Graph Theory and Its Applications, Second - Jonathan L. Gross.pdf

DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN

Graph Theory and Its Applications Second Edition

DISCRETE MATHEMATICS and ITS APPLICATIONS Series Editor

Kenneth H. Rosen, Ph.D. Juergen Bierbrauer, Introduction to Coding Theory Kun-Mao Chao and Bang Ye Wu, Spanning Trees and Optimization Problems Charalambos A. Charalambides, Enumerative Combinatorics Henri Cohen, Gerhard Frey, et al., Handbook of Elliptic and Hyperelliptic Curve Cryptography Charles J. Colbourn and Jeffrey H. Dinitz, The CRC Handbook of Combinatorial Designs 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, Second Edition Jonathan L. Gross and Jay Yellen, Graph Theory and Its Applications, Second Edition 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, Second Edition Daryl D. Harms, Miroslav Kraetzl, Charles J. Colbourn, and John S. Devitt, Network Reliability: Experiments with a Symbolic Algebra Environment Derek F. Holt with Bettina Eick and Eamonn A. O’Brien, Handbook of Computational Group Theory 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 William Kocay and Donald L. Kreher, Graphs, Algorithms, and Optimization 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

Continued Titles Richard A. Mollin, Algebraic Number Theory Richard A. Mollin, Codes: The Guide to Secrecy from Ancient to Modern Times Richard A. Mollin, Fundamental Number Theory with Applications Richard A. Mollin, An Introduction to Cryptography Richard A. Mollin, Quadratics 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 Jörn Steuding, Diophantine Analysis 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

DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN

Graph Theory and Its Applications Second Edition

Jonathan L. Gross Jay Yellen

Boca Raton London New York

Copyright Jonathan L. Gross and Jay Yellen

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20110713 International Standard Book Number-13: 978-1-4200-5714-0 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

PREFACE

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ABOUT THE AUTHORS Jonathan Gross is Professor of Computer Science at Columbia University. His research in topology, graph theory, and cultural sociometry has earned him an Alfred P. Sloan Fellowship, an IBM Postdoctoral Fellowship, and various research grants from the Office of Naval Research, the National Science Foundation, and the Russell Sage Foundation. Professor Gross has created and delivered numerous softwaredevelopment short courses for Bell Laboratories and for IBM. These include mathematical methods for performance evaluation at the advanced level and for developing reusable software at a basic level. He has received several awards for outstanding teaching at Columbia University, including the career Great Teacher Award from the Society of Columbia Graduates. His peak semester enrollment in his graph theory course at Columbia was 101 students. His previous books include Topological Graph Theory, coauthored with Thomas W. Tucker. Another previous book, Measuring Culture, coauthored with Steve Rayner, constructs network-theoretic tools for measuring sociological phenomena. Prior to Columbia University, Professor Gross was in the Mathematics Department at Princeton University. His undergraduate work was at M.I.T., and he wrote his Ph.D. thesis on 3-dimensional topology at Dartmouth College.

Jay Yellen is Professor of Mathematics at Rollins College. He received his B.S. and M.S. in Mathematics at Polytechnic University of New York and did his doctoral work in finite group theory at Colorado State University. Dr. Yellen has had regular faculty appointments at Allegheny College, State University of New York at Fredonia, and Florida Institute of Technology, where he was Chair of Operations Research from 1995 to 1999. He has had visiting appointments at Emory University, Georgia Institute of Technology, and Columbia University. In addition to the Handbook of Graph Theory, which he coedited with Professor Gross, Professor Yellen has written manuscripts used at IBM for two courses in discrete mathematics within the Principles of Computer Science Series and has contributed two sections to the Handbook of Discrete and Combinatorial Mathematics. He also has designed and conducted several summer workshops on creative problem solving for secondary-school mathematics teachers, which were funded by the National Science Foundation and New York State. He has been a recipient of a Student’s Choice Professor Award at Rollins College. Dr. Yellen has published research articles in character theory of finite groups, graph theory, power-system scheduling, and timetabling. His current research interests include graph theory, discrete optimization, and graph algorithms for software testing and course timetabling.

