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.
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
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
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
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PREFACE
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CONTENTS Preface 1.
INTRODUCTION to GRAPHS
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GRAPH REPRESENTATION
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COLORINGS and RELATED TOPICS ' $
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History of Graph Theory
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1.3.2 Trees " " ! L> + A /A:," / ! ?" # % + A /C;," K K + A6@/C:," JI + AA:/ CA;," " # Counting Trees
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34
Chapter 1
() ' AAC" 5) AAC7
INTRODUCTION TO GRAPHS
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Section 1.3
35
History of Graph Theory
() 5) A:7 / Æ +!,
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! " # ! # # " " 1 )
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36
Chapter 1
INTRODUCTION TO GRAPHS
FACTS 5?=CA" ;7 5CC7
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# / & # & + 2 C, # # " #9 # L !
Section 1.3
37
History of Graph Theory
K3.3
K5 Figure 1.3.9
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FACTS 5?=CA" A7
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38
Chapter 1
INTRODUCTION TO GRAPHS
4 " " # # # 8?
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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
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40
Chapter 1
INTRODUCTION TO GRAPHS
? +? ," 2 + ? ," < K 8 =
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Section 1.3
41
History of Graph Theory
() ' C:@" % 4! 5%4::" %4L::7" K L"
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Chapter 1
INTRODUCTION TO GRAPHS
Factorization
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Section 1.3
History of Graph Theory
43
A ' C6" J" 3 %H
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44
Chapter 1
INTRODUCTION TO GRAPHS
5?=CA7 H ? " - L ? " K ="
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Section 1.3
45
History of Graph Theory
5 ;C7 - = &!" % # . " & + C;C," @CP : 5 ; 7 % " # " # " +, + C; ," @CPA 5 7 4 - " J." " @ +K C ," C ,"
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Chapter 1
INTRODUCTION TO GRAPHS
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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 " !
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48
Chapter 1
INTRODUCTION TO GRAPHS
5A;7 H # J " O " "2 ,1;< + ' % ," ? 8 ? H 6 + CA;," # F J" ;P : 5C:7 H #" " J " " / "
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Section 1.3
49
History of Graph Theory
5= 7 4 =" H/# " + C ," ;P :
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Chapter 1 Glossary
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Chapter 1 Glossary
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Section 2.1
2.1
57
Computer Representations of Graphs
COMPUTER REPRESENTATIONS OF GRAPHS
Introduction
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Chapter 2 GRAPH REPRESENTATION
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59
Computer Representations of Graphs
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Chapter 2 GRAPH REPRESENTATION
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61
Computer Representations of Graphs
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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
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 !
#
,
)
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 ' '
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
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 < > , ,
%# $ =*+ *=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,
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,
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
½
*
¾ ,
, ,
+ 3¾ 5 ½* * ½ 6
+
$
½ ¾
#
,
+ 4 ,
*
+
*
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 %
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, # ! / (
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 !* + ) !* + ! * + ) ! *+
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
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
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
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+ , - , ==
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
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 , ,
( , *+, ' , *+, % . ' , *+ %
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( %
* #
. , % *+ , % # % / / , , # # ! # %
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
( ; !' , # # , #
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+
'
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 @
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
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
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
# + % . # # . # , #
' %!#(! .!((
%, %# # %
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 #% # / % , # #
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
+ / * $
* + (
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
- )
*+
# '
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 )
* + 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 +
<
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 & # , #
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
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
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
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
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=
98
Chapter 2 GRAPH REPRESENTATION
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
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
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 #
Section 2.4
101
Recursively Constructed Graphs
Figure 2.4.3
.!!#! ! !# ! ##& """ #
-Trees and Partial -Trees DEFINITIONS
( ' , . , # 5 * + ' & . ,
(
. , .
(
*
+
( / " ' / FACTS
$( $( $( $(
, < .
, , * + EXAMPLES
(
# 4 , #
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
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
) *& *& #
104
Chapter 2 GRAPH REPRESENTATION
Treewidth- Graphs
# % < *, 0<9G1, 0<9G1, 0<=1+ !
# , # % E # % , , > E
& , DEFINITIONS
(
"
" "
( (
) * + *
) *
$ +, # $ # '
* &+ # * & $ ,
(
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# %
#
REMARK
%(
, , , ! ' * + @ , # , , , # * , # # +
EXAMPLE
(
# 4 = 4 , , ' ( ½ ) ½ ) ) ) ) ) # ' 4 =, * + , # 2 ,
$
Figure 2.4.9
$
# " &!!#!
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 +
$
( $
$
*$ + '
$
106
Chapter 2 GRAPH REPRESENTATION
(
(
# %
#
FACTS
$
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$
( 0<=1
( 0<=1 # ' # /
$
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#
#
EXAMPLE
(
# # 4 * +2
Figure 2.4.11
) *& # &!!#!
-Terminal Graphs DEFINITIONS
,
% *# ++ ! # , #
( ) * $ + ' $ ) ½ ¾ , # $ ( ! + ) ½ ¾ # #
EXAMPLE
(
# 4 I # 2
Section 2.4
107
Recursively Constructed Graphs
%(#- !#(! ! & "
Figure 2.4.12 REMARKS
%
(
,
$
)
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+ #
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+, #
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+,
+ #
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½ ½ ¾ ¾ ½ ¾ ½
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½ ¾ ½
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+
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+
$ ½ $½ ¾ $¾
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% +*½ ¾ + & *½ + *¾ +
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Cographs DEFINITION
(
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, &
,
%
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:
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 ! / # / # / # / #
,
, ' , , ,
. , . . %, / ! ( # &
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,
, , , , , , # # % ) # ! # , * &+ # * + !
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 !
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 ! / '
112
Chapter 2 GRAPH REPRESENTATION
REMARKS
%( ! & * + F > # ( < # 0 , 0 , ' ! 0½ 0 0 , 0 %( < *0<991+ ! > E & # , " 7 , , , ' 2 # %# " %( , < " !
%( %# * 4 #+,
' %#
FACTS
$ ( 0:; 91 : 1
$ ( .
$(
,
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.
Figure 2.4.16 $! !# ! " &#
2.4.3 Recognition . , Æ , # "
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
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
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
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
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
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
119
Chapter 2 Glossary
GLOSSARY FOR CHAPTER 2
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%
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, "( # ' $
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! " ( "! (
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½ ¾
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120
Chapter 2 GRAPH REPRESENTATION
( % # , #
, /
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.½ ¿
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"4(*&
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! ( !
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121
Chapter 2 Glossary
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122
Chapter 2 GRAPH REPRESENTATION
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Chapter 2 Glossary
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Figure 4.2.5
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REMARK
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4.2.4 Applications to General Graphs " ! U
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' = >
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Section 4.2
223
Eulerian Graphs
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FACTS
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224
Chapter 4
CONNECTIVITY and TRAVERSABILITY
""& $!"' ( ) *
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225
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226
Chapter 4
CONNECTIVITY and TRAVERSABILITY
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Section 4.2
227
Eulerian Graphs
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Chapter 4
CONNECTIVITY and TRAVERSABILITY
FACTS
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Section 4.2
229
Eulerian Graphs
' ( 7 ! # # ' ( ! ( 7 &
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230
Chapter 4
CONNECTIVITY and TRAVERSABILITY
2
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' 2 #
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# 6 REMARKS
' ( 3 : =( > B : , = > ' 6 ! 0 = >
= > ' ! U : &! , 0 = > = 0 = >> 5 0 = >
Section 4.2
231
Eulerian Graphs
4.2.6 Transforming Eulerian Tours The Kappa Transformations 6 &
6
( -(/10
DEFINITIONS
'
6
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CONNECTIVITY and TRAVERSABILITY
REMARK
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EXAMPLES
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Figure 4.2.8
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Section 4.2
/=>>
233
Eulerian Graphs
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? 9 3 4 7
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Figure 4.2.9 Splicing the Trails in a Trail Decomposition
+ ! 6& ! = > Algorithm 4.2.5: 0"& $!"' (*
#' 5'
: ? ½ + ! ? = > = > ? : ¼¼ : ? = > '?
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References -J310 H J 6 A $ =/31> 74K7/ -2Q/ 0 8 : 2 O Q 2 ! $ % ) =// > K4 -(/40 : . * ( 6 F , ,
$
=//4> 1K -(/30 : . * (
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234
Chapter 4
CONNECTIVITY and TRAVERSABILITY
- (/40 $ * ( , + =//4> 9K -;/0 2 ; ;
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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 * (
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Section 4.2
Eulerian Graphs
235
-(2340 * ( $ 2 ; ! =/34> 7K79 -(+3/0 * ( A + $ , # =/3/> 44K71
%
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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
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$ =/97> 73K7 1 -64 0 + 6 6 +
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. A $ =/94> K -+/10 . + $J% $ =//1> 9K1 -S10 H S % ( '. F!& 6 J! . 11 -Q/90 O Q ( -. 8 $ .&& " F! <& //9
Section 4.3
4.3
237
Chinese Postman Problems
CHINESE POSTMAN PROBLEMS
! "
#
6 8 " L N . $ #
Introduction 6 !* ! * = > " #
!
