M $

>";A " * C ; , F + - ,

./, !&M ( >"C*)?!A " , 9 C , G *, G ) , C 7 , / $ . ??!/, (M >"9?$A " ) 9, 0 " F- , $ . ??$/, $M( .) , ' / >69 A ) 6 , C , ) 9 , ) , (M !" ,00+& . *0, * BU , ) C /, + $!, ) -D , >6'? A 6 " ' , - , '" () . ?? /, ?M $ >G A 7 , * , ) G +, # - - , M . / . B * 5/, + ?(, ) -D , >B&?A * ' ) B , " , 9 ; 6 , ?&?

1

>GFD?A " ' , " G 4 , ) F , G D , # % , ((( ( ? . ??/, M >FD((A " ' , ) F , G D # M # , '" () ( . ?((/, &M $

© 2004 by CRC Press LLC

1042

Chapter 10

GRAPHS IN COMPUTER SCIENCE

> A * , % , M$

!" 2 ,00+&, >?!A # ' , N ,

2

. ??!/, ?M

>D?$A # , ' , D , , M$ # " ) # , + $, ) -D , ??$ >;?!A * ; , + ", / $ . ??!/, !!M& .) D 0 , 1 6 0 " / >;**A 1 ; , * , * ) * , + 0 + + 0 3 # +, ((( /* $ ./, M >B*?$A * BU *0, * 3 , $ . ??$/ . , ' / >B*?&A * BU *0, - 0 3 ,

. ??&/, M! >B?A B 9 B 05 , ' + +, " ((( ( . ??/, !M ! & >G" A * G 9 ."/, ) -D ,

,

>G?A * G, ,

. ??/, M$ >G (A G 5 5, # , " . ?(/, &M (

>GG* A G , G G, *0, # , &M! . / . B * 5/, + ?(, >G)(A B G 5 B ) , 5 , M! " 3 , N ) , ?( >C?!A C ,

- , ?M! 9 5 # ) , + ?!!, ) -D , ??! >C"?!A C " , 1 + , &$M(& . S?/ . '

1 /, + (?, ) -D , ??!

© 2004 by CRC Press LLC

Section 10.3

1043

Drawings of Graphs

>C(&A # C , # , 1 ( . ?(&/, $$M$!( >CG?$A C+ G 0 , , " !& . ??$/, ?M ! >CG?(A C+ G 0 , # - + , . ??(/, (!M >CC?$A 9 C C , , " $ . ??$/, !M$ >CC?&A 9 C C , , ($M . S?$/ . ) F /, + ?, ) -D , ??& >CCA 9 C C , , & ./, $ M($ >CC*9?(A C , # C, ; *7 , ) ; 9,

: ,

. ??(/, M >C*A C ; *7 , D ,

./, &M &(

>C* A F C , B * 5, * , 1 + , M$ . /, + ?(, ) D , >C)?A # C F ) , * + , ?(M !" #$4&, ?? >**?&A B G * 5 * * , # , ( . ??&/, (?M >*)?A * ) ) , , ( . ??/, $!M?

>*)F) ??A G * , )0, ) FU , ) ) , * ), ' ), N , 5 + 4 ,

. ???/, (!M >*J&?A G * # C J , F , " ? . ?&?/, M >F9A ) F 9 , @ , M$ . / . *0, * BU , ) C /, + $!, ) -D ,

© 2004 by CRC Press LLC

1044

Chapter 10

GRAPHS IN COMPUTER SCIENCE

>@)?A # @5 , , G ) ,

/ , B 9 R ) , ??

>#A ; , B # # , , # N*C 3 N ,

$ ./, !!M&? >)?A 6 * 1 ) , ) -D , , ??

,

>)?&A # 5 , B ) , 1 , " + , & M($ . S?$/ . ) F /, + ?, ) -D , ??& >?(A # 5 1 , 1 + , -&. / . ??(/, (M

(((

>?&A B , , V , - , &M! . S?&/ . /, + !, ) -D , ??& >'+?A 1 '+ , ' , , , !?M 1 2 .1 ' ) , /, G # , ?? >'*?A ' ; *7 ,

, " . ??/, !M( >) ??A ) , , & . ???/, &!M

>))D?!A 6 ) 5, C # )0V5, 1 D S , , % 3 +, M . S?/ . ' 1 /, + (?, ) -D , ??! >)( A G ) , ) , * , * +

, ((( 1 )*- ./ . ?( /, ?M ! >)A G ) , (, 9 )4,

5

> (&A ' , @

, $ . ?(&/, M > ? A ' , - , 1 . ??/, &M! > ?A ' , , , ??

" (((

> ??A ' , #+ , & . ???/, !M!

© 2004 by CRC Press LLC

Section 10.3

Drawings of Graphs

1045

>((A ' , , , # , ((( 1 )*- (. / . ?((/, $ M&? >DC A C D , , C , ' , 6 D , " , '" () ./, !M ( >9IA # 9 , 6 I , I , , $M & # . 1# /, + &$&, ) -D , >9 A ' 9 , X , -, - , (M!? 6 ) ) .6)) S/, + !!$, ) -D ,

© 2004 by CRC Press LLC

1046

Chapter 10

GRAPHS IN COMPUTER SCIENCE

10.4 ALGORITHMS ON RECURSIVELY CONSTRUCTED GRAPHS

) ) $ ) ") $) # # # # ! # $ # '

) - - %- -;

Introduction 1 + 4

, - , - , , %- , -; 6 +, 4 , . + /, @ * + + # , , 4 & & + .!/8 ,

+ 6

*, & ,

, , , , , > A DEFINITIONS

3 # 4 . 4/

+ , .

/ " + +

4 + 4 +

+ '

3 " + 3 6 , ' +

7 Algorithm Design Strategy

"Æ + & 3 4 + 4 + 8

© 2004 by CRC Press LLC

Section 10.4

Algorithms on Recursively Constructed Graphs

1047

4 + # - + 4 %+ S %+ 0 , ./

#

%+ % 0 # 5 S + Æ 1 , (

. / + 9 ( T + & + & & , ( 3 , * + , + +

( REMARK

%

3 F + . / + , + .+ + + , , 5, - , / + 6 , + +

10.4.1 Algorithms on Trees DEFINITION

3 + # . / # . / C . #/ # . ½ #½/ . ¾ #¾/

5 7 ½ ¾ .#½ #¾/ # T #½ , . #/ 4 & ; + , 4 + #½ #¾ . #/ + + + + . / , #&&> A Maximum-Cardinality Independent Set in a Tree FACTS

&

3 # + #&&> A 9 ½ ¾

½ ¾ # ( T - #&&> A ( T - #&&> A ( T -

© 2004 by CRC Press LLC

1048

Chapter 10

GRAPHS IN COMPUTER SCIENCE

&3 Æ ½ ¾ 1 , (

( , + + ., / ½ ¾

REMARK

%3 + (, (, ( 5 + M -

+ . / 6 , + + . / ( ( 8 ( 4 # (, , , Algorithm 10.4.1: 5-+#+ ;" 4"" 9 (

3 T .) * / ,3 ( . 0 / 1 ) T

(

( " 1 T ½ ¾

( ½( P ¾ (

( ½( P ¾ ( ½( P ¾(

( ( (

, Maximum-Weight Independent Set in a Tree

; + ) + # 3 ( T - #&&> A ( T - #&&> A ( T -

© 2004 by CRC Press LLC

Section 10.4

Algorithms on Recursively Constructed Graphs

1049

Algorithm 10.4.2: 5-+#+ < 4"" 9 (

3 T .) * / ,3 ( . - / 1 ) T

( .#&&> A/

( " 1 T ½ ¾

( ½( P ¾(

( ½( P ¾ ( ½( P ¾(

( ( ( EXAMPLE

3 6 D ( ( ( +

+ #

0 $- , , + > ( ( ( ( ( ( A ( T !

( T $, +8 ½¿ ) 5 5 , +

+ , +

Figure 10.4.1 5- !" " +- "" $ REMARKS

%3 * + + + , ,

, + . > ?A, > A/ F > B G&(A F-

© 2004 by CRC Press LLC

1050

Chapter 10

GRAPHS IN COMPUTER SCIENCE

%

3 # + 6 , + % 0 , # , ,

%3

4 %+ % + + + 2 % + ,

T .) * / - ) . ) ./ - , . , /+ & & ) + / 6 - /- % / P %+ ( (0 3 0 T ( ( ( /, 3 ( /- 8 (/ - /-

/ 8 (0 3 0 T ( ( ( / - /- 0 8 ½(0 ¾(1 T 0 1 P 0 P 1 / F / . /

10.4.2 Algorithms on Series-Parallel Graphs "+ - + 4 ! 6 , DEFINITION

3 # 2 # . 2 #/

4 + 3

. / .½ ¾ / - . 2 #/ 2 T ½ # T ¾

# . ½ 2½ #½/ . ¾ 2¾ #¾/ - #½ 2¾ 2½ #¾

# . ½ 2½ #½/ . ¾ 2¾ #¾/ - 2½ 2¾ #½ #¾ 2½ #½

# ' . ½ 2½ #½/ . ¾ 2¾ #¾/ - #½ 2¾ 8 2½ #½

REMARK

%3

) - 0 , . /

, -, 4 - Maximum-Cardinality Independent Set in a Series-Parallel Graph

1 , + + , + , + 8 - , + 2 > A #0%> A

© 2004 by CRC Press LLC

Section 10.4

1051

Algorithms on Recursively Constructed Graphs

# 3

( T 2 > A #0%> A

( T 2 > A #0%> A

( T #0%> A 2 > A

( T 2 > A #0%> A

( T

Multiplication Tables for Series, Parallel, and Jacknife Operations

! "

!

"

!

"

!

"

!

"

!

"

Algorithm 10.4.3: 5- ;"$ 4"" 9 9 ' 2

3 - T .) * / ,3 ( .0 - / 1 * T > ( ( ( (A > A " 1 T ½ ¾

( ½( P ¾ ( ½( P ¾ (

( ½( P ¾ ( ½( P ¾(

( ½( P ¾ ( ½( P ¾(

( ½( P ¾( ½( P ¾( " 1 T ½ ¾

( ½( P ¾ (

( ½( P ¾(

( ½( P ¾ (

( ½ ( P ¾( " 1 T ½ ¾

( ½( P ¾( ¾(

( ½( P ¾( ¾(

( ½( P ¾( ¾(

( ½ ( P ¾( ¾(

( ( ( ( ( ) + + +, + +

© 2004 by CRC Press LLC

1052

Chapter 10

GRAPHS IN COMPUTER SCIENCE

EXAMPLE

3 # - + 6 D , 0 - + ( ( ( ( 6 , ( ( 1 4 , + ,, , + ,

Figure 10.4.2 5- !" / "" $

FACTS

&3 @ + - + %, , , , 1,

- + . >)(A/ &3 # - 8 . / REMARKS

% 3 6 6 , +

. $/ -

% 3 6 6 , - -

-4

%3 1 , 7 55 6 , -+ & ., + / 7 55 , , 7 55 + @ , , 7 55 + % 3 ) - # 1 ' 5 , , , # , &+ , >'(!A

© 2004 by CRC Press LLC

Section 10.4

Algorithms on Recursively Constructed Graphs

1053

10.4.3 Algorithms on Treewidth- Graphs DEFINITIONS

T .)* / + .3 0 4 / 3 0 4

) + 4 ¾ 3 T ) ./ * 0 4 3

01 4 1 0 , + 3 3 3 3 + - ¾ 3

3 5 + 3 #

3 #

FACT

&3 "+ - - >)(?A 9 . 3 / T . ½3½ / . ¾3¾ / 3 ) +

#&&> A, ½ ¾ + - + -

- , 3 P 6 , - 3 ) + #&&> A # , >' A T - ' 3 3 ' ( T -

5- ;"$ 4"$ 9 (" 2 3 - T .)* / ,3 ( .0 - / 1 3 T ) 6 ' 3 1 ' 7 +

>'A "

>'A ' " 1 . 3 / T . ½ 3½/ . ¾3¾ / 6 ' 3 1 ' 7 +

>'A "

>'A ½>'½A P ¾>'¾A '½ ' '¾ ' P ' 3 '½ 3½'¾ 3¾'½ 3 T ' 3½'¾ 3 T ' 3¾

( >'A 3 ' 3 Algorithm 10.4.4:

© 2004 by CRC Press LLC

1054

Chapter 10

GRAPHS IN COMPUTER SCIENCE

EXAMPLE

3 # - 6 - , (- +

>' A ' 3 ; + , + (- 0 ,

Figure 10.4.3

5-+#+ !" "" " $

REMARK

%

3 * + , , %, -+ . , 4 /, , L + - . > ?A/

Monadic Second-Order Logic Expressions for a Graph DEFINITION

)

C + + , , + . /,

* )

3

)