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CONTENTS Preface 1.

INTRODUCTION to GRAPH MODELS (( ? # (# % @ (: (1 4 $ ## (3 * ? #! (: ; % 1. () F $9 4 $ 3! (+ ' :( :1

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INTRODUCTION TO GRAPH MODELS 1.1 Graphs and Digraphs 1.2 Common Families of Graphs 1.3 Graph Modeling Applications 1.4 Walks and Distance 1.5 Paths, Cycles, and Trees 1.6 Vertex and Edge Attributes: More Applications

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+ 27, 28 D + ! - +K + 27, 28" *, 8 + 27, 28# J + 27, 28% # + 27, 28& J

+ 27, +9 ' 2 7,! +: ' 2 7,! +? ' 2 7, 28( J + ' ',

' 28 J + ' ,

28 > )

+, '+, 7 +, 28 )

-

# # # $

28

38


28

28"

28# 6 K

28% > )

" 28& )

"

28( > ) 28 > " $ % " 7 * 28 > " $ ( % " 7 * 28 > " $ ( % " 7 * 28 )

!

" ! + ! , + ", H +" ! , 28 > ) + ", 7 " " + ", . + , H + ", 28" > " +, " )

Section 1.5

1.5

39

Paths, Cycles, and Trees

PATHS, CYCLES, AND TREES

F " " %

"

Trails and Paths

+ "

, " I

4

#

292 $ . !

2 .

"

$ . ! 2 . Figure 1.5.1

@! $ 2

!' J

" % " (

Deleting Closed Subwalks from Walks

$ .

$ ¼ . $" $ $" $ ¼ $ $ ¼ $ $ ¼ . $ $ ¼ $ $ ¼ " 4

$ $ ¼ $ ¼ $

297

$ $ ¼ $ $ ¼ 4

$ $ ¼

40


$ " $ ¼ " "

$ . $ ¼ . $ $ ¼ . Figure 1.5.2

! $ * *! $ ¼

!' + ,

$" "

. $ $ $ . . 2 2 $ $

$ "

" " ' $ "

' +

# $ "

$

$ ) $ " ' $ "

" ' $ +

# $ "

$

" =

"

" > 292 . "

/ " +

" " 297

Cycles

" +:*

,

2 3 4

Section 1.5

41


29* ! "

Figure 1.5.3

- ) +

4 +,' 6

6 5Æ +,' > 7 %

) " +3 4 , '

3 . + , 4 . " + ", # +3 4 ,

( > ) ) F

3 4

6 " " +

298, 6 ) )

)

)

F

D +3 4 ,

Figure 1.5.4

= Æ )

297 + "

2 ' +

> 2 ¼

2 # 2 ¼

2 ¼

42


!'

) 299 2

+

/"

, ( * * * * 2

2

2

) "

% 2 +

F

' 3 " 2

' H 2 F ) 299 2

* 2 *

' F 2 *

(

* 2

* * *

2 299

-

Figure 1.5.5

2

Eulerian Trails

2 . . 29:

# ( 298

- 89

Figure 1.5.6

5 6 #

Section 1.5

43


" & '% 6

29:

" K

/" 299 2

: Girth

" 29@ * * 7 2

Figure 1.5.7

Trees

Figure 1.5.8

$

" * 8

"

$ , %

# " 6

G

% 2

44


8 * 29B

Figure 1.5.9

) * * *

0 . % &

+

,

G

+

*7, 292C

Figure 1.5.10

*

+ ' 6 +

*7, Æ

#

(+#+,, 2922

*

3 % %

Figure 1.5.11

*$ $ !