6 , 2 = J! $ J> /7 -270U
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4.3.1 The Basic Problem and Its Variations DEFINITIONS
'
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238
Chapter 4
CONNECTIVITY and TRAVERSABILITY
The Eulerian Case DEFINITIONS
'
= > ! & , # ! &
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Section 4.3
239
Chinese Postman Problems
6 ! ! ! => # !
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DEFINITIONS
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240
Chapter 4
CONNECTIVITY and TRAVERSABILITY
Algorithm 4.3.1: ,$ 5677 #' 5' $ !
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6
! ! @ ! !& A = -A74 0-A74 0> 6 , 6 ! ( = -J 790> ' 2 ! & #
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Section 4.3
241
Chinese Postman Problems
½ ½
6 ! ( 6
= ! 1> ! & !'
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' 6 !
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B # ! A H =-AH90>
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6
FACTS
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242
Chapter 4
Algorithm 4.3.2:
CONNECTIVITY and TRAVERSABILITY
,$ 677
#' ! !
5' $ ! " A
( ? 5= > )&5= > ! '
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Producing an Eulerian Tour in a Symmetric (Multi)Digraph
B = > 6 ! = -A840> DEFINITION
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7"! " $" !"
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# £ ( £ ( # ? £ : ! 5 = > : £ # £ ! !
!
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Section 4.3
Chinese Postman Problems
243
EXAMPLE
' ( , #
#
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4.3.4 Mixed Postman Problems FACTS
' 6 # $ - U 6"("8":"6< = - 970>
244
Chapter 4
CONNECTIVITY and TRAVERSABILITY
' $ - ! # = > # ! ! = -2 H9/0
Deciding if a Mixed Graph Is Eulerian DEFINITIONS
' 6 #
' '
#
# # # B
#
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Figure 4.3.3
$" 89 +"
6 = > B #
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Section 4.3
245
Chinese Postman Problems
! 89 +" 2 $"
Algorithm 4.3.4:
#' # 5'
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¾
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( ( , = > = > ! , 6 # !
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$ $!"' !" !"
246
Chapter 4
CONNECTIVITY and TRAVERSABILITY
REMARKS
'
6 ., & & = @ ! > 5
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The Postman Problem for Mixed Graphs
$ F ! + & ! !
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' , REMARK
' " $ !
# " = ( 4> ! ;! ! M FACT
'
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6 # = > 6 , # ! ( & =-(9/0>
Section 4.3
247
Chinese Postman Problems
EXAMPLE
' 6 B # # ( 4 U ! " # ! F! ! ! ,U ! B ; ! ' U N !
Figure 4.3.5 2" : ''" /!"
" ! ! !
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-AH90 " # @ ! ! M
Approximation Algorithm ES
6 ! # ! 5 M 6 ! = -AH90 -(9/0> 4 C M D Algorithm 4.3.5: "9' $!"' ,
#' #
5'
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N # , : ; M :
6 # 4 /#
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
Section 4.3
249
Chinese Postman Problems
Algorithm 4.3.6: "9' $!"' ,
#' #
5'
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EXAMPLE
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Figure 4.3.7 $ "9' $!"' , Some Performance Bounds FACT
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! EXAMPLE
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A ! A ! ! A
Figure 4.3.8 $!"' , , "& :"
250
Chapter 4
CONNECTIVITY and TRAVERSABILITY
REMARKS
' " ! # 4 =A>
7 =A> *! A# 3 ' # ! A = > ! A ! ! ! A
' 6 A# 3
B !
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C D ! + !& @ B @
6 , ( & =-(9/0> ! ! ! ! ! !
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L = L//0> ! 6 ( /
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6 $
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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>
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
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- 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
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
/
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' '
½ ¾ ½ ¾
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
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 #
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 =(% >
Section 4.4
257
DeBruijn Graphs and Sequences
Necklaces and Lyndon Words
( & J -(J990 & 8 5 B DEFINITIONS
' = B
>
' B
' 0 & ! : & + & #
B
& FACTS
'
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' 6 - = > & ! # ! + - =1> ? 13 - =4> ? /
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EXAMPLES
'
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4 01101 11010
0
1
1
10101 01011
0
1
10110
Figure 4.4.3
'
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6 : ! 1 + 11 8 5 B
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 +
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
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>
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 #
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$$" ! " " & ;"& "$
!
Section 4.5
263
Hamiltonian Graphs
' -$$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>
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 = >
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 :
:= >
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
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
= > !
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 !
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
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 %
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
!" - ;
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
!
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
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 /
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
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
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
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
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
!
! ! ,
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
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 ! #! =!>
! 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
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
! !
°
"
&
? 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 =! ! >
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 #
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
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>
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
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
290
Chapter 4
CONNECTIVITY and TRAVERSABILITY
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319
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' - -
FACTS
: - !- ! + , - + !
:
31 &B6 ; - ! ' ' ' 2 - !
: # < < + =! > ' ' =!- ! > 1 ! + !
:
. "
2K 1 0 - ! + ' . 0!+ < <
: 3E56
- !
< .
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
. 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 , + ' - -
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
#
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
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 ' +
' ! '
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
<
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 ? +
' ' -
"
.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
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
'
·
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
+ '
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 # +
,-+ ' !
/ '
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
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$$0+ $I55
360
Chapter 5
COLORINGS and RELATED TOPICS
3 *1&56 S H + ; D *+ F 1 +
+ S 9 ! ' + 1 + ; + ! ' + 4 5 & 5 XX( .5&50+ I & 3#9B6 S #2 * E 9 + ( , ! + 4$ .5B0+ &I$B 3$6 1 + 8 (+ #$5 !"5 75 5 5 5 B .5$0+ $ I5 3$B6 1 + % + IB S H
% F 8 . 0+ " " + E @ 4 " + 1 .F0+ 5$$+ ; 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$6 ; 4 + 1 ' + S - + ; 55$
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 # $$5 45 .55 0+ 5&I 38 5$6 E 8 + @2 , + .5 4$5 " 5 ( $& .55$0+ I& 38 76 E 8 + @ ' , ,!/
!V ,+ ) 5 .5 4$5 .770+ BI7$ 38 BB6 S 8 ?N + ) + 7 .5 !"5 $ .5BB0+ B I5 M 38M
$6 8M + ;, ' E+ !"5 5 && .5$0+ I$ 38 &B6 ; 8 2+ + + $&5I$5$ ; F?! 1 @N . 0+ 4$ + E @ 4 " B+ 8/ .F0+ 5&$+ !F + 5&B 38 B6 ; 8 2+ 1 F, ' , - ' - + ! *% 5 5 B .5B0+ &I B . *0 38 B6 ; 8 2+ % F, ' + &I ; F?+ D D -N/+ 1 @N . 0+ - 7: + E @ 4 " &+ .F0 5B+ !F + 5B 38 E&&6 ; 8 2 S E/ -+ ; ' + ! *% 5 5 7 .5&&0+ I5 . *0 381 556 4 8 -NY+ T 1/+ E + , ' :
+ BI5& * D . 0+ 4 * !" + (E; @ @ E 1 @ 5+ ; E @ + 555
362
Chapter 5
COLORINGS and RELATED TOPICS
38776 E 8--+ 1 ' + 4$5 # $$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+ $&IB
Section 5.1
Graph Coloring
363
3*5&6 " *+ + J+ + .5 " " & .55&0+ &&I 3* @156 * + S @ + * 1 + F,K ? ' !' + 4$ .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+ $I&7 31 &6 " 1 ' + 1, ! + 5 !"5 5 & .5&0+ BIB5 31 &B6 " 1 ' + ; - ' ! , ' , ! - + * !"5 .5&B0+ &5IB&
364
Chapter 5
COLORINGS and RELATED TOPICS
31 B 6 " 1 ' + @ ' + &BIB$ * " 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
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
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!
+ ' - &
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
" ' - +
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:
; 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 ;
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368
Chapter 5
: : :
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3;156 ; !
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:
;
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Heawood Number and the Empire Problem DEFINITION
:
1 ! " ' '
". 0 <
&C
E + ' -
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6 '
5
& , ,
$& C C
.$& C 0¾
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:
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$!