*

( ( )* T .)* / &

+

T , ) , * 1 5 *)@C . /, . 5/, . 5/

*)@C 1. /

*)@C

1 *)@C + , ./. / ./. / *)@C

© 2004 by CRC Press LLC

Section 10.4

Algorithms on Recursively Constructed Graphs

1055

EXAMPLE

3 ) *)@C X X X . X/ .X / T .+ / .1.+ , / 1.+ , // #7 .+ , + / .+ T + / . / .1.+ , / 1.+ , // MSOL-Expressible Graph Problems

* *)@C . ># C )? A, > ? A, > ?A/ ; + 1).D / .+/ .+ / ..+ D + D / #7 .+ , + // %.D/ .+/ .+ / ..+ D + D / #7 .+ , + // ).D/ .+ / .+ D .+ / .+ D #7 .+ , + // D .D , ( ( (, D / .+/ .+ D ( ( ( + D / 1).D / ( ( ( 1).D /

* ." / . / . / .. " " . T // .+/ .1.+ , / 1.+ , /// ." / .D / .D / . .+ / .+ D / .+ / .+ D / .+ / . .+ D / .+ D // . / .+ / .+ / . " + D + D 1.+ , / 1.+ , /// ; ."/ ." / .+ / . / . / . " " . T / 1.+ , / 1.+ , / . / .. " 1.+ , // . T T /// ; ."/ ." / .+ / . / . / . " " 1.+ , / 1.+ , / . / .. " 1.+ , // . T T // .+ / . / . / .. " 1.+ , // T // FACTS

&3 "+ *)@C- + - ># C )? A, > ?A, > ?A * + , +

*)@C , , , , - - , 5 , & 3 @ *)@C, - > ?A

& 3 .! / - + , + - + P

© 2004 by CRC Press LLC

1056

Chapter 10

GRAPHS IN COMPUTER SCIENCE

&

3 6 , *)@C - 1 + -

+ *)@C > ?A, + - + -0 . ' 5 8 > ?!A

> A/ REMARK

%3

- , - , - , *

6 $ 6 , - + > A , - , -

Two Open Problems

- % 4 , 6 & $& Æ %+ - , + >G A , . Æ

+ *)@C

, - + *)@C

- REMARKS

%3

+ - , 5 + + -0 . > A/

%3

# - + ; , - , - , - , - , - . >9(&A, >9;((A/

10.4.4 Algorithms on Cographs @ - ; , - - , + DEFINITIONS

3

# + 7 + .!/

© 2004 by CRC Press LLC

Section 10.4

1057

Algorithms on Recursively Constructed Graphs

3 + 7 + .! /

3

# ' + + + ' 7 + ' .?/

3 #

3

#

+ 4 + 3

. / # + 1

½

¾

, 7

½

¾

1

½ ¾ , - ½ ¾ ,

5 ½ ¾ .½ ¾/, ½ ½ ¾ ¾

Algorithm 10.4.5:

5-+#+ ;" 4"" 9 ;

3 T .) * / 3 (0 . 0

,

/

1 ) T

(0

"

1

T ½ ¾

(0

"

1

½ (0

P ¾ (0

T ½ ¾

½(0 ¾(0

(0

Three More Algorithms for Cographs

# $ 5 Algorithm 10.4.6:

+*

5-+#+ ;=# ; +! 0#+ ;

3 T .) * / 3 ( . % > A

,

1 ) T

(

"

1

T ½ ¾

½( ¾(

(

"

1

T ½ ¾

(

© 2004 by CRC Press LLC

½ (

P ¾(

/

1058

Chapter 10

GRAPHS IN COMPUTER SCIENCE

5+#+ ;" + 9 ; 3 T .)* / ,3 ( . 0 / 1 ) T

( " 1 T

( ( P ( " 1 T

( ( ( Algorithm 10.4.7:

5-+#+ ;" 5! ; 3 T .)* / ,3 ( . 0 / 1 ) T

( " 1 T

( ( P ( " 1 T

( ( P ) ( P ) .) P )/ Algorithm 10.4.8:

REMARKS

%3

- 4 # ( + Æ ,

½ ¾

P ( .) / P ( .) /

%3

+ 9 + , %, + # ! & ; + , + + . + / 1+, , 2 EXAMPLE

3

# ! ( 6 , - +

0 ,

(0 ( (

© 2004 by CRC Press LLC

(

Section 10.4

1059

Algorithms on Recursively Constructed Graphs

%

%

% 0 !, + 0 , 0 , . / . / . / . /

4# 3 + $$ # $$ $

Figure 10.4.4

10.4.5 Algorithms on Cliquewidth-k Graphs DEFINITION

(((

3 C >5A 4 + 3

#

+

) . / T 2./ >A %-5 1 ½ ¾ %- 01 >A, . / 7 ½ ¾ %- ./ . ½/ %- , . ½/ ½ .½ ¾/ 2.½ / T 0 2.¾ / T 1 ./ . ½/ %- , . ½/ ½ + 0 1

. / #

7

Maximum-Cardinality Independent Set of a Cliquewidth- Graph

# ? 3

>'A T - ( T -

© 2004 by CRC Press LLC

' >A

1060

Chapter 10

GRAPHS IN COMPUTER SCIENCE

Algorithm 10.4.9: 5- ;"$ 4"$ 9 ;=#" 2

)*

3 %- T . / ,3 .0 - 1 T 6 > A 1 T . / > A " > A " 1 T ½ ¾ 6 > A > A ½> A P ¾> A " 1 T . ½/ 6 > A > A ½> A ½> A " 1 T . ½/ 6 > A 1 > A ½> A " > A ½> A

> A 3 > A

( )

' ' 2

'

'

'

' ' '

'

'

' 0 '

' 1 '

' ' 0

' ' 0

(

' '

/

1

EXAMPLE

3 # ? %- 6

! " (- >!A,

> A, >A, >A, > A, > A, > A, > A 0 , +

Figure 10.4.5 5-+#+ !" "" !=#" $

© 2004 by CRC Press LLC

Section 10.4

Algorithms on Recursively Constructed Graphs

1061

A Subset of the MSOL Expressions for a Graph

* + , , %, -+ . 4 / + %- , + 5 *)@C DEFINITION

3

( )* T .) * / *)@C + ) , * , ) *)@C + T , 1. , /, ) *)@C ., , / % 4 ., / , *)@C *)@C - + * + * - + FACT

& 3

"+ *)@C - +

%- > * ' A, + . / +

+ *)@C + + , , , @ *)@C , -

REMARKS

% 3

@ + *)@C + 1).)½ /, %.)½/, ).)½/, D .)½ , ( ( (, D /, *)@C ; + , *)@C + * .*½ /, .*½ /, ; .*½/, ; .*½/ *)@C

%

3 ) + *)@C , 5 + %- ; + , + , + + -0 . > A/

%

3 # %- + -FC >9 ?A

10.4.6 Algorithms on -HB Graphs DEFINITION

3 ,- . ·¾ /- - %-. P / .# > B ' )A/

© 2004 by CRC Press LLC

1062

Chapter 10

GRAPHS IN COMPUTER SCIENCE

FACTS

&3 "+ -; +

T ; , ) L ) , 6 T .) ,* / ) T 7 7 * 7 7 , 7 , 7 , %3 ) ) %.) / 7 , . / * 2 .%./ %.// *

) )

&3 -; - -; , . - / . -/ " S + . / , . / + # , . ) / , ) > B ' )A Maximum-Cardinality Independent Set in a -HB Graph

# 3

>' A T - + ' )

(0 ! T -

Algorithm 10.4.10: 5- ;" 4"" 9

>? 2

3 -; T .) * / ,3 (0 ! .0 - / 1 ) T

>' A ' " 1 T

>' A > A P >- A 3 3 %.) / 8 %.) /, .3 8 / * T ! T ' % .3 / - T ' % .8 /

(0 ! T >) A

EXAMPLE

3 9 # -; 6 $ F T , , , 6 , % -+ 0 0 ,

# , REMARK

% 3 - % -+ + -; ; + , , , 5 + -; * + -; ,

5 Æ

© 2004 by CRC Press LLC

Section 10.4

Algorithms on Recursively Constructed Graphs

Figure 10.4.6

1063

5-+#+ !" "" >? $

A Subset of the MSOL Expressions

* 5 + -; *)@C DEFINITION

3

*)@C T .) * / *)@C + ) , + ) *)@C + #7 . / ) *)@C ., , / % 4 ., / ; + , + 8 + *)@C

.) / ( ( ( .) / .. /9 .

) ( ( ( ) / . /. / .#7 . / 9 . ) ) // . /. /. #7 . / 9 . ) ) ///

REMARK

%3

1 4 ? +, 9, 9 , 9 + , , 1

,

© 2004 by CRC Press LLC

1064

Chapter 10

GRAPHS IN COMPUTER SCIENCE

EXAMPLE

3 *)@C 1), %, D %+ *)@C ; + , *)@C ) *)@C

.) /. /. / .#7 . / . ) ) // % .) /. /. /. #7 . / . ) ) // D .) / ( ( ( .) /.. /. ) ( ( ( ) / . /. / .#7 . / . ) ) /// 1)

FACT

&3 "+ *)@C - +

-; > B ' )A + %- , + + @ *)@C , - +

References ># (!A ) # , "Æ Y +, 6 ! . ?(!/, M ># )?A ) # , , # 5 5, ), # , . ??/, M $ ># C )? A ) # , B C , ), " - , . ?? /, (M ># (?A ) # , # 5 5, C F- 5- ,

. ?(?/, M >C 9 (&A * 9 , " C C , # C 9 , C , ( . ?(&/, $M! > (&A ; C , -, , * 1 , ?(&8 " " . ?((/ > ?!A ' , - 0 - , . ??!/, M & > B ' )A ' , B C B , D ' + , B ) , ' %- 5 , , > ? A ' , ' 5 , # +, # +

, 2 7 . ?? /, &M ?

© 2004 by CRC Press LLC

Section 10.4

1065

Algorithms on Recursively Constructed Graphs

> ?A ' , ' 5 , # +, # +

, & . ??/, !!!M!( > A ' , ' 5 , # +, ) + + , ' , ), N+ # , > C)??A # , D C, B ) , , ???

-, )1#*

> C( A , ; C , C ) , ,

. ?( /, $M & > )(!A , I , C G ) , # , . ?(!/, ?$M? > ?A , - 13 ' 0

4 , (! . ??/, M&! > * ' A , B # * 5 5, N ' , C + 0 % , ./, !M ! > * ?A , * * , * - + - , ? . ??/, ?M( > @A , ) @ , N %- ,

./, &&M

>?&A 6 , # , , N+ N , ??& >"$!- A B " , , , : , . ?$!/, ?M$&

>"$!-A B " , * , + , 6 $? . ?$!/, !M >" ((A " "-* , , 5- , $ . ?((/, !M ?

&

7

>"9 A 9 " , 6 5, " 9 5, ; + - %- , &M ( 8 ,00+9 . / > B G&(A * , ' , B , G, 0 ,

. ?&(/, &&M ?!

© 2004 by CRC Press LLC

1066

Chapter 10

GRAPHS IN COMPUTER SCIENCE

>; ;C 9(&A " ; , ) ;, ' C 5 , G , 9 , C 0 - , ) . ?(&/, &M!& >B @?!A B , ) @ , C 0 - ,

$ . ??!/, !!M &! >G A 1 G , - 5 - + , , >'(!A * ', 0 - 3

, , 1 , ?(! >)(&A )Z , C - F- 5- , '-*#;-L(&, :, , ?(& >)(?A )Z , + * U G 0

, , #5 9 ', ?(? >))($A )Z , ), - -

F- , " 6 ) 5 , ?($ >))((A )Z , ), # - , (;72 (&8 &?M( &( . ?((/ >)?&A B ) ,

7 , 5 .

??&/

>)(A * ) , F- ) - 3 +,

" $ 8< . ?(/, M!

> F) (A G 5 0 , F05, F ) , C - - , ? . ?(/, $M$ >9 ?A " 9 5, 5-FC ,

! ! M$$ . ??/8 + - 6 5

>9(&A D 9 , C 5- + , , N+ , ?(& >9;((A D 9 , ) ;, 5- +

, $ . ?((/, $ M &$

>9;C (!A D 9 , ) ;, ' C 5 , # , ! . ?(!/, M$

© 2004 by CRC Press LLC

1067

Chapter 10 Glossary

GLOSSARY FOR CHAPTER 10 !

M +

3 +

%

,

M +

# #

3 +

M 3 ,

, + :

-+ +

M 0 3 - -

4 0 :

/ "3

+

!# 3

! M

3

0 + +

!7 "½

M - 3

!7 "¾

M - 3 -

" "3

/3

+ &+/

!!" !+

M 3

!!" 3

" : ! /3 " : 3

4

+

+

+ -4

"½

M 3 , , +

"¾

M 3 + -

"¿

M : #3 . ,/ :

+ #,-

" 3

" !+ ?;

M 3

" 3

+

" # 1 +3

-

!:!