Section 1.5


45

- , & ' ' 6

. ¾

( #

2927 3

Figure 1.5.12

. 1 *

7 .8// / 29 % "

K % K % K %

K " " " " "

29 % "

K % K % K %

K " " " 29 #

9

29 # @

"

46


29 J

29" J )

+ 27, " 9 B 9 : ? B 29#

29% "

29& > " $ % " 7 * 29( > " $ % 9 " 8 29 > " ( % " 7 * 8 29 > " $ ( % " 7 * 8 29 > " ( % * " 7 * 8 29 > " $ ( % * " 7 * 8 ++ +))

29 29" 29# 29% 29& 29( 29 29

% % ' * % D + - + K *, - *,K )

J #

29 J + 27, +9 ' 2 7,! +: ' 2 7,! +? ' 2 7, 29 J + ' ', '

Section 1.5


47

29 J + ' , 29"

Æ + ' , 29# J + 29% J )

29& J *, * 29( ) "

29 ) ' /

29 ) " " " " 29 6

) ) 7 29 6 297 29 6 ) 299 29" 6 29: 29#

/" 298

+

, 29% #

* 2922

#

2922K 29&

29( J

$ 29 % K 29 ) "

29 ) "

48


1.6 VERTEX AND EDGE ATTRIBUTES: MORE APPLICATIONS

# & " 2*

Four Classical Edge-Weight Problems in Combinatorial Optimization

$

$ /

-

" $ $ % 6

2:2

+ , " " +

8,

Figure 1.6.1

$ *

" -1

$ % 6

"

2:7 J - +

8,

Figure 1.6.2

Section 1.6 Vertex and Edge Attributes: More Applications

49

" $ % 6

" 2:*

G - #

+

:,

Figure 1.6.3

1

" -* $ % 6

2:8

" +

2*,

Figure 1.6.4

$; *

# %

Æ

47%8 # $ Vertex-Weights and Labels

" " "

" " &

50


" $ ' 59%

" %9 * * 3 N

" -

29

Figure 1.6.5

) A$ -

"" ,

$ $ ' J

" G

#

: < ' G "

- . 7 7

:7

!' I

G

"

" 7 .8// / " 2:

+ 2*2, " +D ' , 2: > " % D $ D K $ $, %

:

2: 6

"

D

K

Section 1.7

Supplementary Exercises

51

2: 6

D

- K 2:

/

Æ

% Æ

K 2:" '

%

. . D

- K 2:# 6

O P%D " 5 P%D /

' / P%D

. D K 2:% 6

'

9C . D

- K

1.7

SUPPLEMENTARY EXERCISES

2@ 7C " :7 / "

* @ D

*K 2@ / * @ " " 2@ ) 9 " @ 7 2@ I

D D

9***77 # 2@ J

,+,

22777** 2@" J

,+ ,

22777** /"

2@9 2@# D K 2@% D + K 2@& % )

K /" 2@( # +78 ' 2 9, +*,K

52


2@ # +78 ' 2 9, 7 " *K 2@ # +78 ' 8 B, K %

K D K 2@ % " +2* ' 2 9, CK 2@ +2* ' 2 9, 2@ ?? )

:7 *2 ( 2@" ) '

2@# ) " 2@% $ % 2@& > "

% " K "

( (

2@( > 9 ) +6

78, 2@ >

8 ( $ " % "

% K /" .

2@ ) "

C+ 8, 2+ 8, 2@

K 2@ 6

8 H 2 ) "

7 " ' ' C 2 ' ¼ ' C 2 ' + , +,

2@ ) * 2@" ) *

Glossary

53

GLOSSARY

' 0 *' " " 0 ' 0 1 ' ( '

! ' *

' "

& . $ . + , . . 2 7 + , . . 2 7 * '

"

+ ,

* '

* '' "

1 ' "

+, .

+, + ' - ,

- 2 2'

"

. C 2 2 . ( H . . . H . *, ' -

( '' 34 !' + ,

* '

$ ( ' ( $ $ '

$ $ $ $ '

'

'

' 2 " !

54


2 ' * * * (

* 2

2 "' "

''

'+,' '+, . ¾ " + ", '

' " "

+ , . # + , . . 2 7 ' "

" ' ' ' ! " " '

'

!