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+ ' 6 " . 0
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8
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:
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&
Section 5.2
369
Further Topics in Graph Coloring
: " . &0 ' - & .F 1 3B760+
? - .34*B60 8
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&
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:
. S 0 # , ' 6 ' , - ' & 6 , & . &3 0+ " . &0 ' ' P Nowhere-Zero Flows DEFINITION
4 < . 4 0 ; " D #" ' 8 :
:
8 8 + .
' - -
0<
.
0
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:
31 6 ; ! ' ' , ,!/ !V , .1 ' '0 ( + # 1 - - , ! ,!/ !V ,A M
/K . ' 0 - !! ,!/ !V ,
:
31 7+ E$&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 - '
370
Chapter 5
COLORINGS and RELATED TOPICS
EXAMPLE
: 1 '
- + -+
.
90 < 9
.9 C 0
9 . 90 <
Z < 9.9 0
FACTS
:
!
. ! # 0 # -
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+ , - . 90 < . ! 90 .! 90+ , Q!K Q!K
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9+ , Æ + .0 9 . 90 < 7
:
,
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- +
. : C 0 < : ¿
½¼ ¾. :
C 0
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: ; ' ( . : C 0 7 ' - .: C < $B0+
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' ' # 1 .;
' ' + . : C 0 / ' - 0
5.2.3 Some Further Types of Coloring Problems Variants of Proper Coloring
9 V ' + , ', ' DEFINITIONS
: ; $ ' - ! : - - ' ' .0
: 1 $ ' '
: ; ' - ! ' ?
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"
: ; ' - ! - ? - -
Section 5.2
:
371
Further Topics in Graph Coloring
1
"
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3! : ; 9 . . 0 0 - ! : 7 .+ C 0 ' .0 .0 + ' - - .0 < .0 : - / 7 + / : .0 .0 / ' : ; ' .0 ., 7 .0 ' 0
- + , ' .. 0 .0 ' 0
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:
0
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:
;
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:
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'
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0 -
372
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COLORINGS and RELATED TOPICS
; -, ! - - E -+ " "
1
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:
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: ; ' < # # # 7 , ( , ' 7 # # + ' .0
: - < . 0 ' + : ¾ # 9
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: " + -
, -
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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
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+ # <.0 .0A ' 2 + & &0!
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' :
# -
.
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:
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;
! : =
( % + - % + , - + + ! 1 , ! ! ' : # - $+ $% - ' S , - $ :
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COLORINGS and RELATED TOPICS
:
(' !
, - + - + ' - $! . $ 0 3 556 E -+ ' ' + $ - ' 3476 .( ? 3;F776
, - ' $! 0
:
. ' 0 (' / ' + ! ! ' '
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: ( - + 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+ ,
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:
; ! < . " 0 - A " ' 9 " < + ' " : ; ' ! : ' ! - + - : ; ' !
'
- !?
:
1 .!0 ¼ .!0 ' - ! + - & + + ) '
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 -!'
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:
3 $6 # - - + !! -!' , ', -
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:
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376
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:
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:
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:
"
1
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0
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:
3@@9 56
P
A
' , '
0
.
.
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Section 5.2
377
Further Topics in Graph Coloring
FACTS
.0 ' - :
+
< ' + . - 0 : .0 ' .3@@9 560+ ,!'
: 3"S@6 1 ' , -
, ' .3"S@60
: 38156 # ' +
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: " .+ ! 0 / ! : 1 !
/ . ! / 3 *1&560
: 3E$B+ #@$56 1 : 38&6 # - + NP! , .0 ; +
NP! , ' ! .34 @ &$60
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- ! + ' ,
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Chapter 5
COLORINGS and RELATED TOPICS
: 3156 - , ! - +
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Section 5.2
379
Further Topics in Graph Coloring
:
% NP! + .38@B560 ' .387760
: 345&6 (' ' + NP! '
:
3F156 S ' + + - , Æ . 0
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:
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:
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
7 1 - ' '
380
Chapter 5
COLORINGS and RELATED TOPICS
: 38D@776 P < NP+ !
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+
3;1 5&6 ; + T 1/+ E +
' + + $ G$$ .55&0+ IB
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 +
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+ .5&B0+ &I&
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+ 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,+ ; !
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+
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382
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COLORINGS and RELATED TOPICS
3"88EM 76 F 4 " + 1 8 2+ 8 + F EM + ; !
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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&
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5
384
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COLORINGS and RELATED TOPICS
315&6 @ E 1+ @ 2 .2 : 30!
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K + B .5B 0+ $BI&B 345&6 8 4 % + 5 + D
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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+ ;
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386
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COLORINGS and RELATED TOPICS
38156 4 8 -NY T 1/+ ; ' + + 7 .550+ 5&I7 381556 4 8 -NY T 1/+ *2 ' + + .5550+ &IB
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+
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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 + @+ %
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+
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, 7
+
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+
+ 7
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) +
$
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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
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+ ; ' + + $ .5 0+ B7I5
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+ E + 5I * 8 . 0+ + "+ 5&7 31576 T 1/+ ! + I 5 31 5$6 T 1/ E + % ? ' *+ 5 .55$0+ $5IB
+ $ .5570+
H
+ * 1 +
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+ + .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
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+ 7 .550+
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/
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+ $
= 2
! +
) + + B
39 76 * 9
+ D ' + * ) 2 + D E @ D
@ BB+ - S+ 77 3T76 X T+ : -+ 7
+ 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 %-
!
.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 ) ' ' , ' ' )
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1+ -
:
< . 0 - - ' ' 6
:
/ '
; 6 ' -
1 / ' - '
: 38 + 6 (' + / ' ' - - '
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 ! ' < ' - >
392
Chapter 5
COLORINGS and RELATED TOPICS
REMARKS
: ' + # $ - ' - !
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3" "SS556
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+
+ < ?
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Two More Formulations of the Maximum-Clique Problem
9 ' - - ' ,+ ' , E
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:
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
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!@ ' '
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- ' Æ
' .+ + 3; 8ES55+ " "SS5560 % ' ' Æ - # 5
? ' ! ! + ?.0
? , .0 (' @ ' ! + @.0
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.0 ('
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: 3" F56 1 ! + .0 ?.0 < . ¾ 0
?
@.0
'
!-
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.0 ?.0 < . . 0¾ ¿ 0 REMARK
: # 7 . 0
' ' S 3F5B60
H .
Some Results Involving Maximum Degree
@ # - ,2+ ! ' ' , % J.0 ' # --, ' ! , ' ' J.0+ 3D176 # , '
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+
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: 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
< . 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
' ' ! - # $
396
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COLORINGS and RELATED TOPICS
DEFINITION
: - + ' !? , FACTS
:
3E E $ 6 1 ' !-
¿
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.@ ' 3@60
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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 , ' !
Section 5.3
:
397
Independent Sets and Cliques
"!! , , ' - !
/ + + +
Ü$
F ,-+ ' ! +
!! + + ' ! '! ! !! 3* @76
:
1 - Æ !! ' !
,
@+ + ,
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:
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! ' 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
,
,
)
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COLORINGS and RELATED TOPICS
REMARKS
: - ' + - 2 , F ,-+
'
: 1 ' !
- (
- ' 38 *B&6 ; -+ - ' - 38 *B&6
:
E ' - / + ' + ' ! ' 3" F56 3#6
D -
'
, ' ! 3ET-6 - D !
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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
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
+ * S+
400
3B56 # -+ 1 I S (+
Chapter 5
COLORINGS and RELATED TOPICS
/, .5B50+ 57I$7
3576 # -+ 1 I S ((+
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3EE776 + + E + ( E + , 2 + / .7770+ I& 3ET-6 + + E + ; T- - + , 2 + +
3W76 + ; + ; W + ; ' % / S + 5 .770+ I 7 3F5B6 E FN
+ ; ' + *,/
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B
+ - S+
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: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
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
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/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
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 $
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+
39B$6 F @ 9'+ 1 ' ' + ) 7 .5B$0+ I&
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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
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+ )
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: 9 ' ! - ½ ' ' - ' , -+ + ½
Section 5.4
405
Factors and Factorization
FACTS
;+ V ' !' ? @3 "
: 31&6 2& ' $-: ; !' ' ' ' 6 .0+ #( . 6 0 6 + , #( . 6 0
'
' 6 , - ' - : 3SB56 &2& $-: !'
- !! !
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: 3@&+ @&$6+ 3D& 6 (' ,!' ' - + !'
: 3@&$6 (' ! ' - + ' ! ½ + !' : 3#F E$ 6 ('
!'