M 3

! +! "- M 3

+ 2

© 2004 by CRC Press LLC

1068

Chapter 10

GRAPHS IN COMPUTER SCIENCE

! +! #+ M 3 + 7 + + 2 3 + 7 8 % M 4 + . >A ( ( ( /3

!=#

!=#"

# ) . / T 2./ >A %-5 1 ½ ¾ %- 0 1 >A, . / 7 ½ ¾ %- ./ . ½/ %- , . ½/ ½ .½ ¾/ 2.½ / T 0 2.¾ / T 1 ./ . ½/ %- , . ½/ ½ + 0 1 M 3 , 7 + -4 M 3 % , + +

!#

!# ! =#

! M 4 +3

# + 1 ½ ¾ , 7 ½ ¾ 1 ½ ¾ , - ½ ¾ ,

5 ½ ¾ .½ ¾ / ½ ½ ¾ ¾ 3 -

+ M &+ 3 7 + 3 3 + 3 + + 3 < =, , . / 3 3 -

, +

+ + 3 + + + ½ M 3 -4 , ¾ M 3 -4 + + , 7 2

!- " ! " ! !#" !# "

# " # # " : ! "/

" : / " : /

© 2004 by CRC Press LLC

1069

Chapter 10 Glossary

" : "/ M 3

+ + + -4

"!" M + 3 + %

, M + 3 + "/3 + &+/ "+ M 3 + "! / -3 + 4 "! " M 3 +

) , + + , ,

"! M + + 3 - "+! "3 ,

+ + ./ ./ ./ +

./ ./

./,

"+ M 3 + + +

"

"+ / - M : 3 + + , T +

#,- "+ M : 3 "+! +3 Æ % 4 , !!3 % , + , "!+3 , /#3

+ , !+3 , ++#+ 3

, , - , 3 , , "+! 3 % 8 , + + "+! ++3 + + + ;993 -- -- , +3 -")) ( M 3 + % + " # # / 3 , 5 + , 5 +

© 2004 by CRC Press LLC

1070

Chapter 10

GRAPHS IN COMPUTER SCIENCE

/! / "3 4 : + -3 + + : + " M 3 +

) , 4 , ,

6 3 +

+

# +

/" " M - 3

" "3 + , , +

+ !! M 3 + +

+ M 3 + +

! M 3

% , +

"3 -

0 +

+""" " " +3 ./ 4

.,

/ +

0&.,/ M + , : 3 % + ,

++" "+

+

"" M 3 + - 7

; 3 ' ) .'/ ) . / ) .; / , + ,, ; , , '

! + M

- M 3 + 3 + + "#! 57 ! 3 * 5 +

-

/ ! M 3 +

+ +

" "3 +

0

M 3 + ,

/ / -

/#! M 3 ,

+

+! 1 /! +! M 3 +

+

+! M 3 +-" 3

© 2004 by CRC Press LLC

1071

Chapter 10 Glossary

+"! !" " ! 59.@3 +

+ .+ / .+ /8 4

, 59.@ M T .) * /3 *)@C + ) , * , ) "!+ M 3

( ( ( 0 , 7 + . /

"" 3 + M 3 % "3 0 +

3

+

# 3 + +

3 +

+

"3 "3 " M 3 4- 4

#/+ M

3 +

"! !!##3 +

+ ., , /, % 4 ., /

" M 3 + 4

-+ "3 !# !#!" !3 4 . 4/ +

,

8 + + 4 + 4 +

#

% 8 , + , "T #, . / +

"#! 6 3 : +

" ! M 3 +

! "3 5 +

! 3 . -/ +

; 3 ./ ; ., ; 7 + ; /8 ./

) .; /, 7 ;

+ M

© 2004 by CRC Press LLC

1072

Chapter 10

GRAPHS IN COMPUTER SCIENCE

+"+3 , 4

-4

3 + +

M + 4 3 4 ! + +3 + +

7 %

/ &

M 3

+

#

#

-

7 M 3 + #! M 3 + 3

.) /

:! ! =#3 % ,

,

.) * /

. .) ) //

-3 + : "3 - !+ 9; M 3

+

!+ 9; M 3 -

+ + 8

&

!!" " 3 + + +

+

M 3

,

! #+ ! "1 ! M -

3 +

3 ,

#+3 + 7

+-" 3 + + +

3

+

, " ++#+ 3 ¾ . ¿/ . " /

Ê

Ê

, "1 !# ":"3 + # . /8 7 7 "3 " M

3 5 + -

8

© 2004 by CRC Press LLC

1073

Chapter 10 Glossary

" 3

!!" 3

+

+

#/+

M

3

+

#" 3

+

+ ,

#" "3

-

+

#" " 3 - ! M +

3 + +

"3

+ 8 ,

+ 0 , + +

A "+3

© 2004 by CRC Press LLC

Chapter

11

NETWORKS and FLOWS 11.1

MAXIMUM FLOWS

11.2

MINIMUM COST FLOWS

11.3

MATCHINGS and ASSIGNMENTS

11.4

COMMUNICATION NETWORK DESIGN MODELS

!" # ! $ # % GLOSSARY

© 2004 by CRC Press LLC

11.1

MAXIMUM FLOWS

! " # $ % " &"

Introduction & ' ! " ' '

( ) ! * " ' * " '' ! + ! !

,- ' + ! ! ,- . + ! ! ' ' ' + ! ! " ) * '' ' ! !- " '" / " + + ! ( , 0- ! 1 2 134 " ! " ''

1 * ! / *! " + 25 364- 2%*734- 28794- 2564- 264- 2:3;4

11.1.1 The Basic Maximum Flow Problem )" - + ' - < ' " ' , - ! " < ! ' " DEFINITIONS

=

> ? @ !' * - ! * - - ! * =

A ' * - " ! " ? @ * ? @ ?5 @

=

' " = ? @ ? @- " ? @

? @ > ? @ " * ? @ ;- " ? @

© 2004 by CRC Press LLC

" =

= -

-

? @ ? @- + ,-

? @ >

? @

=

) - ! * + , > ? @ / + " * EXAMPLE

=

' " + , '' ! + " , '' ! ' - + '' ' ) !" * " '' " + /

Figure 11.1.1

Figure 11.1.2 FACTS

=

+ * '' ' * " + . + " - -

? @ >

? @ >

? @

= ! + / * " - !* + ? ' " !*@

11.1.2 Minimum Cuts and Duality '

© 2004 by CRC Press LLC

( + "

Cuts in a Network

' " " ! ? @

( $ ? 9@ ? @

DEFINITIONS

=

8 > ? @ ,- " ' " , " * ' ' " , ? @ " " - ? @ "

=

- - " ' " " " " - -

>

=

? @

+ , " *-

=

EXAMPLES

=

> ? @ ? @ '' !

' " > ? @ B ? @ > C ? @ , ' ' "

Figure 11.1.3

=

! > ? @ ? @ " !#!$

" ' ; '' !

Figure 11.1.4

© 2004 by CRC Press LLC

! !#!$

Weak Duality DEFINITION

%

= : * + - - " + " " + , - > ? @ ? @

FACTS

=

* ' " ! * +

= ?) "

@ 8

- ? @ >

+ , -

= 8 + , - '' ? @ > + + , - ? @

?

. " , 2 4@ EXAMPLE

=

+ ? '@ ! # ) ' - ? @ > 9- + - - . ? @ B ? @ ? @ > B # > 9

Figure 11.1.5 ? @ > > 9

11.1.3 Max-Flow Min-Cut Theorem ' + ' -

' - ' " &$

' ( ' ' DE !F ?5 @- ( " * * ' !' . ( " !- !F ?5 @-

© 2004 by CRC Press LLC

* * " !' " G 0 ' The Residual Network and Flow-Augmenting Paths

! * + , * *'$ ( " !'- '"- " / ! ' DEFINITIONS

&=

8 + , > ? @ ? @ " ? @ ? @ " ? @ ;

8

" - " ?1" - !- > @

=

: * + , > ? @- > ? @ * - " , "= " ? @ - " ? @ - ? @ - ? @ > ? @ ? @H " ? @ -

? @ - ? @ > ? @

=

: * + , > ? @- " , ' , - I - ! * I >

? @

REMARK

'=

) " " / ' I ' *

FACTS

= 2 ! 4 8 + , > ? @- +! ! ' ' I 8 / "=

? @ B I

? @ > ? @ I

? @

" ? @ ? @ " ? @ ? @ ? @

" + , ? @ > ? @ B I

= 2 ( " 4 8 + ,

+ " " ! ! '

% ( ( )

= 2##- %5#9- #94 ! * ,- * " + . ' "

© 2004 by CRC Press LLC

EXAMPLES

=

+ , ' ! 9 * 3 ' ! ,

Figure 11.1.6

* " "*

' " ,- ' ! < +! ! ' " ! + 1 " '

, ? @ ' ' ? @ , C I >

=

)" + ! * ! 9 * ! +! ! ' % ' #- ! + ! 7 1* - ? @- " +! ! ' +

Figure 11.1.7

+! ", # -$

' '' " # + !" " +! ! ' ' !

11.1.4 Algorithms for Maximum Flow ) ' ' + ' '' '' / +! ! '- ( $ - ( A / +

© 2004 by CRC Press LLC

Ford-Fulkerson Algorithm

/ ' + ! - , 294-

= * ' + ! +! ! ' " , Algorithm 11.1.1:

* " ,

( = + , > ? @ + ( = +

) ( ? @ > ; " ? @ , A ! ! ' 8 ! ! ' ' I

1 + ! +! ! ' ? 9@ =>

J' ,

Æ " , ! '

! ! ' - "

' )" ! ! ' - ! ' 2K74 L*- " ' ' ! / ! - % D' 2%D74- ( 2 7;4- ! ! '- ! " ' / " ' ! ? @ " ' % . / ' " + * - ? @ - ! ? @ " '* ' 2564 ! ! + * ' 2D74 ? , , ! +@ " ! ! '' :! & 2:&364

Preflow-Push Algorithms

* '' ' ! +- ( *'$( ! :! 2:674 :! 0 2:664 ) ( $ !- ' + * * - ' + M

N * - ! & * ( ' " * * ' ! + ! M N , ' ' ,' , " ! DEFINITIONS

=

* " +- " = ' " =

? @

? @- " ? @

? @ ? @ ; " * ? @ ;- " ? @

© 2004 by CRC Press LLC

=

8 '+ , > ? @ * - ? @- ! * ? @ > * ? @ ? @ ? @ ; *

: * '+ - " - - - / " + ?/ 3@ 5 - , > ? @ / " ?/ ;@

=

8 '+ , > ? @

/= ?@ > -

=

?@ > ;? @

? @ B " ? @ !'

=

! * '+ " - ? @ , " ? @ > ? @ B Algorithm 11.1.2:

. ." ,

( = + , > ? @ + ( = +

) (

? @ => ; " ? @ ? @ => ; " ?@ => ? @ => ? @ " ? @ ' ,

A * * 5 * * )" ? @ + ? @ I => ? @ ? @ ? @ => ? @ I ? @ => ? @ B I )" ? @ ? @ => ? @ B I % ? @ => ? @ I % & ? @ => ? @ = ? @ ? @ ; J' ,

@ ! ' " " ' " " ! ! " ? @ ? !? @@ '+' !

?

2:664 " '+' ! ! " D !- & 0 2D &34

© 2004 by CRC Press LLC

) ' ! ' " '+' !

" ! ' " ! ! ' ! ! ' ! ' ! - , / " !' '" ' ' 5 - (

- . , / * , " - ' 2:374

11.1.5 Variants and Extensions of Maximum Flow A + * " + ' * *!- - !- 2 13- %* 34 FACTS

&

= * " ? @ * ? @ " ! + !

*- ? @ * ? @ ? @

' ? @- ? @ * ? @ ? @ ' ? @

= 2 ' ' ,4 5'' * + , ' ' ,- A / + , " ! " A ' ' , - ? @ ? @ > " - ? @ ? @ > " , + , ' + " ' ' , '

=

+ / ! ' !' ? ! 5 @

=

@

)" - $ *' ( ? 5

=

2+ !4 , ! ! ? @ ' ? @ ! ' ? @ " + ) +- + - + '' ! '- ! + ? @ > ? @ > . * ! ? @ > + ? @ A " * ! '' ' ? @

=

2 +4 , + * ) ' / + !'- '* ! ?- !- 2 13- %* 34@ Multicommodity Flow

' ' " + "

'

© 2004 by CRC Press LLC

DEFINITIONS

=

-

=

' ? @ - ,- " +

> ? @ !' * -

- ! * " =

A ' * " ? @ *

? @

! !

%=

" ! >

+ , > ? " ! " ! =

" =

? @

@

? @- " ? @

? @ > ? @ " > ! ? @ ? @ > " > ! ? @ ;- " ? @ >

!

Variants of Multicommodity Flow Problems DEFINITIONS

&=

) - ! *

+ , - , "

+ A +

=

)

- ! *

+ , - ' * " " "

+ , ' "

=

) - ! *

+ ,- ' "

- ! * A /- "

- + " * ? @ ? @ ? @ (

)" . + ! ?* !

' !@- *

+ '

* ' * '!

! Æ ! ' '' ' 2853O3#- :D364

)" . + !- *

' C ! '' * " ' ' ?5 2$334 " *@ C !