'

' "

$ " $ ( " $ " ' " ! +, '

$ ' "

( ( ' % $ '

+ , '

' '

' ' + , " ' ' " ' ' C 2 ' ¼ ' C 2 ' + , +,

Glossary

55

. + ,'

+ , /

'

'

"

'

* +'

"

$ " '

* '

" ' '

( '

(

1 '

"

'

,+, ' " ,+, ( " '

( ' $ '

$ '

* ' " ' '

"

34 !'

" $

" ' ' ' + "

",! '

56


+ ' * 2C " Q )

' " ' * E '

+,' " * " " ' " ! $ $ ¼'

$ ¼ " $ '

$ ' . $

'

" + ,' + ,

$ ! ' ' '

*! $ '

$ ' '

' ' 1 ' " " ' + , 1' 1'

! 1 1 ' "

$ " $ ! " " '

" " '

%

Chapter

2

STRUCTURE AND REPRESENTATION 2.1 Graph Isomorphism 2.2 Automorphisms and Symmetry 2.3 Subgraphs 2.4 Some Graph Operations 2.5 Tests for Non-Isomorphism 2.6 Matrix Representations 2.7 More Graph Operations

INTRODUCTION

! ! " # $ # # # # %

& ' ( ) 57

58

Chapter 2

STRUCTURE AND REPRESENTATION

* + ' ! , # # # -. # / # Æ Æ

2.1

GRAPH ISOMORPHISM

Æ !

! Structurally Equivalent Graphs

$ -00

Figure 2.1.1

, ' ' 1 ,

Section 2.1

59

Graph Isomorphism

-0- !

Figure 2.1.2

$

,

' # 0 4

- . 2 5 3 6 $ # 0 ' - 2 4 , # 7 (0) ' 7 (-) 7 (2) 7 (4) , # '

8 ' ' " Formalizing Structural Equivalence for Simple Graphs

9 # ' 8 ' () ( ) '

() ( ) ' '

# ' 8 # ' ' ' () ( ) '

7 # # ' 8

# 1 ! # 8 8

$ -02 %! # '

60

Chapter 2

Figure 2.1.3


$ -03

Figure 2.1.4

8 ' # ' ' 8 , 9 % 0: # ; 3 $ -04 ' ' ' , ' : - ' 3 .

Figure 2.1.5

!"# "$#% &

' # - $ -0. ' ' '

Figure 2.1.6

(# " $"% & "#

Section 2.1

Graph Isomorphism

61

Extending the Definition of Isomorphism to General Graphs

# # ' ) 8 # $ -05 () 7 7 0 - 2 ' '

Figure 2.1.7

&& *&#

# ' 8 # () ( :)

() ( ) () # # ()

# ( ) # # ' 8 7 Specifying an Isomorphism Between Graphs Having Multi-Edges

9 # ' 8 ' 8

() ( 8 8 ) 8 ( 7 8 8

(

, # ' ' % 7 # ' 8 1 ' 8

' () ( ) ( ) 8 , # ' 8 '

8 () ( )

62

Chapter 2


< ' # ' # ' 8 '

8 ,

+ # ' $ -06 / # ' # ' < 0-

Figure 2.1.8

Isomorphic Graph Pairs

! # #

(

9 8

' 8 ( ( )) 7 ( ) (

# # ( ) ( ( )) 7 ( ) ,

, ' 8 () - , $ -0= 6 0 2 # # ' # # ' ,

Section 2.1

Graph Isomorphism

Figure 2.1.9

63

.& &

/ $ -00: *> + . = 2 #

Figure 2.1.10

! 01 &

Isomorphism Type of a Graph

, 7 ( ) 7 ( )

7 (? @) AB# 7 / 7 , $ -000 %

( % 03)

Figure 2.1.11

& $#

Isomorphism of Digraphs

#

64

Chapter 2


() () ( ) 2 $ -00- C

Figure 2.1.12

3 & $

The Isomorphism Problem

, # %! # ' # # < D # ' ! #

+% % ! /$ % ! ! , +% /$ +% ( )

, +% ! /$ ! ! < ! +% !