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! . 0 ' - ' D .0 . 0+ !' .; &+ D .0
. 0
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' 1
K !' 1
- !' ' '
: 38&6+ 3D &6 1
: . ; 3;&60 D
.0+
' -
B . 0 .0
('+ '
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' , - .0+ !' -
F ' - - !' REMARKS
: ; ' Æ ' ; -
39 576 ; Æ '
- !' . -+ 32 36!' . ,00 ' 3 % 8BB6 . -+ 385760
: 1 , - ' !'
) F ,-+
' ' !' + , , -
? 3S5+ S5$6
406
Chapter 5
COLORINGS and RELATED TOPICS
The Number of 1-Factors DEFINITION
: ; ' C !' ' - '
, L - C .# ' + 3D SB$60
`.0
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:
D
- !' 1:
.0 38 56 !' A
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:
! !' + .0 Z !' + .0 ('
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:
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:
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:
,+ 8 1? 38155+ 8176 - . 0 , + !' ) + '
: % `.0 , ' #Æ .# + 3D SB$A B60 ( , + S'Æ `.0
:
1 - ' `.0
,
1-Factors in Bipartite Graphs
( ' + ' !' ,
+ S F 3F 6+
8M 38M
+ 8M 6 DEFINITIONS
' .0 - '
/ ' -
: ; ' - '
: 1 ' -
: : .0
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
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 ('
+ !'
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
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
Section 5.4
411
Factors and Factorization
FACTS
:
3DD5B6 ('
! ,!' +
3 6!
'
:
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+ 3
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('
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6 #( 6 ¾ +
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C 6! +
C 6!' '
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:
3
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'
!
- +
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:
' +
@ 3;5B6 ' Æ ' ' . 3;57+ FF8D576 ' )
(
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0!
38B+ 857+ DB56 ' ' ' - '
:
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.
0!'
# ' ' , , 2 V ,
' 3#455+ #455+ #455+ #47+ 8 @ 56
Factors in Random Graphs 1 - ' !
9 ,
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DEFINITIONS
:
D
' -
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7 ,
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:
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1
0 ' '
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S
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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 , , -
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
Section 5.4
415
Factors and Factorization
CONJECTURE
'5$62& 3
Æ ½
½
.
0
C
B6: ('
C
+
,
-
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<
½
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C
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Æ
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:
(
Æ , '
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:
@ 3S176 ' ' !' /
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' -
<
3
.
0 <
½
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9
- '
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:
3&B6+3D &&6 D
- '
.0 < -
.0
¼
'
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½
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@ '
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<
<
C
C
1
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+
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2 ½
5 2 5 Æ 5
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.
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:
# #& - 2 ,
' 3 8M
B6
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 -+ '+ +
Section 5.4
Factors and Factorization
417
: 1 ,+ 8 1? 31855+ 1876 - ) , + !' ) + ' + 2 , - !' +
: ; 3;B 6 -
' '
.( 0!'
!' ' ' ,
+ . ¿ 0
'
' ' ( : ; ' ) !' + ' + , ) '
4 3 4 &76 (' !' !'+ - .@ 3 SB760 (' , ' ! ' $ +
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+ + , ! ' , ! ' + -
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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
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: 342& *' $- 38M $+ 8M $6: (' +
¼ .0 < J.0 1
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*// 38*776
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- ' .@ 3*760
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S! F - - )-
S!
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" ! S
418
Chapter 5
COLORINGS and RELATED TOPICS
: 3 15&6 1 " ! S S! ,- " , .@ 3; W5B60
: 3"F 5&6 1 ) ' / ' -
! , !' !' , ' /
: 39 B6 (' - $+ -!
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$
: " + 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
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
420
3;56 * ; + () +
Chapter 5
COLORINGS and RELATED TOPICS
5
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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&
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
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!' / ' +
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3 B56 W , F
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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
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$
)
.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
= 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
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$
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 + ' ' +
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 +
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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 ' !
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5 .5&&0+ $
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3@ B 6 8 @ L+ #2 + > B& .5B 0+ &I& 3@ B56 8 @ L+ 1 !' ' + - F - .5B50
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C , *
2 ;9% 8.A8% + = < F+ 5B+ $I&
3@ B6 # @ /+ % ' ' +
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+ .5B0+ I
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+ 1 ' / ' + 7&I
+
31 6 9 1
+ ;
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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
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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 - - ' ; + + -
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COLORINGS and RELATED TOPICS
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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
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COLORINGS and RELATED TOPICS
EXAMPLE
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C ? C¼ " < ¼ : 1 " ' ' , '
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435
Perfect Graphs
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' , ' ' L ! # + - ' ! 1 ' . . 00½ + ' .. 00 ! ' ' + '
' . . 00½
' .. 00 1+ , " ' + .. 00½ REMARKS
: D -N/ 3D &56 - @ ' ! - ½ < ( . 0 : 1 .. 00
. . 00
/ ' ! (' S ' + .. 00 < .. 00+ @ .. 00
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.. 00 . . 00 G. . 00 G.. 00 ' , ' , ' . 0 ' + . . 00 < .. 00
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Tour E
A
E C D
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COLORINGS and RELATED TOPICS
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Section 5.5
437
Perfect Graphs
:
K ' ! ' - .0 - .0 REMARKS
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1
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DEFINITION
: 3D &56 1 & 1 H.0 ' ( ) -
438
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COLORINGS and RELATED TOPICS
FACTS
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2 ,
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1 ' ' '
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( - , ) F? 3F &6 1 - + ,
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439
Perfect Graphs
DEFINITIONS
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440
Chapter 5
COLORINGS and RELATED TOPICS
: # @ 3" 56 "/ - 77 ' ' - 1+ 2 , L ' ' ; .
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441
Perfect Graphs
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/ ' " DEFINITIONS
: ; ' ' '
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442
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COLORINGS and RELATED TOPICS
: ; 0 < . 0 ' , ½
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,
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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 + ; !
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444
Chapter 5
COLORINGS and RELATED TOPICS
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5.6
445
Applications to Timetabling
APPLICATIONS TO TIMETABLING ) ( % & " *+ ,
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447
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457
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Applications to Timetabling
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Applications to Timetabling
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465
Applications to Timetabling days
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COLORINGS and RELATED TOPICS
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467
Applications to Timetabling
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Section 5.6
469
Applications to Timetabling
Some Characterization Results FACTS
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470
Chapter 5
COLORINGS and RELATED TOPICS
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471
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References 3;FM5B6 ; @ ; + 1 E 4 + * FM 2- + + - S+ 55B
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Chapter 5
COLORINGS and RELATED TOPICS
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473
Applications to Timetabling
3 856 1 "
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COLORINGS and RELATED TOPICS
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475
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GLOSSARY FOR CHAPTER 5 $- -"
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COLORINGS and RELATED TOPICS
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Chapter 5 Glossary
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Chapter 5
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5 0 ' 7
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COLORINGS and RELATED TOPICS
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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
Section 6.1
6.1
485
Automorphisms
AUTOMORPHISMS
!
" # $%
Introduction &
' ( ) ) ) $
* + , - ) ) + !( !, $ ) ) . / 0 + , $ ( ) % ( ) . % /( ( % ( ) )
1 + , ( )
6.1.1 The Automorphism Group DEFINITIONS
2
( + , + , +, +, + ,
+ ,
2 ( ( )
+ ,
( ) + ,
)
2
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 ) + , + , $ - ( & - ( &
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
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 ,
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
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
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 & (
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
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 : < :
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< ) " &
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( (
2
6=! 8 > % ) ) ( C ( ) &
2 2 2
61!8 5 +,
+,
6 "8 > % )
6A) "8 3 ) ) 2 3 + , ) )
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 + ,(
&
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
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
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
Section 6.1
499
Automorphisms
DEFINITIONS
2
%+ ,,
+
2
& )
+ , %+ ,
+ ,
%
( ½ + ,(
+ ,
%
REMARK
2
$
(
) % %
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(
$
9 +5 6K1!!8,
FACTS
2
7 % ( : 2
<
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% ( &
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½
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7
% ) 7
2
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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 ( B
2
+ ,( + +,, + 9
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 & + , ( & )
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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
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"
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 !
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 (
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Chapter 6
ALGEBRAIC GRAPH THEORY
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6.2
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Chapter 6
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509
Cayley Graphs
DEFINITIONS AND NOTATIONS
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Chapter 6
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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
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511
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Chapter 6
ALGEBRAIC GRAPH THEORY
6.2.5 Factorization DEFINITIONS
2 2
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513
Cayley Graphs
6.2.6 Further Reading REMARKS
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514
Chapter 6
ALGEBRAIC GRAPH THEORY
6IZ"8 I I I J Z ( E
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515
Cayley Graphs
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516
6.3
Chapter 6
ALGEBRAIC GRAPH THEORY
ENUMERATION
! I I I I I
5 # N I
I 5
Introduction $ - I ) C :
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- ) C
$( ( : & ) # : ) ?! - S + 6S "!8 > , 7= 6=8 & (
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( ) 6=!8 ( - % :( : - % 65 ?8
6.3.1 Counting Simple Graphs and Multigraphs DEFINITIONS
2 ) ( ½ ¾ ( ) )
)
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Section 6.3
517
Enumeration
2
. ) .