+ ! ( , $( (

=

) - *

' * " ' C-

" '' ! " ! ' , * 2D39- D537- ::364

© 2004 by CRC Press LLC

References 2 134 & D

0- 8 ! - P 1 -

- L- 33

25 364 AP ,- AL ! - A& ,- - P A 5- C O,- 336

5 0*-

2:374 $ ,, $ :!- 1 ' ! '

" + ' - 3?@ ?337@- 3;G; 2 7;4 % - ! " " ' " + , ' - !" #" ?37;@- 77G6; 2 ::364 O (- C :!- : - 1 ! ' + ' - ) $ % &' ( ( ( % ( ?336@- 3;G33 2%D74 P % & D'- '* !

Æ " , + ' - )( % ! 3 ?37@- 6G9 2%5#94 % - - % 5- C + ! ,- *+, *% *-. 3 ?3#9@- 7G3 2%*734 5 %*- / - ' 5 - 373 2%* 34 % / - P & %* % ,- ,,- 33 2#94 8 & - P & ,- + ! ,- )" % !" 6 ?3#9@- 33G; 294 8 & - P & ,- - J * - 39 2##4 & , : ( !- ' " + ,- + 0 1( ?3##@- 77G6 2:D364 C :! P D - ' ! "

+ " ' , ! ' - ) $ % &' ( ( ( % ( ?336@- ;;G;3 Q - & % 0- C, + ! 2:3;4 $ :!- % ) D- 8Q 8*Q (- L P E - 5 0*- - $ 20 *-0(- '' ;G9- 5' !$!- 33; 2:674 $ :!- ,Æ % 3( ( - )- !- - P 367 2:&364 $ :! 5 &- + ' )( % ! # ?336@- 76G737

© 2004 by CRC Press LLC

2:664 $ :! & % 0- '' + ' - )( % ! # ?366@- 3G3; 2D74 $ D(*- ! + , " '+- !" #" # ?37@- G7 2D &34 $ D !- 5 &- & % 0- " + ! - )( % 7 ?33@- 7G7 2D394 P D !- 5 ! ' +- ) $ % &4 ( ( ( % ( ?339@- 96G77 2D5374 5 : D ' 5 - ) '* '' ! " ' + ' - ) $ % &5 ( ( ( % ( ?337@- 9G9 28794 % 8 8- ! - L- & A - 379 Q - 2853#4 8 !- ,- 5 , - 5 - % 5 !- '' ! "

+ ' )( % ( #; ?33#@- 6G 2564 L ' D 5 ! (- - L- 36 2564 5 & % 0- " - )( % ( 9 ?36@- 9G3 2$;4 $ $( - - 5' !$!- ;; 2O3#4 C O!- & ( ! * ! '! ) $ % 6 !- *! ( # ?33#@- 7;G76 2K74 C K- %Æ " % D' ! " ' !

+- )( % ! 3 ?37@- 6G3

© 2004 by CRC Press LLC

11.2

MINIMUM COST FLOWS

/ 1' ! " # % " &"

Introduction + '" " , + ! '' - - + ! " , ' '

! ' ' - " !-

- !' !- - !- ' - ! ! '

+ ! ( , ' " ' ' = Æ ! / ! +- ! ( " !

+ ' C , MC, N 0- ! - 1 '* !

*! " +- '' - ' 2 134 1 * ! / *! " + 2 5 36-:3;-) 5;;-5 ;4

11.2.1 The Basic Model and Definitions DEFINITIONS

=

> ? # $@ !' * - #- ! * ' " = #

- " =# % - ! '' * $ = % / $? @ > ;

=

? @ + , > ? # $@ ! * $? @ > ; " $?@ > $?@ ;

=

> ? # @ " > ? # $@ , * > - # > # ? @ $? @ ; ? @ $? @ ; ' " /

© 2004 by CRC Press LLC

? @ ? @ > $? @ $? @

" ? @ " > " >

#

=

=

' #

% /

+ , * /

?

@

" =

? @ " ? @ # = ? @ ; " ? @ #2 ? @ ? @4 > $? @ " 7

= = ? @

=

+ + , > ? # $@ * $?@

= %=

+ * ! +

+ " '' * $ * !

5 - ' - ?& @- ?& @- ?& @- $? @- ' *

$ "

REMARK

'=

) / *- / " ' " ' / ! ? @ > ; " ? @ # < " ! + " ' * " ' + " * " ' ! ' , + * ! * * " * ' '' !

EXAMPLE

=

' " + , '' " ! '' , ) ' " ' " + , /! ! (1,2)

c

2

(2,3)

[2] a

e [−4] (2,1)

c

2

[2] a

e [−4] 2

(3,2) (1,2)

(4,3) (2,4)

d

(1,3)

(2,3)

2 f [−3]

[5] b

d 3

[5] b

# * "-

Figure 11.2.1

" ! ! =

f [−3]

?@ ' > ( > # ? " * - ' *- ,@ ?@ ) > ?@ ?! @ ? @ * > ?@ ?@ ?! / ' @ ?@ + > $? @ ?! ! " '' 2 4@

© 2004 by CRC Press LLC

?@ , ?"@ - >

?@ ?@

#- ?, @ > ?@

?!@ ?' ( ) @ ' " ' ! ! ' " / * , ' * ( ! ? @ ) ASSUMPTIONS

= =

' '' ' "

C ! *

FACTS

= =

. - '' " +

' " ! " * *- * ' '' ? @ ! " ' ! ? @ ? " @ ?" @ ? " @

' ? @ ?" @ ; / '

=

' " ! )" ? @ # ! * * = " " - $ ! ? @ > ? @- ? @ > ? @- $? @ > $? @ ? @- $? @ > $? @ B ? @ )" ! ' + * " ? @ ? @ * " ? @ > ? @ ? @

=

+ / !' ' ! '' " ' - ' * !' )" + !'- + !' "= " ' ? @- ? @ > ; ? @ ? @

=

?" @ + + , " / ! + + , ?5 ' ' , #@

= + ' . * ' , ! / ' ? @ ()

= " ! !' ' ( ' ' " + * - * =

" $ > . > $ $ > ; " - / +

( . J ! $ - - # # ? @ - > - > -

> ? # $@ + ! , ?5 5 @ =

" ! * * - / ' !

! " * J ! $ > - $ > - - -

+ !' ' , ! ' * + ?5 5 ;@

© 2004 by CRC Press LLC

! = ! * * - " - ! !- / ! ' " J ! $ > ! > $$ > ; - - ! *- +

' , ! ' * + ?5 5 - !F @ =

! * ' !' > ? #@ ! - / ! " ' J ! $ > - $ > - > - -

+ ! ! , ! ' * + ?5 5 @

=

+ ' ,

Residual Networks DEFINITIONS

&=

: * + , - , > # + > # ; C , + " ? ' @ - - , " > ? @ , ? @ - " > ? @ + ? @ ' " #

=

'

=

# " " ' * ' " ! !

+ , ! & # "= " " ? @ - ? @ > ? @ B &- " , ? @ ? @ > ? @ &

= #

' "

" ! . .

FACT

%=

8 + B / ? B @? @ > ? @ B ? @ ? @ +

EXAMPLE

=

! , " + '' ! ! (−1,2)

c

a

(−2,2) (2,1)

(2,1)

e

(−3,2)

(4,3)

(1,2) (2,2)

d

(1,3)

(−2,2) b

Figure 11.2.2

© 2004 by CRC Press LLC

(−2,3)

f

"* ,

11.2.2 Optimality Conditions DEFINITION

=

: * ! / = B / /

% -

* / =>

FACTS

&= )" +- ! * = )" +- " ,

=

+ " "

)" ! * - / = % ' " * '' ! 0,F ! ? 5 ;@ / ; " ? @

=

+ ! /- *

? @ ? @ > ? @ "

? @ ? @ > ? @ B / / " ' ? @ > B / $ ( " " (

= ?& ' @ )" ! / = % ; " ? @ - + = ? ' 5 ,@ )" ! / = % ; > ; > ; > ; ; + ) - /

REMARKS

'=

' " ' = "

? @ " '' ! " / ' " - " ? @ ! '' ! - ! )" ! *- !-

)" ' *- !- +

' " - * / "

'=

"

EXAMPLE

=

! ! ! ! * , '' ! ! 1 " " ! + !

© 2004 by CRC Press LLC

- ! !' , ! + "

+ , ! e (−3,2) (1,2) d

(1,3) f

, ! " !$!

Figure 11.2.3 2

c

2

[2] a

0

(−1,2)

3 c

a

e [−4]

(2,1) (2,1)

2 2

d

[5] b

Figure 11.2.4

(4,3) (2,2)

2

3

0

b

5 e

(3,2) 2 d

(−2,2) f [−3]

(−2,2)

(−1,2) (−1,2) (1,1)

(−2,3)

f

3

! " / "* / * * - "

A Basic Cycle-Canceling Algorithm

" " ! ' ! " / ! + 3- - Algorithm 11.2.1:

0"! )$! )! ,

( = + , + ( = +

" + ' , A ! * ! * ! + ! " ) '

FACTS

=

)" !- " + !- ! / ! + ! )" !- ! " +

=

C! * " ! ! " 2#6-#94 ?5 " !! @

=

+ " " ! * / ' ) - ! ! / !

+ " - !

© 2004 by CRC Press LLC

?@ K 2K74 " " '

! ! . ?* @ ! ?@ / ! ! ! " 0 " + - / + " ?( !?')* @@ ! 2634 ? @

/ ! ! $ ?

( ? @ @ ' / + " ?(' !?') @@ ! 2:634

REMARK

'=

" " - ' ! '' " 2) 5;;4

11.2.3 The Dual Problem .- ' * ! + . * + '

* - ! ( ! ' ' ( "

+ ' ' " ' DEFINITIONS

=

" + ' ?@ / + , (

" B

50 B /

=

/ B " ; " ;

' " * " /

FACTS

/ $

" ? @ " ? @

# #

" " "

Ê- ' ?/ " @ " " " Ê " B / $ B > ;

%=

: * > ; / "

* /

&= " + = " / ' " " " + * R " B R > ; =

)" + R " ! ; - " / ! R - R$ > $ ? @ > ; R > R B R - !

R > ? #R@ R ! * " " + " R $ , R # > ? @ > ;

© 2004 by CRC Press LLC

11.2.4 Algorithms for Minimum Cost Flow * " ! ! * + ' 5 ! - , '" ( &- '!

!

! ! " ' ' ! = " +- '* ' ? " @ " ) - ! = ?' @ ! !' ! ' " ' , , , " H ( $ ! ( *'$( + ! 5 ) ' '" ' ! * ' = ( / ! ' " ' / ( !' ? ' " '' @ - ! ! ' ' + ' A Transshipment Problem Associated with a Minimum Cost Flow Problem DEFINITIONS

=

8 > ? # $@ + , ! * , ' ! > ? @ ? @- ? @- ? @ * ! / ' - ; ;- ' * - '' / $? @ > $? @ >

=

" ' , > ? # $@ ' ' , ! ! ! * ! " / ' - B

FACTS

=

R ' , 8 + , R " + R "

" + " " R R > ? @- > - > - R > *- " R . " = 8 + , ' "

' , )" ' - B ! + ' " = 8 + , ' " '

, ! / '

+ ' " A Primal-Dual Algorithm

% D' !* / ' ! / + 2%D74 ! " ' ! * ! ' ' / * ! " ' ' ' ' ,

© 2004 by CRC Press LLC

DEFINITION

%=

: * + * - - ? @- '' + " - - ? @ > $? @ 2 ? @ ? @4

! ? @ " - + - * /- ? @ ? @ ! ! ' " ! ? @ ! ! ' ! Algorithm 11.2.2:

)#!$ 1!, ,

( = ' ' , > ? # $ @ + ( = ' ' ' /

) ( /? @ > ; " H ? @ > ; " ? @ #H I > ¾ A I A ?@ I ?@ I 5 ?@ I ?@ I ' ? @ " ! + I ! ! ' J' ' /? @ > /? @ B ? @ " I => I

I " ! ' I > I

FACTS

=

! ! - ! * '- / '

" !

=

A ! - " - / '

-

=

" ! I' '

' ! ! /

' ?'?! + @ ?' ( ) @@ *- " + , ' * ( !- " * ' "

' , ? / 9 7@ . ' B ( " ! / + ' ' , ? - - / ! , @ . ( B ' - ?(@ 5 ' " ' ' " " * " ! ? " / 9@- " ' " ' ! ! " * !