4 ,5 !

"

-0 -0 -0 -0 -0' -0)

3 # 3 #

4 # . # 4 # # . # #

# $ !

"

-0+

3 # #

Section 2.2

-0-0/ -0 2

Automorphisms and Symmetry

65

3 # # 2 #

2 #

% !

"

-0 -0 -0 -0 3 # # # -

& ' ! !

-0 '

-0 )

-0 +

-0 -

-0 / E # 4 # -02 E # = #

2.2

AUTOMORPHISMS AND SYMMETRY

# ( ) 1 7 ( )

66

Chapter 2


1 # #

8 # ( ) ( ) # Permutations and Cycle Notation

, () ' # F ( ) ' F ( ) ' 7 : 0 0 # # 3 '

0 2 3 4 . 5 6 = 7 5 3 0 6 4 - = . 2 0 5 - 3

7 (0 5 = 2)(- 3 6 .)(4) Geometric Symmetry

# / B $ --0

Figure 2.2.1

$ 0-:Æ ! # ()( ) ( ) B # # ()()( ) ()( ) #

Section 2.2


67

G # /

()( )( )() ()()() 0-:Æ ()( ) ( ) -3:Æ ()( ) ( ) B ()()( ) ()( ) B ()( )( ) ()( ) B ()( )( ) ()( ) # -3 + -02 # 2 # # ' # '

# 8 /# # # # ' $ --- # ' , # 1 B B 06:Æ H #

Figure 2.2.2

& &

Limitations of Geometric Symmetry

! B $ --2 H 4 (: 0 - 2 3)(4 . 5 6 =) (: 4)(0 6)(3 5)(- 2)(.)(=) - B ( # . =)

( /# )

68

Chapter 2

Figure 2.2.3


(

Vertex- and Edge-Transitive Graphs

#

' /# ---

# 2 # ) # ( /# ) + # ( /# ) - H # ( /# ) / / !(F " ) # , # ; # , $ --3 $ # 2 # 5 !(02 8 0 4) ( /# )

Figure 2.2.4

& !(02 8 0 4)

Vertex Orbits and Edge Orbits

Section 2.2

69


(/ # -) 2 /# --2 ( $ --4) #

# 8 0 6 3 . - 5 2 4 8 0- 56 23 4. -2 -4 25 45

Figure 2.2.5

24

6

(

-02

(

% -04

8 # #

# / # # 7 F # < ( /# ) 8

% 03 04 # How to Find the Orbits

1 # (, !

# D # ' ) , --0 --- # #

70

Chapter 2


, $ --. # : # - B (:)(0 3)(- 2) 0 3 - 2 * 0 3 - - 2 0 3 - 2 # : 0 3 - 2

Figure 2.2.6

3 #

-2 2 :0 :3 - 2 # B :0 :3 0- 23 02 -3 (:)(0)(3)(- 2) 0- 02 -3 23 -2 :0 :3 0- 02 -3 23

1 3 $ --5 (: 4)(0 3)(-)(2)(.) ! < #

Figure 2.2.7

3 #

1 ! : - 2 4 2

0 3 . ' $ --6

Figure 2.2.8

3 #

, # : - 2 4 0 3 . " :4 -2

Section 2.2

71


4 ,5 %

--

--

--

--

& #

--'

---

--)

--+

( !

--/ -- !( 8 " )

-- 2 -- H

-- H # 7 % #

-- !(= 8 0 ) -- )

)* + !

-- ' !(02 8 0 ) -- +

' # !