% ) 2 ' . +, . +, .
2
. .
)
FACTS
2
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2
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B
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5$*" * - 2 " "
B B B B B B
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B
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Chapter 6
Table 6.3.2
ALGEBRAIC GRAPH THEORY
1 " *$*" * - 2
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519
Enumeration
Table 6.3.3 /* -
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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"" * *
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6.3.2 Counting Digraphs and Tournaments DEFINITIONS
2 ) ( ½ ( ) ) )
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521
Enumeration
2 " ,
Table 6.3.5 5$*" *
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B
B ?B (B (" (B "(! B !!(B (?!B !(? B "(! (B"(!
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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
522
Chapter 6
ALGEBRAIC GRAPH THEORY
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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
*
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2 7 E
Figure 6.3.4
- 2
6.3.3 Counting Generic Trees DEFINITIONS
2 ) ( ½ ¾ ( ) ) ) )
2 &( ( %
524
Chapter 6
ALGEBRAIC GRAPH THEORY
2 + , FACTS
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5$*" "
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3)
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525
Enumeration
2 ) "
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½
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2 6 * 6E"82 Æ " $+, )
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2 7 B 0
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Figure 6.3.5
- 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!?( (
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! " ? B ! " ? B ! " ? B ! " ? B
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2
0 % 7 ( 0 ( 0 ?
2
7
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
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+, 4
½
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H " H ! H
2
6A =8 Æ 0 +, )
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2
I + , : % ( & ) ½
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4 H H H H ? H " H
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H
+, + , H " +, +, H +, H + ,
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4 H H H H H H H
528
Chapter 6
ALGEBRAIC GRAPH THEORY
2
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2+, +,
+, 4 2+ , H +,
½¾
+,¾
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2
7 0
Figure 6.3.6
# - 2 "
2
% 7 ) & 0 ( 0 ( " 0
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2
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2 & ) ) > ) ) % ) )
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REMARKS
2 2
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Section 6.3
529
Enumeration
Table 6.3.10 #
" # " # -
! " ? B ! " ? B ! " ? B ! " ? B
2 "
+ ,
+ -,
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? " ! ? "B ("" (! B(? ("? B( "(" (? ?B(! (!"( " (!("B (?B( (!?!("" ?("?( B(("B !(B( (?B(B!( ((" (! B( B(B!(!? !(!((! ? !((B""( B ""( ( (? ?(!"(?(?B (?(?!(""(" (B(?B(!"B( "(? (!!(!(? ( !(!"(?(? ? ("("B(!(
2 A E )
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 ? !
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 +""?,( !
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6=8 7 =( ) ( ( ( ( ! !" +?,( M 6= ?8 7 =(
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6=!8 7 = > # ( #
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532
Chapter 6
ALGEBRAIC GRAPH THEORY
6=?8 7 = ( ) ) ( ( B +??,( M 6 8 > J )( > ) ( 6# "8 K 1 # (
! !( = ( 1 ( ?
6E"8 E( ) (
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2 ? +?",( "M??
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" +? ,( BM
2 # !* # !*
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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 > )
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5
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6.4.1 Basic Concepts and Definitions
( ( )
) I 7 - ( ) ) % 7 ( 6Q??8 65?8
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4 ½ ¾ (
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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 + ,( ) , + ,
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 $
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
& )
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
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 ^,
+
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 ( "
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+ ,
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+ ,
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
) ) ) )
) ) >: (
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
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
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 ) ( (
$
,
&
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
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+ ,
)
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% ,
& +
% -
,
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
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
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
) : ) :
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( @ &( )
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
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!
&% !
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
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!
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 (
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
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,,
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
& )
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 ( (
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
$ & & ( &
% (
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+ < , )
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
" + * ',
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+ , $
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+
,) ,
, +
,
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
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H +
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,C 4 D
+ 6 ,
2
¾
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,) H F
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¾
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6 H 6
6 ( ,¾ ( ,¿ ,¾ H ,¿ 4
,¾ 6¾ H ,¿ 6¿ 4
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5 6¾ 4 6¿ ( ,¾ ,¿ :
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EXAMPLE
2
¾
7 (
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¾,(
H +
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6, H + , 4 ¾ H +/ ,)
H ,)(
6¾ 6¿ 4
FURTHER READING 7 ) &
6 B8
6AI J"?8
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
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 ) ( ))
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
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,
2
6
4 ; + ,
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( " &! (
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!
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?
*
(
" !
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 , ))
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
+ ,( %* "
9
E'JN 5> +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
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
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 -
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
&
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3 E )
, ( (
E+,
+
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+
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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
)
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
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
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 ( ) + ,(
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+ 6 8, + ,
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9 +
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,
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
9)
9 ( 9 9)
582
Chapter 6
$* #
+9 , "
"
"
C +9 , " C , +9 ,
+9 , "
"
9½ 9¾ (
+9½ , +9¾ ,
2
+9½ , +9¾ ,
4
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+9 , "
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5
9 " (
ALGEBRAIC GRAPH THEORY
2
C , +9 ,
9¾ (
+9½ , +9¾ ,
4
EXAMPLES
2 9 + , 4 9 + ,( ) ) 2 9 + ,) 4 9 + ),( ) ) )
(
( )
2 @ 4 @
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2 @ ) 4 @ ½
½
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½
4
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& 9 (
9 + ½, 9 + ¾,
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) ) - 9
FACTS $
2
(
9)" ,
+
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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
,
+
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,
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 ( (
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 )
+ ,
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2 +>U , 6>?8 3 )
+ , + + , ,
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, : )
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2 6J '??8 > ) + ,
2 6#BY8 > ) C
) P + 6E&B8,( OPEN PROBLEM
4! 3 9 ) N 9
+9 , Y
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
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
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
(a)
4
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 $* 3*""# > * " I
>&
@
@
)
@
@ 7 7
I
>&
@ @ 7 7
@ 7 7 9
+ , 9 + ,
+
,
+
@ 7 7 9 9
,
2
)
CONJECTURES
9 )
588
Chapter 6
ALGEBRAIC GRAPH THEORY
1! + * I 9 6 !8, 7 % % ( % &
) 1! 7 % % ( /
) ( % &
/ REMARKS
2
7 " )) (
Æ 7 ? 7 " & 9 ) 7 " ) &)
2
7 ) ( > ( &
7 +>, ) % ( ( G 6GBB8 7 +,( ) & &
* I 9 > & 5 I 9 I ) ) 1 61B8 ) ( ( 1 61B8
6.6.10 Wheels, Whirls, and the Splitter Theorem 6 8 % ) ) ) - 5): ( C 6 8 5 ( C ( ) 5 65"B8 ( ( ) J
6J"8 DEFINITIONS
2 7 ( 0 ) & 9 ) + , &
2
7 ( 0 0 0 & ( -
2 3 9
B
B
)
9
B
9
2
/ - ½
/ ( ' ( ' - ' H
EXAMPLES
2
7 0 ( ( 9 +0 , 0 9 +0 , 0
Section 6.6
589
Matroidal Methods in Graph Theory 1 5
1
6
1
(a)
(b)
6
5
(c)
6
5
2
3
2
4
Figure 6.6.5 :;
3
2
3
4
0
4
" :$; 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
# .' $ *""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 ' (
' 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 ,
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 ) (
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 ? ) ) '
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
)
# ( + ,
"
+ , + ,
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595
Matroidal Methods in Graph Theory
+
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61B8 K 1 ( A * 9( " +BB,( MB
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Chapter 6
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6 K??8 3 A K- ( ) ) ( +???,( ?M
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597
Matroidal Methods in Graph Theory
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Chapter 6
ALGEBRAIC GRAPH THEORY
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GLOSSARY FOR CHAPTER 6 "?% 3 M
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Chapter 6
ALGEBRAIC GRAPH THEORY
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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
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TRIANGULATIONS
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GRAPHS AND FINITE GEOMETRIES
! # # $ GLOSSARY
7.1
GRAPHS ON SURFACES
Introduction ! "#$ % ! & ! ' ! #!! ! !
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* ! + ! ! ! &
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! . + ! #! ! ! .
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Chapter 7
TOPOLOGICAL GRAPH THEORY
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!
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"* / " ¼$ ! ! ! " $ ¾ , ¾ , ¾ 3 * / " $ #!! ! ! !