+ ' ?(?! + @ ?' ( ) @@

! / , , / ' / / ' ? / ' @- " ! ! ! = ? @ ; " ' " ? @ I ) / ! - " ! ! ' I' ' I ' ,-

" I - ! * ' I

© 2004 by CRC Press LLC

A Push-Relabel Algorithm

' " + ? '+ ' ! 5 @ A / " + ' ) ! :! 0 2:634 DEFINITIONS

&=

' ? /@ / # 0 " ! * ? @ 0 " ' * '

= =

? @

" ' * ' ? @ ;

: * + , - ' " ? @ ? @ B ? @ ?& @

=

? @

! +

' '? @ " ! + ? @ . ? @H Algorithm 11.2.3:

." ' - ,

( = + , > ? # & $ @ + ( = + - ' ' /

) ( /? @ > ; " H ? @ > ; " ? @ #H 0 > ) A 0 5 ? @ " ! ? @ ; A ? @ ; 5 ? @ ; )" ? @ '? @ ? @ => ? @ ? @ B ? @ % ? @ /? @ => /? @ 0 0 => 0

0 " ! ' 0 > 0 FACTS

%=

)" ! * /? @ > ; " - ? /@ ) ' )" + ? /@ 0' " 0 - +

&=

" " 0'- ' ? /@ 0' - 0' ! ' " 0'- + " '- +

= = = =

, " ! ! / ! ! !

' ' * 0' ?'@ ! ' ' 0'

© 2004 by CRC Press LLC

' " ! '

- " ! * - :! 0 2:674- ' , % , 2% 664 !! ! / * '! ' !' " - !

! ! 0 " ! ' - " ! ' ?' @ ' 0' J ! ' :! 0 2:674 ' " ! ?'( !?' (@ !?') @@ ' '* - - ( " - *' -&- ( "$ - ! ' ' * 2:374 Strongly Polynomial Algorithms

! * + ' - ' ' 2 13-) 5;;-5 ;4 ! ' * ' ' ( " ' / ( ? ! ' ' ( " ' @ * + 26#4 " / ! + . ! ' ! / + * ! ! " :! 0

! ' 2:634 " ! ' ! 1 21664-

' ! + / !- ' ?( ! ' ?' ( ) @@

11.2.5 Extensions to Minimum Cost Flow Convex Cost Flows

5' * + ' '

- " ' Æ + ! DEFINITIONS

=

Ê Ê

: * ' * " = " - + ( +

+ , #-? @ * "

=

* / ' * * / ! !

* / " " FACT

=

) - ,' " * ! ' * ' " ! * + ' + '

© 2004 by CRC Press LLC

) - "

'' " * " " ! " ' + ! ' +- ! ' -

" " " ) - ' * ! . " '!

!- ! * ! " + ?!- ' ! ! !' ! 2 694H

! ! 2D 374H ! ! 2374@

Flows Over Time

* ' ' - '

*' DEFINITIONS

=

+ , > ? # 1 $@ ? @ # 1 1 ' " ' + ? @ + *

=

( " 8!

" = 2; @ ?2@ " + ?' @ ! 2 * - / ?2@ > ; " 2 * 2; @ + * / " ! =

Ê

= ?2@

-

; " ? @ # 2 2; 4

7

= + ! ? @ 2 * 2 B 1 -

" 3

?2@ ?2 1 @ 2

2; @- $ > ;

> ;

= + " = ?2@ > ; " 2 - + ,- -

?2@ ?2 1 @ 2

87 9 = ?2@

=

> $ "

- " 2 2; @ #

) ! - " + * / ?@ =>

%=

2 1 @H

?2@ 2

$ > ; " (

( + * "

$ ! + *

© 2004 by CRC Press LLC

&=

? @ + , + " " ' ? @ ?@ > 4 ?@ > ; )

- " ' + ? +@ 4

=

8 + , > ? # $@- S " ' + + S " + " " ? @ #- " 5 S- 5 ? @ ; " ? @ ;- + ? @ > 5 ? @

! * +* , - + , ! ! 1

' + 5

S- 4 ?5 @ * " 5 - ?5 @ ' " ' ! 5 )" S + ' " + + , " - 1 ?5 @ " " ?5 @

FACTS

=

+ > ? # $@- + ' S " + ' ' - S (

=

R ! , 2#64 8 + , ! / ' ? @ ? @ > 5'' R S + ' " 1 ?5 @ " ' + 5 S / + * "= " ' + 5 S- + ! ?5 @ ! + / ?5 @ 4 ?5 @ " ; 1 ?5 @ + * " ?5 @ * 21 ?5 @ @ , 2#64 + *

+ * (

=

J , ' ' ,- + ' ? #@- ' " " ' ' , +* ' +* ' L*- ! * ' ' 2L;;4

%=

! +* ( C 2DA3#4- " / 0 ;- ! ( ? B 0@

" +* ( " ' ? ' ' 0@ 25,;4

Flows with Losses and Gains

! + ' ,!

5 ' " / - '' !- * " ' - ! + ! . *' DEFINITIONS

=

T

> ? # 5 @ , > ? # $@ ' ** ! " 5 = '' " $ > ; " 5 ?@ ; " " " " +

© 2004 by CRC Press LLC

Ê

- 5 ?@ )" 5 ?@ - H " 5 ?@ - ) !- ! ! " " ! , +- ! " " *

Ê /= = ? @ ? @ " ? @ # = ? @ ; " ? @ #7

= 2 ? @ ? @5? @4 > ; "

=

" = #

=

+ ! ( " + ! ! * ''

=

?@6?@

(

+ !

FACT

&=

' " + ! ! ( ' " + 2197774

+ ! *

'!

! * ! C - ' ! " + ! ( ' ?' ,@ 2:34- ! ! ' 2:P 1374

- ' ! " + ! ! ( " ! ! "

+ 2A;4 ) ! ' . " ! ' ! " '

References 2 134 & D

0- 8 ! - P 1 - " L- %! <- CP- 33

Q - C A F 2634 % ! - *! )" (" 6 ?363@- #73G#6 2#64 & % - 1 ! ' - 1( " " !" 9 ?3#6@- 67G3; 2% 664 , P % , - ' " , + ' - !" $ ?5 @ ?366@- ;G 2374 ,- 8 ,- !- 0 " ' * , + ' - *! )" " 7- ?337@6#G67;

© 2004 by CRC Press LLC

25 364 A P ,- A L ! - A & ,- - P A 5- C O,- 336

5 0*-

2%D7;4 P % & D'- '* ! Æ

" , + ' - )" ! 3 ?37@- 6G9 25,;4 8 5,- + * ! $ % :; ( !- *! ( # ?;;@ 99G7# 2#94 8 & - P C, + - & '- 3-

! 3#9

2#64 8 & & ,- ! + " +- + " 9 ?3#6@- 3G 2:374 $ :!- Æ ' " ! + ! - )" - ?337@- G3 Q - ! " 2:34 $ :!- 5 , - % ! ( ' - !" " + " 9- ?33@- #G6 2:634 $ :! & % 0- !

! ! * - )" " (" !" 9- ?363@- 67G669 2:674 $ :! & % 0- !

* '' - !" " + " #- ?33;@- ;G99 2:664

Q - & % 0- C, + ! $ :!- %

$ 20 *-( ?- 366@- ;G9

2:P 1374 :"- K O P - P 1 - !! !

! ' ! " ! ( ' - !" " + " - ?337@- 73G6; Q - U , ' - !" " 2L;;4 L'' % + " #- ?;;;@- 9G9 2) 5;;4 5 )- 5 ,- 5 !- & ! * ' * ! ! " +- !" " + " #- ?;;;@- 79G; 2D 374 $ D(* 5 ,- " '

* ' ( ' '' - *! )" (" 9- ?337@- #G7# 2DA3#4 D ( : P A! !- += ' - * 8 :''<9 3G- 8 C ' 5 - 3;- 5' !- - 33# 2 694 - 5* ! ! + ' * 0 * ' C+ ? - 36@- !" $ (" C 9 ?369@- 7G3

© 2004 by CRC Press LLC

21974 D 1!- 1' + !

,- )" * " 6 ?397@ ;6G7 21664 P 1 " ! ' + ! - " + " ?33@ 6G#; 25 ;4

5 0* - 5' !$!- - ;;

Q - ! ' ! - 26#4 % #- ?36#@- 7G## 2774 D '- 1 + ! ' +- *! )" " !" - ?377@- #;G#9 2A;4 D A- ' ! " ! (

+- !" " + " 7- ?;;@- #G#3 2K74 C K- , ' " '

+ ! - !" $ " # ?37@- ##G99

© 2004 by CRC Press LLC

11.3

MATCHINGS AND ASSIGNMENTS

! ! ' :' ! C ' :' &"

Introduction ) !'- ! ' . / ! " 0 ! * ! ! ( ! ! !' ' ( ' " '' - " ** ! ' ' ! " " 0

11.3.1 Matchings ! / !'- ! ! ! " * " '' - * !- - ' ! - ! ' - ! DEFINITIONS

=

8 > ? @ !' * ! %

!

= =

> ? @ -

" ' 0 !

) " * * ! * )

=

> ? @ ! - * " ! " -

=

-

= =

?- @

%=

'" ! "

" H #

? @ " ! - " ! - - " ! - ?- @ >

" ! - * ! ! ( -

"

! - * ! ! !

& * ! - > ? @- ! - !- ! - ! $ " !H * ? @

&=

%* *

© 2004 by CRC Press LLC

-

' " !

=

A ' ! - - ? @ " ' " ! " " ! " ! " !

=

" *

' ! " ! ' " *

'

! - ! ' ' " * ! !- ' ' !

EXAMPLES

=

! !' ! ! - > ? @ ? #@ " ( H ! ! ! " * * - " * # * & * ! - - * 9 " * - ! ! ' " 9 ! * > ? @ ? @ ? @ ? #@ ?# 9@

! - ( !H ! - > ? @ ? @ ?# 9@ " ( ( !- " '" !

2

3

1

4

6

5

Figure 11.3.1

! , ,#

=

! !' 9 * ! - > ? @ ? #@ ' ( !- " ( !' * '"

! 1

2

3

4

5

Figure 11.3.2

6

"2 ! , " # !

=

) ! !' " ! - ! ! ! " ! - > ? @ ? #@ ?- @ > 7 & * !- ' > ? @ ? #@ ? #@ ? 9@ ! '

? @ > 7 B # > ' ? @ ? @ ? @ ? #@ ?# 9@ ! ! '- 0 ! " * 9

1 1

2

2

Figure 11.3.3

5

5

3

4 7

6 4

© 2004 by CRC Press LLC

1

5 3

6

! , , * ,#

Some Fundamental Results FACTS

= )" - ! " > ? @- " * " " * -

= )" - ! - -

= ?A, @ ( " * * " '' ( " !

= =

%* ! ! ' " !

)" - ! ! ! ' ' - -

< - I ! " ( - B

< - I , ' / ! -

= ? ! ! @ - ( ! " " ! ! ' ' - ?5 2634- 2#74- 2C&#34@

= )" - ! ! ! ' ' - -

?- I @ > ?- @ B ? @

%

= 5'' - ! * ! ! ! ! " / ( ! )" ! ! ' " ! ' - - - I ! ! ! ! " ( ! B

&=

8 - ! ! ! ! " / ( > !- ! ! ! ' ' - ? @ ?@

? @

=

. " 6 3 ! ! ! ! ! =

! - " ! " " ! " * ! ! ' * - ' * EXAMPLES

=

) ! - ' > ? @ ? @ ? @ ? #@ ?# 9@ ! ! ' ! - > ? @ ? #@ ! - ' " ! ! - > - I > ? @ ? @ ?# 9@ ( ! - - ( ! ( !- ? @ ? 9@ ? #@ ? #@ ? 9@ ? @

=

) ! - > * * " - - ( " ! - / > 1 - > # ? @ * * " !' ! - (

! - " !' / - >

=

! ?@ ! - " ( - ?- @ > 7 5 ! ? #@ ! ! !- - ! ! " ( & * - - ! ! ' > ? #@ ? #@ ? @ !

?@ > 9 B 7 > - ! ! ' > ? 9@ ! ) * / ! ! ! ' * - ) !

© 2004 by CRC Press LLC

6- - > - I > ? #@ ? @ ! ! " ( -

?- @ > ?- @ B ?@ > ; ? ! ?@@ & * - * ! ! ' " * 9= 7 > ? @ ? #@ ?# 9@ > ? #@ ? #@ ? @ ? @ ? 9@

7 > ? @ ? #@ ? #@ ? @ ? 9@ 7

9 > 9 >

?7 @ > # B B 9 >

?7@ > B

?7@ > B 7 B

7 ! ! ! ' ? 6@ - > - I7 > ? @ ? #@ ? 9@ ! ! " ( - ?- @ > 3 ! ! ' * - * ! * !- ? ;@ -

! !

1 1

2

2

7

6 4

5

3

4

1

5 3

5

1 1

6

(a)

Figure 11.3.4

2

2 7

6 4

5

3

4

5

1

5 3

6

(b)

( , ! ," "2 " *

REMARKS

'=

9 ' ! 2#74 & K C 1 & 2C&#34 ! ( 63 '' " P 2634

'= '=

'' * " ! '* 234

234 " * ! ' - ! ' '

11.3.2 Matchings in Bipartite Graphs ' !' " '' ? ! ! ' 0 , ! 0 * @ 5 * 2 134- 2 1&3#4 2:3#4 28694 " '' '' ! " ( ! ! ' !' DEFINITIONS

=

8 > ?8 !