-- - --2 -- -- --)

/# --0 /# --2 /# --4 /# --5 $ --=()

-- / -- -- --' --+

/# --- /# --3 /# --. $ --=( ) $ --=()

72

Chapter 2

Figure 2.2.9


--- / H /# --3 ( ) H - B () H () H --/ H /# --2

2.3

SUBGRAPHS

H # # $ 042 I ! % 5 ( ) ! , /

Section 2.3

73

Subgraphs

$ -20

Figure 2.3.1

7 8 &

A Broader Use of the Term “Subgraph”

? @

$ -20 # $ -2- $ () % ()

Figure 2.3.2

&

Spanning Subgraphs

7

' , $ -20

) $ -22

74

Chapter 2

Figure 2.3.3


% 2 3

(

!" # $%&

( #

!" # $%&

+ , $ -23 # 0- # /

Figure 2.3.4

Cliques and Independent Sets

" " ' " # ' # ' ?@ # &( ) - $ -24 , # 2

Section 2.3

75

Subgraphs

Figure 2.3.5

*&

" " ' " ' '( ) 8 &( ) '( ) , -3 Induced Subgraphs

$ ( (( ) # ( (

7 ( 7

)() (

/ , 2

2 # $ -2. #

Figure 2.3.6

& & & #

$ % (%) % # %

7

)() %

7%

, $ -25

76

Chapter 2


& & & # Figure 2.3.7

* ( ) ( 03)

$ -26 2 2

Figure 2.3.8

+$

8 , -3 Local Subgraphs

( ) # ( )

$ -2=

Figure 2.3.9

&

8 () ( ()) (

() ' ( ())

Section 2.3

77

Subgraphs

$ -20: 3 + '( ) 7 3 '( ) 7 - &( ) 7 - &( ) 7 2

Figure 2.3.10

$& -$#

Components

,

,

? @

'

5 # $ -200

Figure 2.3.11

&

# # ! C # #

, ( ) " # ( ) ( ) 7 ( )

, + ? @ % 3

78

Chapter 2


Application to Parallel-Computer Computation

! #

! 6 6 2

! 02 ( 0-) , 02 6 6 2-

8 , ! ( ) ! ( ) ! * ! % 0: , % 04 ? @

! 4 ,5 , ,

,

%

-2 , 7 F % 7 - , 7 F % 7 ) - , 7 F % 7 ) ! . -2 , 7 F % 7 , 7 F % 7 - , 7 F % 7 ! .

,

,, , ! ( ) ( )

%

,

%

-2 , 7 F % 7 ) ! -2' , 7 F % 7

-2 -2)

, 7 F % 7 . , 7 F % 7 .

,# , ! ( )

(

-2+ ( 7 -2 2 ( 7

(

-2- ( 7 -2 ( 7

-2/ ( 7 -2 ( 7

Section 2.3

79

Subgraphs

,, ,& !

-2 -2 )

-2 -2 +

-2 '

-2 -

,( ,

)+ -

) + )+ -

)+ )+ -

-2 /

-22

-2

-2

,, , !

-2 -2 -2' -2)

/# -20= /# -2-: /# -2-0 /# -2--

,# ,' !

-2+

-2-

,( ,, !

"

-2/ . # # -22 3 # # -2 4 # # -2 . # # -2 < J -2 H -2' H # -2) -2+ E - J -2- H 8 /

80

Chapter 2


-2/ H 8 $ # # , 7 (, ) -2 2 H 8 $ # % 7 (%) -2 9 ( , # & (( ) (, ) 7 (( , )J -2 9 % / & (%) (/ ) 7 (% / )J -2 % # -2 % 8

-2 ' H H -20 (< 8 H

) -2 ) H H -2- (< 8 9

)

2.4

SOME GRAPH OPERATIONS

% # ' , % 3

, K Deleting Vertices or Edges

, # # 7 7 0 )() * ( ( ( , 7 7 * % % % #

Section 2.4

81

Some Graph Operations

# #

Figure 2.4.1

Figure 2.4.2

& #

&

Network Vulnerability

!

% 4 # ( ( ( ) # #

, $ -32 ! !

! # # ( #)

Figure 2.4.3

& &$#

% % ( )

$ $ -33 F

! F !