" $ ¾ , ¾ 3
Section 7.1
613
Graphs on Surfaces
¼
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Chapter 7
TOPOLOGICAL GRAPH THEORY
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Surface Operations and Classification DEFINITIONS
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615
Graphs on Surfaces
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Chapter 7
TOPOLOGICAL GRAPH THEORY
7.1.2 Polygonal Complexes DEFINITIONS
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617
Graphs on Surfaces
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Chapter 7
TOPOLOGICAL GRAPH THEORY
7.1.3 Imbeddings DEFINITIONS
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619
Graphs on Surfaces
FACTS
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620
Chapter 7
TOPOLOGICAL GRAPH THEORY
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Section 7.1
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Voltage Graphs
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TOPOLOGICAL GRAPH THEORY
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Voltage Graphs
677
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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
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Chapter 7
TOPOLOGICAL GRAPH THEORY
References A/ FD /0 !
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Chapter 7
TOPOLOGICAL GRAPH THEORY
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AG 7-D < G < ; 7+ . ! ! \" $0 &
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THE GENUS OF A GROUP
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Extremal Graph Theory
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ANALYTIC GRAPH THEORY
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Random Graphs
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ANALYTIC GRAPH THEORY
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Ramsey Graph Theory
857
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859
Ramsey Graph Theory
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ANALYTIC GRAPH THEORY
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ANALYTIC GRAPH THEORY
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ANALYTIC GRAPH THEORY
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The Probabilistic Method
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Chapter 8
ANALYTIC GRAPH THEORY
GLOSSARY FOR CHAPTER 8
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* ' * ' * ' * DEFINITIONS
3
( * + . 5
+ . * 1
3 4 * & +. 5 & +. 5 +.&
+ . 5 + + .. 4 & /$ * + . 5 + . & " # + . & & * 1 $ & + . & "#& * '
"# 5 1 " #
3 A * B + . 5 "#& ( $ $ ' ' $ $ "#
##
3 ( 1 $ $ $ " # $ $
EXAMPLES
3 B 4 $ ; & 5 5 +. 5 5 +. 5 & * 1 < * 1 4$ & * 1 $
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--. ( * $ $ $ ' * '
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 *
$
! "# $
$ '
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
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
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 # % & " # %
#
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
& 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 -
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* ' " #
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 -
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=
!'
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
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
&
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3
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"
3
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-
'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
( '
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& $ % (
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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 # & #
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 @#
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 $
* + '-.
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 "# & & % "#
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
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
( #
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 & % " #
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
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"#
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 &
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 ( #
)
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 "#
$ $ $ $ + . $
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 * %
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 ;;.#
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-@-
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
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-
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;
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
910
9.3
Chapter 9
GRAPHICAL MEASUREMENT
TOLERANCE GRAPHS ! " " B
Introduction $ * 1 * ! 2 $Æ
' &
& 1
B & * * N ' $ * $
(
( $ $$' ' * '& $ ' !$1
9.3.1 Intersection Graphs ( ' ' * & * ! ' * ' Ordinary Intersection Graphs DEFINITIONS
3 A 5 ' & T"#& * * 1 ! ' 5 / 5 ( 1 ' $ 5 T"#& ' ' ' ' "# 5 $& $ "# ' 5 C 5 T"#&
3 & "#& $ % ' $ ' $
3 ( ' ' 5 $ $ * ' $ " # $ " # 4 $ $ $ 1 /$ * 5 ' 3"#
6 & /$ /$ 1 $ %
Section 9.3
911
Tolerance Graphs
EXAMPLE
3
$ 5 5 & 5 & 5 & 5 & 5 5 T"# 4 $
Figure 9.3.1
( 1
FACTS
3 3 3 3
" 2 ) H # +. 6* '
& "# 5 3"# +LC;=. 3"# >% +6@@. 4 ' 5 "#& "# +6@@. 4 * '
REMARK
3
' $ $ +
.
Interval Graphs
B * 0 $
%
' +;. $ ' +, . ' $
* $ +4 =.& +=-.& + .& + =. :' )$ $ DEFINITIONS
3 ( '
*
3 ( * * *
3 ( * * ' 3 (
* $ & ' * & $
!
3 ( $ $ % 2"# 0 " # " # 2"# " # 2"# * "#
912
Chapter 9
GRAPHICAL MEASUREMENT
EXAMPLE
3
4 4 $ & *
Figure 9.3.2 " #
# #
FACTS
3 +A ,@. ( * ' $ $
3
+ @. ( * ' $ $ *
3 +@ . A /$ * 3 "# * " # $ * " # * $ $ Chordal Graphs DEFINITIONS
%
3 (
$ $
'
&
3 ( ' $
3 3
( * 1
"% ' # 0 &
$ $
( * & * 1 $ $ * 1 $
&
REMARK
3
$ : $ +,A . + .& 1
FACTS
!3 %3
+,$;& ;& C;=. ( ' $
+4$@& ;-. ( '
Section 9.3
913
Tolerance Graphs
Competition Graphs DEFINITION
3 A # * 1% % 2 4 2& ' 4" # 5 " # 2& 5 4" # 4" # " T"# ( 1 # $ 5 T"# ' $% * # EXAMPLE
3
4 $ #
Figure 9.3.3
( #
REMARK
3 E ' $ ' E +E;=. % $ '
' B 1& # $$' ' +;=. $Æ
$ * & $ % ' $ * /$
1 * ' $
"
A$= & L # $ * %
2
+ . ' FACTS
&
3 + $,=& A$=. ( '
' * 1% "# 5 $ /$ * & & 0 5 /
3
+ $,=. ( ' '
3"# "#
3 + $,=. ( '
* 1 "# 5 & /$ * $
5 /
3
+$ =. ( "# ' /$ * % & " & 1 & 5 & $ #
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 * ' & % $ ' % /$ *
/$ $
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 $ % * & $ *
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 * & ' ( & $ $ $ $&
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 "#
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@-
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 $ & ! # & ( &
& =-
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;=
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@;
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 )
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 $
$ $
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
+=-.
*
%
* ' ' 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 &
* *
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 ' & * * *
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
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 " "# #
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
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 " #
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# " #& $
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
& $$'
*
FACTS
3 +E;-. B
/$ & , 1 "/ # / 5 '
, Æ 3 +EE =. B
& , "Æ #
Section 9.4
933
Bandwidth
EXAMPLE
9¿ $ 4 $ = , 1 * ¿ & Æ 5 & , 5 ""# # 5
3 4 % ' $
Figure 9.4.8 5 + ) +
Æ 5 , 5
Number of Vertices and Edges for Arbitrary Graphs FACTS
3 + $,= . , "$ 7 #
3 +, $=. ,
" # =$
& , & $ " # " # $ & $
!3 +AC ;%. B
%3 + $,= . B , &
$ " #+" , #. &3 + $,= . B , & $ " # + " #. 3 +(A 6 . A , 5 " <# - 5 < 5 * % & " <# & $ & < " , # & " "<# < " , # $ $ * ,
Number of Vertices and Edges for Graphs with no
FACTS
3 +E=. A " , # 1 $ $ %* 1 * , * , " , # ,
2 & & , & " 7 #, " 7 #, $ , 7 " 7 #, 7 " 7 #, & $
3 +,E $4 J --. A
934
Chapter 9
GRAPHICAL MEASUREMENT
3
+,E $4 J --. A 2 & 5 " 7 #, 7 =, = 01 $ - = 5 & , " 7 7 #" =# $ , " #& $
Radius and Diameter DEFINITIONS
3 & "#& $ $ * 1 * ' * 1
3 & "#& 1 $
' * FACTS
3 3
+EE =. 4 ' & , Æ "Æ #
+E;-& E L;. 4 ' & " # "# ,
"#
3 +E = . 4 ' & , 1 " "¼# # "¼ # 1 % $ ) * $ ¼ * * REMARK
3 '
* $ 4 @ 4 $ % $ * 5 & 5 & , 5 &
* $ $
Figure 9.4.9
5 + ) + 5 5 , 5
Vertex and Edge Chromatic Number DEFINITIONS
3 "* 1# & >"#& $ $ $ ' 3 "# ' &
& ' " # 5 ' "#
!3
& >¼ "#& $%
$ $ ' 3 $ "# ' &
* 1& ' " # 5 ' " #
Section 9.4
935
Bandwidth
FACTS
!3 %3
& , >"# +, $=. 4 ' & , > "#
+E L;. 4 '
¼
Vertex Independence and Vertex Cover Numbers DEFINITIONS
%
3 & "#& ' * $
&
3 & ="#& % ' * $ * ' *
FACTS
&3
+E;-& E L;. 4 '
&
"# , "#
3
+ ;@. 4 '
& , ="#"#
Girth, Vertex Arboricity, and Thickness DEFINITIONS
3
& "#& 2 $ '
3 & ; "#& $ $ $ "# $ * $
$ ' $
3 & $ $
&"#& $
FACTS
3 3
& , " "# #" ; "# # 7 . B &
+, $ . B +, $
, +" "# # " "#. "# 7
3
+, $=. 4 '
& &"# 1 ", #
936
Chapter 9
GRAPHICAL MEASUREMENT
9.4.6 Related Concepts $' * *
' Bandsize
, 2 $
O' +AC
.