=

9 @ ' !' ' * - ( !- !

? @ " " *

)" 8 S? @ > 9 9 0 * "

=

? 8 @ " > ?8 9 @ ! * " 8 ! " - 5 ! " 8 9

© 2004 by CRC Press LLC

APPLICATIONS

=

! ' ! ' ' * ' ' 5 ' * , ! " " " ! " ' < ' < ' ' !' > ?8 9 @- 8 " 9 " ' ! ? : @ ' : !

' ! " !

=

' '' ! ' 0- 0 ! / '' !

! ? ' * @ " '' " 0 : ! * ! ? !@ * ! * ! ! ' ' !' > ?8 9 @- 8 " '' 9 " 0

=

* " ' 0 ? @ " * " !' " 0 , ! / " * 0 5'' 8 > 9 > ' ' " 0 B I )" I Æ - % ! * 0 F ' * " ! 0 ? * " *@- ' ! 8 9 ( * " %

" ! ! ' ' ' !' > ?8 9 @- ! ? : @ ' ! ' ' ! ! " ! ! * " % !

! ! " ( ' '* ' ? @ ' ! " * B I

FACTS

=

?DE !F @ ' !' - ( " ! " * * " ' !' ! . / . ?5 23;4@ ?LF @ > ?8 9 @ ' ! " " S? @ " * 8 ) - ' ! ' *

=

" * 8 0 . " * 9 ?5 23;:334@ 5'' ! 6?@ ! 6?@ > 9 @ " 8 9 ' ! ?5 2:334@

= ?8

EXAMPLES

=

) ' !' " ! # - > $ * * "

- - > ? @ ? @ ? $@ ?# @ ( ! ! DE !F - - > - ! # > # * S?#@ > $ 5 S?#@ # - LF

' ! ' 8 > # ) " -

! - * ( #

© 2004 by CRC Press LLC

Figure 11.3.5

1

a

2

b

3

c

4

d

5

e

) " * ! ," -# ,#

%=

) " ! 9- ' , ! , ' - , V '- ' , ( ' ? @ ? @ ? @= ' ) " , ' ! ' V - - - ! V H " - " " Æ L " , ! , ! V . " ! V " " DE !F - ! ' !' > ?8 9 @ 8 9 # H ! ? : @ V : ) - ' ' ' ! * ! ' * *

1

2

3

4

5

1 2 3 4 Figure 11.3.6

! ""- * - 3 "

REMARKS

'=

DE !F LF * " "

'=

( ! ' ' !' "

+ ' ' , * ! + ! ? @

'=

! ! ' ' !' "

+ ' + , * !

+ ! ? @

Bipartite Maximum-Size Matching Algorithm

! - 9- ' ( ! " ' !' > ?8 9 @ % ** / / " -

© 2004 by CRC Press LLC

! " * 8 * " * " ! ! ! ' ?'(@ ? 2564@ Algorithm 11.3.1:

0# ( 12 (! ,

( = ' !' > ?8 9 @ + ( = ( ! -

- => ; => 85% A C1 ; 8 , " " * " " => 8 , => << < => &J% A << < " ! ! ' # => ? @ " )" # , > ! ! ' ! ! ' 2W4 - => - I << < => 85%

% , ! " ! ! ' => # " => ? @ - # )" " > << < => 85% ; => &J%

REMARK

'=

! ! ' ' 2W4 *- ! "

9 * * " ?0 @ / " # * # - $ " 8 9 " " - " *

EXAMPLE

&=

! / ( ! ' !' " ! 7 A ! ! - > ? @ ? $@ " ( - ! 7?@ - " > # > $ 5 * " # - ! " > # > 5 # "- ! ! ' > ? @ ? @ ? @ -

! ' - > ? @ ? $@ ? @ H ! 7?@ ' " > - # > $ H " > - # > H / " > - # > " > C " ! ! ' "- ! ( ! - > ? @ ? $@ ? @

© 2004 by CRC Press LLC

a

1

a

1

2

2 b

b

3

3 c

4

c

4

(a)

(b)

( "2 ! , -# ,#

Figure 11.3.7

Bipartite Maximum-Weight Matching Algorithm

! - 6- 3- ;- ' ! ! " > ?8 9 @ % / ! ! ! ' * ! - ! ' ' * ! !" ' " ! ?' (@

* ! ! " ! ' " " * 8 * : ! ?: @

Algorithm 11.3.2:

0# ( 4 , (! ,

( = ' !' > ?8 9 @ + ( = ! ! -

- => ; => 85% A C1 ; 8 " " " * " 8 8 ?: @ => ; " : " ?: @ => A " > # => ! ? @ - " )" ?@ B ?@ ?@ => ?@ B # => # " => ! ? @ - # )" ?@ ?@ ?@ => ?@ " => " 8 " * ?@ ' )" ?@ ; - => - I

%

; => &J%

© 2004 by CRC Press LLC

EXAMPLE

=

! / ! ! ' !' " ! 6 )" ! ' !- / ! ! ' > ? @ - ?@ > 9- !

! ?" ( @ - > ? @ - ?- @ > 9H ! 6?@ " > * $ ' ?@ > - ?$@ > - ?@ > #- # > $ J ! ! ? @- * ' ?@ > " > C " '

- " * ?@ > # ' !

! ' > ? @ - ? @ > # ! - > ? @ ? @ ?- @ > H ! 6?@ - " > * $ * ' ?@ > - ?$@ > 5. ' ' ?@ > - ?@ > - ?@ > - ?$@ > - " * $ ?$@ > - ' ! ! ! ' > ? @ ? @ ? @ ? @ ? $@ ? @ > ! * ! ! - > ? @ ? $@ ? @ ?- @ > - ! 6? @ ' 3- ! " ! ! ' != ? @ ?@ ?@

4

1 2 3

1 4

a

1

b

2

5

6 5

c

3

(a)

Figure 11.3.8

4 1 4

a

1

b

2

5

6 5

c

(b)

3

4

a

1 4

b 5

6 5

c

(c)

( , ! ," "2 " / / *

11.3.3 Matchings in Nonbipartite Graphs ! ! ? ' @ !' ! " ! ( ! ! " ' !' ' ' " M N DEFINITIONS

=

5'' ! ' " " * !' > ? @ * " ' " 0 ! * !H " !

=

5'' ! ' " " * * * ! ? @ 0 * * ? @ . -

=

$- ! ? @

+ ' ! *

+ + " ! ? $@

* " ' ! *

© 2004 by CRC Press LLC

FACTS

= !

= =

+ ! ! B ! !- " ' !'

?% F @ 2%9#4 5'' !' " " ' ! + ! ! ' " "

=

?: 5 ( !@ ! - 9- ' ( ! " - " " !- " * " - " ! ! ' + ,- ! !' Algorithm 11.3.3:

5 ( 12 (! ,

( = :' > ?8 9 @ + ( = ( ! -

- => ; => 85% A C1 ;

, " * * , * , " ! A ! ! ! ' " 8 ? @ ! , ? @

/

)" * , " * % " ! ? @ ! ? " @

0

)" * ! < ! ! ' "

1

)" * ! + " 5 , + * * $ )" ! ! ' " - => - I % ; => &J%

%=

! '' % 2%9#4 " ?'@ '* ' " ! ?'(@ H 264 2:3#4

&=

( ! ' !' " ! ! " : 2:794- ?' @ - ! " $( 2 $6;4- ?( '@

=

' ! . " * ! ! ! '

© 2004 by CRC Press LLC

! !' / ! - ** ! - *' % 2%9#4 " ?'@

=

) '* ! " ! ! ' - ! ?'@ ?'( ! '@ ' *H 2 134 2:3#4 ?'( B ' ! '@ ! ! * : 2:3;4 EXAMPLES

=

) ! 3?@- > ? @ ? @ ? @ ? #@ ! ! ! '- ' ! - > ? @ ? #@ & * ' - * # * * 5 ?# @ ! 0 ! * * - + > ? @ ? #@ ?# @ " 1 - 7 > ? @ ? @ ? #@ ?# @ ? 9@ ! ! ' * - - I > ? @ ? #@ ? 9@ ! " ! (X " ! "

( C * ' 7- * * * # 9

=

5 , ! + * ' ! 3?@ ' !' ! 3?@ ' > ? @ ? $@ ?$ 9@ !

! ' ! ? @ ? 9@ "- ! ! ' 7 > ? @ ? @ ? #@ ?# @ ? 9@

4 1

2

6

3

1

2

b

6

B

5

(a) Figure 11.3.9

=

(b)

1 , - "" +

! '' ' !' ! ;?@ 5'' ! - > ? @ ?9 6@ " ( *

* : = " * # 7 , *- *

9 6 , " " * # 7

)" " ! ? @ '' - * , * *H " ! ? @ ! ? @ " )" " ! ?7 @ '' - ! ! ' > ? @ ? @ ? 7@ " J ! ! - > ? @ ? 7@ ?9 6@ " ( H ! ;?@

* . = " ( " ?*@ * #

)" " ! ? @ '' - * , * *H ! ? @ ? @ " % ! " ! ? @ ?7 9@ , 9 * 7 6 * %! ? @ ? 7@ ?7 9@ ?9 6@

© 2004 by CRC Press LLC

)" ! ?6 7@ - '' - + > ?7 9@ ?9 6@ ?6 7@ ,H ! ;? @ ! ? @ ? @ ? @ ? $@

)" " ! ?$ #@ - '' ! ! ' ? @ ? @ ? @ ? $@ ?$ #@ " ' ! ! ! ' > ? @ ? @ ? @ ? 7@ ?7 6@ ?6 9@ ?9 #@ ! - I ' ! ? @ ? @ ?# 9@ ?7 6@- ( !H ! ;?@

1

2

3

4

1

2

3

6 5

6 7

5

7

8

8

(a)

(b)

1 1

2

4

3

2

3

4

4 6 5

5

7

b 8

(c)

Figure 11.3.10

(d)

+", ,

APPLICATIONS

=

" ' ! " * !

: ' " +

!! *

' + ! ! !' * ' '

! ' ' ' " ' ' " + ! ! " " ' ' ( ! '

=

* , " " J . " " * , ' ! * ! " / " ! " " ' " '

! ' " " " !

" ! ! ' C - / !' * ' ! " * ! ' "

' " ? " ! ! @ ! " ! ? : @ ! * " " ! ! ! * " : ) * ! ? @ ' ' " ! ' * * ! "H ! ? @ ! * Æ ! ! * ! ! ! " !

! '" ! '* ' ! " " " *

© 2004 by CRC Press LLC

References 2 134 & D

0- 8 ! - P 1 -

- L- 33

2 1&3#4 & D 0- 8 ! - P 1 - & &- '' " , ' ( - '! G6 - ! - - : C ?%@- ! - CL- 33# 2#74 !- !' - $ " " " ?3#7@- 6G6

" ?J5 @

23;4 D !- * ( - L P** - 33; 2%9#4 P % - - - +- )( % ! 7 ?39#@- 3G97 2%9#4 P % - ! ' ; * - )( % + % =( ( % 93 ?39#@- #G; 2:794 L C :- Æ ' " % F ! "

! !'- )( % ! ?379@- G 2:3;4 L C :- " ! !

, !- $ % : ( !- *! ( # - 33;- G 2:3#4 L :- !- '! #G - ! - : C ?%@- ! - CL- 33# 2:334 & : - # ! 8! - 33;

A

28694 8 8*Q(

- ! - CL- 369

@ ! " / ! 2 $6;4 5 $ $ $( - ? ! ! !'- $ % .: ( ( ( % ( - 36;- 7G7

2C&#34 & K C 1 & - ! " * " !'- $ % ! ; ?3#3@- #G3 2564 L ' D 5 ! (- - L- 36 2634 P - !E !'- ! # ?63@3G; 234

- ! X '= " Q DE ! '# ! ;; ?33@- 77G3

© 2004 by CRC Press LLC

234

- ! * ' , != MN Y- '! 7#G P : - P A D- 8 $ U ?%@- 1( 2 / >- % # ! ##- CL- 33 264 & % 0- # (( - 5) - 36

© 2004 by CRC Press LLC

11.4

COMMUNICATION NETWORK DESIGN MODELS !" # ! $ # %

: C, ! J ' C, ! 5* * C, ! ' C, ! &"

Introduction

, ! ' 0 '' " !' ! , ! ' < +* ' <

'' -

' " ?" '- - '- -

@ ! , ? ' @ " ? ! " * - * - Æ @ ' " A ' - , ! ' ! ! " !- ! - ' '' *- * ' , ! ' ! " C ' !' ' - " '- 5 - * ! 5 - ' - '

" ! , ! ' ) '- / ' ! , ! " ' 5 * ' " , ! '* " ! ' *- 0 * , ! ! ' '' !

11.4.1 General Network Design Model Preliminaries DEFINITIONS

=

!' > ? @- " ? * @ " ! !!'

! - ! " ! ' , ! > ' > (

© 2004 by CRC Press LLC

=

-

?

!- "

" * ?!@ ;?!@ 8 "

>

@

=

- ' - ?! '' - /' - @ , ! ) - " - " " " ' ) !- ' + " ) !