DEFINITION
3 A ' $ ' & "#&
$ N $ ' ' * ' "# 5 "# 3 ' $ FACT
3
+6 C = . 4 ' & , "# "#
Edgesum (Bandwidth Sum)
$ 0 +@. 6 $ $ + ;-& B;& B;@& EE =& E==& KC & K$$ & 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 (
'
! $ "#
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
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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 ' !
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B " 8 " ; / 0 " ) J ) " ; " 1 2 ( 0 0 ")0 3 0 3 " 1(1,1)
3(2,2)
2(3,3)
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6(6,4)
Figure 10.1.9 . # ! ' '
"B " 3 ")0 0 0
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Figure 10.1.10 ( 1 # 2 / ' .
B 2() 3 3 " "
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4 %"
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1
1
1
1 {1} an SC
2
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{2,4,5}
{2,4,5}
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4
5
6
6
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Figure 10.1.11 . # ! ' '
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4 .
B ' $5 /@1 00 REMARK
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a
b
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d
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{c}
e
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Figure 10.1.12
Bridges and Cutpoints of an Undirected Graph
+
? / 1 "
DEFINITIONS
B ' (
"
B B
'
'
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B 9 $ 0 0 ) $ 2!" " "0 )( ) . 3 3 "
0
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TWO EXAMPLES
B " 3 3 0 . # " . & 0 ) + " 0 " "
:3. " 0 0 1
2
11
12
14
Figure 10.1.13
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
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! B " ? / 1 1B 0 ) 0 " B 3 " " B B )5 . )5 0 /
1
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MORE EXAMPLES
B " 03 " (" ) "
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 % ' "
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Figure 10.1.15 /
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DEFINITIONS
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B
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Figure 10.1.18
! ! ! 4 5 3 % '
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4 ) ) ) 3 " : > 0 " 3 . " ". 4 DEFINITIONS
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a
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B " 0 : > 3 ) '" 7'"# =B 0 /1
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Figure 10.1.20
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FACTS
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Figure 10.1.22
#'(
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1
2
3
4
5
6
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FACTS
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B
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0 ")0 . 3 ) " " 4 (
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B
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! B 0
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FACTS
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10.2
DYNAMIC GRAPH ALGORITHMS
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Chapter 10
GRAPHS IN COMPUTER SCIENCE
DEFINITIONS
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987
Dynamic Graph Algorithms
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Chapter 10
GRAPHS IN COMPUTER SCIENCE
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Section 10.2
Dynamic Graph Algorithms
Figure 10.2.1
989
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56 56 . !
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GRAPHS IN COMPUTER SCIENCE
FACTS
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Figure 10.2.2
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Section 10.2
991
Dynamic Graph Algorithms
REMARK
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APPROACH
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Chapter 10
GRAPHS IN COMPUTER SCIENCE
APPROACH
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Dynamic Graph Algorithms
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GRAPHS IN COMPUTER SCIENCE
REMARKS
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Dynamic Graph Algorithms
995
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Chapter 10
GRAPHS IN COMPUTER SCIENCE
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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 ! " & ( '
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Chapter 10
GRAPHS IN COMPUTER SCIENCE
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10.2.3 Minimum Spanning Trees 0 ( ! ! . ! ( ! !
Section 10.2
Dynamic Graph Algorithms
999
! 9: > 0 ! 9:D ;<> ) - ! Decremental Minimum Spanning Tree APPROACH
& ! ! ' ! ' ( ! ! ( !' ' ?- Ü ! ( ! - ! ? ! ( 0 5 ! 6 ' ! 2 ( ! & ' ! ( ! !
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1000
Chapter 10
! H
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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
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Section 10.2
1001
Dynamic Graph Algorithms
( - ! ' " G " 9 #> G 9 $ > H ! 9$ #> ' % G %9 #> G 9 #> ! 9 #> 2 ! !
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1002
Chapter 10
GRAPHS IN COMPUTER SCIENCE
DEFINITIONS
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Section 10.2
Dynamic Graph Algorithms
1003
/ 0 9&>'
Æ A = ' ) ( ! Long Paths Property
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Chapter 10
GRAPHS IN COMPUTER SCIENCE
REMARKS
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Dynamic Graph Algorithms
1005
FACTS
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1006
Chapter 10
GRAPHS IN COMPUTER SCIENCE
/ D ! ! 9D ;;> ( ( 56 5¾¾6 ! ! 56
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Section 10.2
Dynamic Graph Algorithms
1007
! / ! 7 ' ! ./ 5 6 * 5 6 ' ! ' (
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1008
Chapter 10
GRAPHS IN COMPUTER SCIENCE
½ ' 0' ½' ½' '½ ! ' ( ' ' ' 1 ( !/ ! 7
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Dynamic Graph Algorithms
1009
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1010
Chapter 10
GRAPHS IN COMPUTER SCIENCE
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Dynamic Graph Algorithms
1011
RESEARCH ISSUES
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1012
Chapter 10
GRAPHS IN COMPUTER SCIENCE
References
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Dynamic Graph Algorithms
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Chapter 10
GRAPHS IN COMPUTER SCIENCE
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GRAPHS IN COMPUTER SCIENCE
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Chapter 10
GRAPHS IN COMPUTER SCIENCE
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Chapter 10
GRAPHS IN COMPUTER SCIENCE
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Chapter 10
GRAPHS IN COMPUTER SCIENCE
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1023
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Chapter 10
GRAPHS IN COMPUTER SCIENCE
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Chapter 10
GRAPHS IN COMPUTER SCIENCE
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10.3.4 Complexity of Graph Drawing Problems $M ( 0
1 + 2 6 , , F- F , 6 .M/, Æ 9 Æ
+ -
1028
Chapter 10
GRAPHS IN COMPUTER SCIENCE
( $$ (+ !+- / + /#"+ " + " " $ C#)) @6 '#;)
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,
4
#
Section 10.3
1029
Drawings of Graphs
( $$ (+ !+- / + /#"+ " + " " $ C#)) @6 '#;)
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1030
Chapter 10
GRAPHS IN COMPUTER SCIENCE
( $$ (+ !+- / + /#"+ " + $ C#)) @6 '#;)
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Section 10.3
Drawings of Graphs
1031
%3 + : +
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+
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, + % .! P /
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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
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
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60 19
23
7
26
25
8
9
58
37 42
3
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62
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51
49
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6 5 48
10
2
61
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44
53 30
15
22
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4
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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
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41
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Figure 10.3.3 4+""" $ 5+#+ ! 6 7 6 - ! ! ! 8 6 +" ! # ! " !7 /! " - " . /$ ! ' " " / ++#+ #+ / "$ " . " " / "#!" " + " / " + / #!$
* + + + F- , ) >)( A + % >FD((, GFD?A 6 + BU *0 >B*?&A
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 + + . / + -
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&3 , , D >D?$A - . / . / + . / +
1034
Chapter 10
GRAPHS IN COMPUTER SCIENCE
&
3 6 , C , 9 >69 A +
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9 >9 A % + - . / +
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+ -8 - + , + -
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+ - . / + Graph Drawing Checkers
5 + + DEFINITIONS
3 # +
5 5 + 4 + + % 2 ., , >C?(, *)F)· ??A/ + < = 5 +3
3 5 @ , 5 5
3 3
!Æ + 5 Æ
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REMARKS
%3
5 + 1, 4, 4 # , , + + 6 , + 9 4
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 , +
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5 9 >9?&A8 4 + + 5 >?(A 5 , ) , >)?&A8 + G >G?&A, 9 >9?(A8 - , , B , 9 >B9A8 5 F 9 >F9A Experimentation
* + * , + ; + , + + 0 , + +
1036
Chapter 10
GRAPHS IN COMPUTER SCIENCE
+ % + FACTS
&3
#
5 , # , >#A >A
&
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Fixed Parameter Tractability
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3 # W 4 # , $%& , + W . ./ / , 0, 0, ,
FACTS
& 3
1 F- , + + ,
.
Section 10.3
1037
Drawings of Graphs
+/ ; + , > A 4 % + 4 +
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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 ,
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
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1 /, + (?, ) -D , ??! >B9A * , ) , B B , ) G 9 6 ,
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1040
Chapter 10
GRAPHS IN COMPUTER SCIENCE
. / . *0, * BU , ) C /, + $!, ) -D , >C?&A , # , C , # , ' , " , 6 D , C D , #
,
& . ??&/, M! >CA , # , C , # , ' , " , 6 D , C D , 3 # ,
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. ??(/, (&M( >C9 A " , C , )G 9 , - , @ . S/, >C9A " , C , )G 9 , 5- - , , , ), N+ C , >*?$A * 5 * ; * , # . K/
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Section 10.3
1041
Drawings of Graphs
>)?A * 9 ), - 0 , (&M? " .
, ?? >D?$A C D , # , ? . ??$/, ?M!?
>9A D 7 +V ) 9, # Æ 4
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.
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1042
Chapter 10
GRAPHS IN COMPUTER SCIENCE
> A * , % , M$
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2
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>D?$A # , ' , D , , M$ # " ) # , + $, ) -D , ??$ >;?!A * ; , + ", / $ . ??!/, !!M& .) D 0 , 1 6 0 " / >;**A 1 ; , * , * ) * , + 0 + + 0 3 # +, ((( /* $ ./, M >B*?$A * BU *0, * 3 , $ . ??$/ . , ' / >B*?&A * BU *0, - 0 3 ,
. ??&/, M! >B?A B 9 B 05 , ' + +, " ((( ( . ??/, !M ! & >G" A * G 9 ."/, ) -D ,
,
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>GG* A G , G G, *0, # , &M! . / . B * 5/, + ?(, >G)(A B G 5 B ) , 5 , M! " 3 , N ) , ?( >C?!A C ,
- , ?M! 9 5 # ) , + ?!!, ) -D , ??! >C"?!A C " , 1 + , &$M(& . S?/ . '
1 /, + (?, ) -D , ??!
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,
: ,
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>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 ,
1044
Chapter 10
GRAPHS IN COMPUTER SCIENCE
>@)?A # @5 , , G ) ,
/ , B 9 R ) , ??
>#A ; , B # # , , # N*C 3 N ,
$ ./, !!M&? >)?A 6 * 1 ) , ) -D , , ??
,
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(((
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, " . ??/, !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!
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 ,
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
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 -
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 -
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-
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
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(
( ( ( ( ( ) + + +, + +
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 , +
. $/ -
% 3 6 6 , - -
-4
%3 1 , 7 55 6 , -+ & ., + / 7 55 , , 7 55 + @ , , 7 55 + % 3 ) - # 1 ' 5 , , , # , &+ , >'(!A
Section 10.4
Algorithms on Recursively Constructed Graphs
1053
10.4.3 Algorithms on Treewidth- Graphs DEFINITIONS
T .)* / + .3 0 4 / 3 0 4
) + 4 ¾ 3 T ) ./ * 0 4 3
01 4 1 0 , + 3 3 3 3 + - ¾ 3
3 5 + 3 #
3 #
FACT
&3 "+ - - >)(?A 9 . 3 / T . ½3½ / . ¾3¾ / 3 ) +
#&&> A, ½ ¾ + - + -
- , 3 P 6 , - 3 ) + #&&> A # , >' A T - ' 3 3 ' ( T -
5- ;"$ 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:
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
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 /
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Chapter 10
GRAPHS IN COMPUTER SCIENCE
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1057
Algorithms on Recursively Constructed Graphs
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Chapter 10
GRAPHS IN COMPUTER SCIENCE
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1059
Algorithms on Recursively Constructed Graphs
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1060
Chapter 10
GRAPHS IN COMPUTER SCIENCE
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1062
Chapter 10
GRAPHS IN COMPUTER SCIENCE
FACTS
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Chapter 10
GRAPHS IN COMPUTER SCIENCE
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Chapter 10
GRAPHS IN COMPUTER SCIENCE
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GRAPHS IN COMPUTER SCIENCE
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GRAPHS IN COMPUTER SCIENCE
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Chapter 10
GRAPHS IN COMPUTER SCIENCE
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11.2
MINIMUM COST FLOWS
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Introduction + '" " , + ! '' - - + ! " , ' '
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11.2.1 The Basic Model and Definitions DEFINITIONS
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11.3
MATCHINGS AND ASSIGNMENTS
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11.3.1 Matchings ! / !'- ! ! ! " * " '' - * !- - ' ! - ! ' - ! DEFINITIONS
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Figure 11.4.5
$" !, "-$
Diversification and Reservation Model [DR]
) ! * - + (
" ! =
2 Z ' ! " ,- 2 > ; ' ! ? ! ' @ - - , 2 > - 2 > : : - ! : , ?2@ > ? ?2@ ?2@@- ?2@ ?2@ " !- ' *- ' ! 2 ?2@ " " ' " ?!@ ;?!@ ' ! 2 $ % &% ( ! %
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REMARK
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! ' ! " ?&@ ? ! ! ?!@ ;?!@ @ + " Æ ?@ * / " ! ! ! * ! 5 2534 2 :PA364 " " ! ' '' " * !
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' : *
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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
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
2 64 - 1' 5 " C, C
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)- J' - )- !- - 363 2 394 - - *+! )( (- 6 ?339@- ;G6 2CA664 : 8 C 8 5- C O,- 366
A- * $ - P A [
2134 P 1
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GLOSSARY FOR CHAPTER 11
, #
G * != ' ! "
, , #
G * != ! ' " * " *
, , # , -
G + ,= ' "
!#!$ G , > ? & @= I ! * I > ? @
-! * ! ? @ G = -* * G "
" = ' "
" ' ?'@
- "" =
! " 0 ! * * " ! '- " *
!! G + , > ? # $@= + " '' * $ ! *$ ! G
, > ? @=
" * ? ( ;?!@
3(4@ ?!@
! ! !$ ? ' # @= ?!- ''
- /' - @ , !
! ! =
!' > ? @- " ? * @ " ! !!'
! - ! " ! ' ,
-!- = , ! ! ' ! - ! !! ""= ,- - " ! ! " -
, ,
-
"- = , Æ ' ! " ! ' '
! " > ? # $@= !' * - #-
! * ' " = #
- " = # ! '' * $ = % / $? @ > ;
% -
- = + , > ? # $@ ! * $? @ > ; " $?@ > $?@ ; > ? # @ " > ? # $@ , * > - # > # ? @ $? @ ; ? @ $? @ ; ' " /
* *
? @ ? @ > $? @ $? @
" ? @ " > " >
#
! G ' ! ' ? @ "
= " * ' " ' H -
!#!$ = " ' " ! "
- - >
? @
= " *- - =
* G " + ' = - /
G * ! '= * * ?
! "" ? @ G * = ? @ > ? @ ? @ !$= ?!- /' @ ! " " !@ " " '
! - * * !
G + , > ? @= " = ' " =
Æ
? @ ? @- " ? @
? @ > ? @ " *
? @
;- " ? @
G + , > ? # $@= " = #
% /
? @ " ? @ #7
= 2 ? @ ? @4 > $? @ " = ? @ ; " ? @ # - = + * ! + ? @ - = + ,- - ? @ > ! "" ! = + ! " +
! , - - > ? @ ? @ = ? @
> ? @ = !' * - ! * - " ! * ( " = Æ ," G ! , T > ? # 5 @= " = # Ê
/=
? @ " ? @ # = ? @ ; " ? @ #7
= 2 ? @ ? @5? @4 > ; " = ? @
-
= + ! ( " + ! ! *
-
! " = + ! (
''
?@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 * ! '= * ? " !@ " " '
# = * " +- " = % " =
? @
'
? @- " ? @
? @ ? @ ; " * ? @ ;- " ? @
*!- ! ? @ , " ! * += ? @ ; "* !#!$ ? @ G " ? @ , " ! * +
'+ = ;- / ;
"* G " + ,= ;- / ; "* G " + ,= ;- / 3 ! G ' ! ' ? @ " =
" * ' " ' H -
!#!$ = " ' " ! "
- - >
? @
= " *- - = > ? @ = !' * - ! * - " ! * ( " = Æ " * - "" = ' ! ! * 1
# - = ! * ! !' " * -
/ - / ! !'
" !
G
,= . ' " ! ' !
Æ
"" # = + , > ? # $@ * / ' -
-
"" ! *= ' , ' ! > ? @
? @- ? @- ? @ * ! / ' - ;- ;- ' *H '' / $? @ > $? @ >
! # =
' ' , !
! ! * ! " / ' - B - - > ?@ ?@
"#- # - =
+ '
'
! G " !' = " * ! " , # G * ! - = " ! " "
! " ! " ! - ? @