=

)" " + "

- ' ( " " "

-

8 H

)$

> < " ' " " * ) - > <- )

)

?! *@ ! " ! " " ' )% ! : % ! " > <- % ! % ! ?! *@ " ! "

! " : ! : $ !

=

' ! % ' - " - -

=

' - -

=

"# ' ! "

* " ( ( 0 . ** ! * - " * ! * "# * ! *- $# * ! ? - * @

' * " '! / * ?8- ) )@ " SUMMARY OF NOTATION

C, > ? @- ' ( ! 5 "

- ! > %

! ! ?!@- ;?!@- '- * $ ! " ! " : ! : 5 " " *- > < % " * ' )% / % ! ! : % -

© 2004 by CRC Press LLC

General Edge-Based Flow Model

, ! ' / * ! '!

* - '' - , '! / "=

* ! "

! ! : " : ! * &% ! " " " * % % ! : - & ! = ! - '' * . " ; " * - * % ' " ! '!

! ?" '- * ! ' " ! ! * & ! / @ ( ' * " & ! F &%

General Model: Edge-flow [GM:EF]

( 0 =

! & !

!

! %

% ! &% ! B

!

B $ ! !

" > ?!@ ! > " > ;?!@ ! ! & ! ; ) &% : % !

! %

) &% : % ! ! %

-

?@

! & ! &% * %

; : ! ; &% ! =% ! ! : > ; !

!

?%@ ?%@ ?%@ ?%#@ ?%9@ ?%7@ ?%6@

REMARKS

'=

?@ *' "

?%@ ?%@ + "

" ! ' - ?%@ + ! - ?%#@ '! " ! ?%9@ ! ?%6@ / ! * ! .

© 2004 by CRC Press LLC

'=

" ' " ! ( " ! ! " ?%@ ?%@- ! * ' * * " '

'=

?%@ ?%@ + "

!H - " ' ) - '' + "

! ! : - " ) ! + "

! ! " : - ' " " =

!

) ! ! :

'=

2:=%4 " + ! '' . ! + "

+ * ' ! '- + 5 '' ! . +

. + ! '

'=

! !' " < ! - " '- '' - /' -

'=

) . * + * - ' M * ?@N " ! General Path-Flow Model

) " ! ! + * / " 2:=%4 * '' " ! , ! ' ( $ + * A ' ! : ! , '' ? : @ ?: @ * ! ! :

* " ! * "=

! > - " ' "

! ?!@ ;?!@ + ? '@ "

! ' 6 " ! " +"

! " ! ?!@ ;?!@ ' > $ !

!

5 ( * 6 . 75(6.8 ( 0 =

! %

% ! &% ! B

6 B

6 > !

6 ) &%

!

© 2004 by CRC Press LLC

%

% !

!

6

)%&% ! : %

6

&% * %

-

6 ; : ! ; &% ! =% ! ! : > ; EXAMPLE

=

5'' * ?' > 7@- " * ?8>@ ' ) > ) > ;- * > * > - "

#

9

7

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

;

#

;

;

;

;

;

;

9

;

;

;

;

;

;

7

;

;

;

;

;

;

) ! - ! * " " - , ' " ' ' " ' 2

3

1

4

5

6

Figure 11.4.1

© 2004 by CRC Press LLC

7

# " , !#! *

REMARKS

'=

C - - ! *- ' .- ' ! ! ' " " "

'" C #- 9- 7- ' - $Æ % Æ . ! % " Æ ! " -

'%=

/! !

, ' Æ - " ' ! " ' , ,

, ! '

11.4.2 Uncapacitated Network Design ) '' - ! " ' =

?@ )" )- ! ' " ! ' ' + ?@ )

!- /' * !

' - ' ' " '' ?@ A ! ! '! " , ? - " - ! ' + @ DEFINITIONS

%=

%&' , !

" ' ! " '

&=

8& " ! ! '! '! " ! ! .

=

8 ) ) ( ! ! '! " ' ( ' )" 8 8- ' *- '!

! - 8 8 " "$ "$

=

" ( ! ? !@ '! ) . ?"' "$ ( @"$ (

Uncapacitated Network Design [UND]

" *- , ' ! ' " ' , " '' " ' ASSUMPTIONS

! ? @ " A

- ' !- * + $ ! $ ! ! : - ) >

© 2004 by CRC Press LLC

A * % > " - '' " $ ( & B !

0 =

! !

!

! !

* ?@

&

!

! & :

! !

?JC@ ?JC@

; : ! & ! > ; :

!

?JC@ ?JC@

FACTS

A + (> ' ?!@ > !- ;?!@ > ' " - * " ?' *@ ' !

=

!

'

= )" * " ! * & ! ,- * ! ' JC / ! '- "

! ! $ ! !' ? #@- # " ? : @ ?: @ " & ! > :

= &' ! !!! M N ?JC@ ?JC@ !!! & ! ! ?JC#@

!

& !

: !

?JC9@

. * ! '! - 2JC54 L*- '!

! ! ! & ! * " 2JC54 ! '!

! " 2JC4

=

5'' " + =

" ! : H

?@ $ ! > $! > $ ! " ! ?@ $ ! B $! ; " :

) 2A634- "

! ! ! - ' 2JC4 ! ! + * !- -

½ B ¾ ! !

& ! ! ! ?! @ > ?! @ ;?! @ > ;?! @ :

?JC7@

= , 2A634 *' * < * '' * " " 2JC54 ?JC7@ J ! ?2(@ '*

© 2004 by CRC Press LLC

'" !' ! - ! '

= 2A634 ! !' " " 2JC 54 ?JC7@ - *!- " ! ' 2 364 " ' , ! *' ! !' "

EXAMPLES

=

' !'

/ ?@ > ?@ > ?@ > ;?@ > ;?@ > ;?@ > 8 / . + . ( " !- - > > > $ > $ > $ > ; 5 ! & > & > & > > > > ! * " " '!

! " " 2JC4 L*- " '!

! " " 2JC54 ' '!

! " " 2JC54 ?& > & > & > > > > > > > > > > @- - - 2JC4 ! " 2JC4

=

?JC7@ ! '!

! " " 2JC54 -

, ?@ > ?@ > ;?@ > ;?@ > % ' %- > > > - $ > $ > $ > ; '!

! " JC5 ! * H " ! ?JC7@- '!

! ' ! & > & > Multi-Level Network Design

' , !

' ! / " !' ' - < ! " " * ! ! ! ?*@ " $ & ' $&' '! , ! '' ! DEFINITIONS

=

)

$&'

' - " *

> <- ' $ ' *

?!'@- ! !' " * ! " ! ? '@ 0 * ! " * ! ! 0 * ' " %¼ : %¼¼

! ' " * ? @ A + " '

) ! * " ! ! ! $

!

=

: " , % ! -

8C '

" ?% ! % ! @ > <- " !

© 2004 by CRC Press LLC

=

= : * ! !' " * / - / ! !' ) ' -

Multi-Level Network Design Model [MLND]

) " " 8C ' 2 34- * &% ! . " * " ! : ; A /

! > ' "

!- ?!@ > ' ;?!@ > ! > !

( 0 =

$ % &%

! %

! !

* ?@

&% ! % % $ &% ! : ! = ;?!@ %

! % % $ ; ! : ! &% ! > ; : > < !

¼

¼

¼

¼

FACTS

=

5 ! ! *- ! ' 8C /

%=

> <- ' / " * ?@ / " *

&=

A < > ! > ;- 8C ' . * 5 ' / ! / ! 5

=

5 8C ' ! ( 5 ' - 8C ' C ' C < > - > ? @

'' ? @ ! > ! > ; " : 2134

MLND Composite Heuristic

" ! ' 2 34 " < > , " * *' '" " * * ' *

( / 5' ! ?@ = * * - / ' ! ? @ ! ! 5 & ! > ! " ! "

( 0

5 1* ? ,@ = 5' ! ! "

$&) ' ! ! ! J ! !

© 2004 by CRC Press LLC

! ' ! - ! - * 5 ' 5 ) * " ! " 5 * " ! ! " $&) "

)"

> '" " 5 ( 0- '" " 8C ' * " '' > B " !

!- " ' - * "

< > - ! ( ' * " ! 8C ' ? ' ?$ @ @ < > > > '' - ! ( * '*

'" " # 2 394 REMARKS

'&=

' , ! ' - M (N ! " 8C ? - ' ! '' @ '' " ( " * ! ! ' ! ;3 ' " ' " < > 9 ' " ' " < > #

'=

8C ! ( ' ! * * . '- . !' / * " !% " '' / ! !% ! ( * < > - ! > - ! >

11.4.3 Survivable Network Design (SND)

, ! ' " ( '' * "

'- ! ?' ! " @ ?' !

@ " '

* "- , ! '* !

' ,- ,

"

* " " ! " , ' < , ' "

! ' " 5 ' "

" " ?! @ * - ! " , ' ! '' "

Uncapacitated Survivable Network Design [SNDUnc]

'! ! * ' " $ - " ! " , %! * 0 ' ?!F @ 7 ! ! " !' ' 9- / ! + , ;

© 2004 by CRC Press LLC

DEFINITIONS

=

, " " * " ! "

=

( : > : - ! * ! ! - " ! 0 ' :

=

A * . - ' )$*+ , ! '

%=

8 > ? @ , '' ' " / - - " ! /

> : :

# + - 2# + 4 " ' ! *

# : +

2# + 4 > : REMARKS

'=

, * . / 9 * ' * ' " : " ! - *

: "

'=

5 !' - * .

- - ! > !

'=

A : , - ! - "

- ! ' ) - ! > ; " : ' ?5 @ , - . - Cut-Based Formulation of SND [SND-CUT]

)" - " " * * , ! ?5C@ ' ! * (

! &

!

! & !

0 = & !

! & !

& ! > ;

© 2004 by CRC Press LLC

" '' '

" :

FACTS

=

)" ! > ? ! @- ' 5C '

=

! ! * . " - ! ($ '' - 2A : $3#4 *' '" " & * *H 2::5A 34 '* ? B B B B ( @ EXAMPLE

=

! ' " * * , > ' 4

1

5

2

3

6

8

9

7

11

10

Figure 11.4.2

12

Connectivity requirement between pairs of

nodes = 3

pairs of

nodes = 2

pairs of

nodes = 1

1 19 # -

SND Iterative Rounding Heuristic

2P;4 *' '!

! * ! - ' - ? !@ '" " * ' !

! * " " 25CJ4 ' ! ' / " ! 5C

( /

' & ! : '!

! " 25CJ4 ?

= % ' " / - " & ! * * 5' @

( 0

! : & !

- / & !

REMARK

'=

J ! !" ! - P ' " 25CJ4- & ! * & ! ! * P ' " ! * '! " ! * * " * ! '! - / M*N !

© 2004 by CRC Press LLC

- '* / , 5C '"

8 > ? @ , . ! / *- # +

8 2# + 4+ " ' : * . - 2# + 4+ > : # : + !

FACTS

=

" ! . ,

0 ' * ' " : " !

"

=

! ! & " '

& !

" = 2 ? "@4+ >

2:53;4 2:534 " " - *' '' ! - *'

" ! ' '' " 85 , ! ' ! " '

Flow-Based Formulation of SND [SND-FLOW]

! + * - 5C ' ' " ! , !

* ' " ! > ! - ?!@ > - ;?!@ > : - < > ( 0 =

;- /

!

!

! & !

* ?@

!

& ! ! & ! ; ! & ! > ;

: ! : ! :

REMARK

'=

2 ;4 / * * . ! + " ' < * " 5C ' Survivable Network Design: Bounded Cycles

" 5C " * !

- - 85 , ! ' ! L ! ) -

© 2004 by CRC Press LLC

! " . ! " < ! ! ' '* ! - ' * ! ! ! ' /

A " ! ' . = ! : - 9 ! "

: " ! ! : ) 9 ! - !, * "

) ( " ! ? ' (@ " '

, & :

!

! , # , & : : :

!

! , # ! , , > ; : ) 9 ! !

¼ ¼

¼

¼

REMARKS

'=

( 28;;4 ! $$ '' " ! "

, 0 ' * ' "

'=

" , ! , " " ! ? " (@ ' " "

?!- ?2 34- 2:364@

11.4.4 Capacitated Network Design ) "- * ' '' ! ! " ' '' . * "

H ' * ' ( ' * ' ! ' * "

! ?- " '- 2 64@ A " ' * *

Network Loading Problem [NLP]

)

!- " * < < ' " " * ' - - ' " *? B@ " ' " ' " *?@ " - > < *- ?@ " " ! A " ' & $ # &$# " < ' " " " ? /*@ ' '-

!- %/ %1 " * " " 6 ' " - 6 " " / " ! ! ' * *

© 2004 by CRC Press LLC

" ' " ! * " ' " * " C - '- ,* ' !

) " 1! 9 : *, . -

Figure 11.4.3

Network Loading Model [NLP]

" ! " C, 8 ! < > ) > ) > )

(

!

? ! & ! B ! & ! @ B

!

$ ! !

0 = * ?@

& B )&

!

!

! & B )& : !

!

! ; ! : ! & ! & ! > ; :

EXAMPLE

=

!

? - > @ < > ) > ) )"

- !! '= ! ? ) @) " +- ! ? ) @) " + " ! ' ?2 634@ ? ! @ ) '-

! -

- > ) >

! "

© 2004 by CRC Press LLC

! "

Figure 11.4.4

# "

# "

FACTS

=

2 $3#4 2 :E 3#4 *' ' ' '' " * ! , ! ' - 2 $34 '* ' ' ' " " '

=

!

, ! ' " ' +

- " ' ( + C ?2: 5364@ Capacitated Concentrator Location [CCL]

! "

, ?

, ". @ . ! ? @

' - ' ! , ,

DEFINITION

&=

) ' ?**$@- ! *

? @ ! - - " ' " ! - ! ! " ' : - - ! " ! : A " ! ! * ' -

= " ! : - . ' - ! - " : ! :

8 ! . " : - ; & ! . " * : -

Capacitated Concentrator Location Model [CCL]

?8@ ' " " ! ; ! '! = (

© 2004 by CRC Press LLC

& !

! & ! B

!

! !

&

!

! >

?8@

: ! !

?8@

&

! : & ! & ! ! & ! > ; : ! > ; : -

?8@

& !

?8@

¼

?8#@ ?89@

REMARKS

'%=

) " 284- / " - 8-

- 8

' 8 !

" 8 ! .

'&=

' ! ( " ' , ! ' " , ! !

!

'=

'

, ( :664- 2:34@ ) ' - '

" " ? @- '- "

! "

! !- ! > '

" ! ! : ! + . ( A

" ? @ ! " ' ! " ? @- ? @ " " " ! ! ' '' / ' " ' ( '' * *' " '

(

' ?

Survivable Network Design (Capacitated)

A ! ! ' ,- " '* ! * * * ! 0 '- , ! ' ,' ' " ?!@ ;?!@- ' " " ! "

! A '' - , ' # # - '* * * - ! ' ? ,' ' ' ! , ! '@ , ", ! * * " " ' ! * *

' , A " '' / #$ ?5L&@ ? '* ' ' ! " @ " ' + ! - !

© 2004 by CRC Press LLC

DEFINITIONS

=

%-? + ?5L&@ , " !' " < 5L& ! ?

@H ! 5L& * ! * ? ' ' @ ' 5L& ! " 5L& % ! 5L& ' ! + L - ' " 5L& ! 0 ' "- ! + ! ! ! ' ! ! ! ! " ! " < ! ' 5L& ?- " '- 25A5Q8:364@

=

# @ +

' ! ' + "

! " Æ + ! ! ? ' ?!@ ;?!@@ - * ! ' ' , " " > "

! " ! "

EXAMPLE

=

2P :A394 ! # < * / * '- > ?@ > ;?@ > - >

Figure 11.4.5

© 2004 by CRC Press LLC

$" !, "-$

Diversification and Reservation Model [DR]

) ! * - + (

" ! =

2 Z ' ! " ,- 2 > ; ' ! ? ! ' @ - - , 2 > - 2 > : : - ! : , ?2@ > ? ?2@ ?2@@- ?2@ ?2@ " !- ' *- ' ! 2 ?2@ " " ' " ?!@ ;?!@ ' ! 2 $ % &% ( ! %

0 =

- !

-

6 ?2@

)%&% !

6 ?;@

: 2 Z

%

6 ?;@ >

6 ?2@ > >

Æ

! !

?&@

!

?&@

! 2 Z ;

?&@

! ?!@ ;?!@

Æ ! > ?!@ ;?!@ 6 ?2@ ; : ! 2 Z &% ! ; ! :

6 ?;@

?&@ ?&#@ ?&9@ ?&7@

REMARK

'=

) " *- ?&@ ' ?&@ " "

' ! ? " @- ?&@ " > "

! ' ! " ?&@ ? ! ! ?!@ ;?!@ @ + " Æ ?&#@ * / " ! ! ! * ! 5 2534 2 :PA364 " " ! ' '' " * !

© 2004 by CRC Press LLC

References 2 :664 D , :* - L % : " '! ! " 8

C,- ! - ?366@G 2 :PA364 *- :E - P- J & AE- 5* * C, = 5 - *,,, ( ! - 9 ?336@- 66G3 2 :A364 *- :E - & AE- %Æ C, 5 " 8 8 - % + - 79 ?336@-G; 2 L3;4 : P & .- 8 L ! - ' - ' & ! - ""+"," # ( $ - 33; 2 34 , - D , - J ! L' : *

C, !- *+! )( (- - ?33@- 3G;# 2 34 , - 8 ! ! " * C, !- ! - ; ?33@- #97G #6 2 34 , - 8 ! - ! L A " " 8* C, ! - ! - ; ?33@- 69G697 2 374 , - 8 ! - C, !- ) = - F - Æ - 5 ? @- P A 5- C O, ?337@ G 2 364 , - 8 ! - ! ! L

5* * C,- + - 9 ?336@- 9G9 2A634 , - 8 ! & A!- " 8!5 J ' C, !- + - 7 ?363@79G7; 2 :E 394 ,- 5 '- 1 :E E , - ' ) "

C, - ! $ 6 ?339@- 77G33 2 5 34 ,- U ! 5 8* - ' 5' ! - 33

! "

2 :E 3#4 , 1 :E E ,- ' % ' Æ !

- ! $ - 96 ?33#@- G7

© 2004 by CRC Press LLC

2: 5364 5 '- ) : 5- 5 5 , ' ) - # ! - 6# ?336@- 9#G3 2%A 994 8 & % D A - 1 ' ! 5 !- *=! )( - # ?399@- G7 28;;4 (- 8Q Æ - 5* ! C, - + - 6 ?;;;@- 699G677 2:6#4 :* - ! 8!! ! " ( C, !- *,,, " ("- 1 ?36#@- 7G#7 2:34 :* - '! ! "

C, X 8 ! - % + - ?33@- 7G7

2:364 8 :* - J ! $ &/ " ' ! 8 " 5' ! 5 L' - *+! )( (- ; ?336@- 6;G66 2:34 V : P - 5* * C,- 8 !

! & '- ! $ - 9; ?33@- #G99 2::5A 34 V : - :!- 5 , - 5 - % A - '' ! " C, ! #- ?33@- G 2:53;4 :E - 8 5- )! ! " ! * - *! )( # ! - ?33;@- #;G# 2:534 :E - 8 5- ' & ! ! " ! !

C, 8 * - + - ; ?33@- ;3G; 2:5394 :E - 8 5- ! " 5* * C, ) $ ! " ? + ! - 1 - 8 ! - 8 : 8 C ? @%* 5 - C- ?33#@- 97G97 2P;4 D P - '' ! " : ( 5 C , - - ?;;@- 3G9; 2 $34 8 ! - & $ - * L " ' C, ! - ! $ - 9; ?33#@- G#; 2 $3#4 8 ! - & $ - ! 5* ! " ' C, 8 ! - + - ?33#@- G#7

© 2004 by CRC Press LLC

2 64 - 1' 5 " C, C

&. ) ( % / # $ L ? @- CL- 93G77- ?36@- 93G77 2 ;;4 - 0 " C, 8 ! - ,( )( % + - ?;;;@- #G#9; 2 634 - 5 " ' C, ! ''

)- J' - )- !- - 363 2 394 - - *+! )( (- 6 ?339@- ;G6 2CA664 : 8 C 8 5- C O,- 366

A- * $ - P A [

2134 P 1

- 33 2P :A394 J - P- *- :E - & AE- 5* * C, = 5 - ' 5 396- DK "E )" ,- - 339 2&374 5 &!* 8 ! - C, * - ) = - F - Æ - 5 ? @- P A 5- C O, ?337@ #G# 25A5Q8:364 5 - A & 5Q! - 8Q- : (- ! ! ! " 5* * 5L\51C% C,- ) ( $- 5] 5 ? @- D

- C- ?336@- 7G96 2534 5 : - ''

5* * C, !- ( ! - 96 ?33@-3G97 2A : $3#4 A - V : - - $ $ $( '' ! " : ( 5 C, - # ?33#@- #G# 2A64 & A!- '' " 5 :'- ! $ - 6 ?36@- 7G67

© 2004 by CRC Press LLC

GLOSSARY FOR CHAPTER 11

, #

G * != ' ! "

, , #

G * != ! ' " * " *

, , # , -

G + ,= ' "

!#!$ G , > ? & @= I ! * I > ? @

-! * ! ? @ G = -* * G "

" = ' "

" ' ?'@

- "" =

! " 0 ! * * " ! '- " *

!! G + , > ? # $@= + " '' * $ ! *$ ! G

, > ? @=

" * ? ( ;?!@

3(4@ ?!@

! ! !$ ? ' # @= ?!- ''

- /' - @ , !

! ! =

!' > ? @- " ? * @ " ! !!'

! - ! " ! ' ,

-!- = , ! ! ' ! - ! !! ""= ,- - " ! ! " -

, ,

-

"- = , Æ ' ! " ! ' '

! " > ? # $@= !' * - #-

! * ' " = #

- " = # ! '' * $ = % / $? @ > ;

% -

- = + , > ? # $@ ! * $? @ > ; " $?@ > $?@ ; > ? # @ " > ? # $@ , * > - # > # ? @ $? @ ; ? @ $? @ ; ' " /

* *

? @ ? @ > $? @ $? @

© 2004 by CRC Press LLC

" ? @ " > " >

#

! G ' ! ' ? @ "

= " * ' " ' H -

!#!$ = " ' " ! "

- - >

? @

= " *- - =

* G " + ' = - /

G * ! '= * * ?

! "" ? @ G * = ? @ > ? @ ? @ !$= ?!- /' @ ! " " !@ " " '

! - * * !

G + , > ? @= " = ' " =

Æ

? @ ? @- " ? @

? @ > ? @ " *

? @

;- " ? @

G + , > ? # $@= " = #

% /

? @ " ? @ #7

= 2 ? @ ? @4 > $? @ " = ? @ ; " ? @ # - = + * ! + ? @ - = + ,- - ? @ > ! "" ! = + ! " +

! , - - > ? @ ? @ = ? @

> ? @ = !' * - ! * - " ! * ( " = Æ ," G ! , T > ? # 5 @= " = # Ê

/=

? @ " ? @ # = ? @ ; " ? @ #7

= 2 ? @ ? @5? @4 > ; " = ? @

-

= + ! ( " + ! ! *

-

! " = + ! (

''

© 2004 by CRC Press LLC

?@6?@

= + , > ? # 1 $@- ? @

1 1 ' " ' + ? @ + * #

= - / 9 * ! ? @ G =

*, " G " ! - = ! " !' -

! " G " !' = * " ! , T > ? # 5 @= , > ? # $@ ' * * # 5 = Ê '' " $ > ; " H

# 5 ?@ ; " " " " + - 5 ?@ , +- ! " " *

! "- ! ? @ G , " ! * += ? @ ? @ ! * *, " G " ! - = ! " - ! * ! " G " !' = * " ! ! , G !' = " ' 0 ! - ! # G " ' !' > ?8 9 @= ! * " 8 H 8 $

"2 = ! - * ! ! ( - - , = ! - * ! ! ! ?- @ - # ! G " !' = ! * " - "2 = " ! ! - , = " ! " ! ! # - = ! * + , > ? @- / + " -

*

! *$ # - = "

- / + " * ? @ ? @

? @ (

! *$ G

+ , > ? " ! > " = Ê " ! " ! =

&@ =

? @ ? @- " ? @ ? @ > ? @ " *

> !- ? @ ? @ > " > ! ? @ ;- " ? @

> !

! *$ > ?

&@ = !' * - ! * ( " & =

A ' * " ? @ * ? @

! ! -

** G * ! '= * ? " !@ " " '

© 2004 by CRC Press LLC

# = * " +- " = % " =

? @

'

? @- " ? @

? @ ? @ ; " * ? @ ;- " ? @

*!- ! ? @ , " ! * += ? @ ; "* !#!$ ? @ G " ? @ , " ! * +

'+ = ;- / ;

"* G " + ,= ;- / ; "* G " + ,= ;- / 3 ! G ' ! ' ? @ " =

" * ' " ' H -

!#!$ = " ' " ! "

- - >

? @

= " *- - = > ? @ = !' * - ! * - " ! * ( " = Æ " * - "" = ' ! ! * 1

# - = ! * ! !' " * -

/ - / ! !'

" !

G

,= . ' " ! ' !

Æ

"" # = + , > ? # $@ * / ' -

-

"" ! *= ' , ' ! > ? @

? @- ? @- ? @ * ! / ' - ;- ;- ' *H '' / $? @ > $? @ >

! # =

' ' , !

! ! * ! " / ' - B - - > ?@ ?@

"#- # - =

+ '

'

! G " !' = " * ! " , # G * ! - = " ! " "

! " ! " ! - ? @

© 2004 by CRC Press LLC