82

Chapter 2


&$ 9# 8 ,

# ( ) LM # L M (

) $ # $ -34 LM L M C '

Figure 2.4.4

Figure 2.4.5

LM L M

( '

(

9 ,

%

+ + ; H -30 , ' ' (

H -30 ( ) , ' ( ) 7 (( )); 0

Section 2.4


83

The Graph-Reconstruction Problem

9 7 #

1 # ! #

$ -3.

#$ &

! 8 1 - # - ( ' ) ! 8 # # < HN I * & 0=30 ' , /# -33 ! $ -3. ( )

! Figure 2.4.6

Figure 2.4.7

5

& :

8 7 7 ( 7 ( 7 0 7

#

84

Chapter 2


'

(

, # 7 * - - 0 7

- , ' ' )

(

$ $ # , ) ) (

+ % -34 ! " ! 0 ' # 8 + *I L*55M C' LC55M # ' 0: Adding Edges or Vertices

#

) $ -36

Figure 2.4.8

# # # 8 .

K 1

Section 2.4

85


Graph Union

#

7 ( ) 7 ( ) # ' # + '

$ -3= # #

Figure 2.4.9

&

' - $ -3= - - Joining a Vertex to a Graph

# # ' -5 , # ' # ; ' # # # # , ' ; # ( # ) , , F ,

Figure 2.4.10

4$ , 7 ;

' ( +

(

$ O ' # F O 7 (( ; ) ; )

86

Chapter 2


$ -300 7 (O - ) ; ; ) , # - 2 3

Figure 2.4.11

0

Edge-Complementation

9 , ( ) # ' ' / $ -30-

Figure 2.4.12

( 7 7 ) C# # # ' 8 " 7

- &( ) 7 '( ) '( ) 7 &( )

8 " -5

Section 2.4

87


4 ,5 % %& !

-3 F - -3 F -3 F ) - -3 F -3' F # . -3) # H ( 0-) -3+ H %' % !

-3-

-3/

-3 2

-3

% %&

! !

-3 -3

/# -36 /# -30:

-3 -3 '

/# -3= /# -300

% %(

! !

-3 ) -3 -

/# -36 /# -30:

-3 + -3 /

/# -3= /# -300

%$ % !

"

-32 -3

3 #

3 #

% %& !

-3

88

Chapter 2


-3 -3

-3'

-3) < ! J (E !) -3+ < !

J (E !) -3- % ( ); 7 # -3/ H -32 9

H # -3 9

$

# -3 4 #

E # $ # # E # E # E #

E # -3 4 #

; # -3 9 # H 8 ; # -3' H 8 $ ; -3) Æ # # -3+ E '

Section 2.5

Tests for Non-Isomorphism

89

-3- H . ( ) -3/ H # # -3 2 9 H

2.5

TESTS FOR NON-ISOMORPHISM

, -0 8 1 # ! # '

# ! ! 1 ( ) ( )

' 1 -0 , A Local Invariant

' 8

(

( )

(

% -04

' $ -40 8 0 0 0 - - 2 + -0- # #

2 < 0 0 - ' 0 - - -40

90

Chapter 2

Figure 2.5.1


; $ *&

'

$ -4- - ( -40) + 2 2

Figure 2.5.2

; $ *&

Distance Invariants

9 , 7 ! ( ) 8 , ! (, ) 7 ( ) ( ) ( ) ( ) ( )

' ) ) (

, ' ( ( ( ( ( (

(

-4- ' , ' * 1( ! & !& 1 , '' ( ( (

% -42 ' ' $ -42 *> 6 # 2 #

Section 2.5


91

; $ *& ! 7 -6 # ! # # # $

$ *> ; 0 6 # # # : 2 # . #

# : - 1 '() 7 3 '( ) 7 2 Figure 2.5.3

Subgraph Presence

') * ' (

(

9 8

( ) ' '' $ -43 2 1 2 $ % -4. 2 < 4 2

3# && $ % -$#% $& % 1 '(2) 7 3 '( ) 7 2 Figure 2.5.4

Edge-Complementation

# ( # ' )

92

Chapter 2


'+ '

(

+ ' ' ') $ -44 ( -: -6 ) ' 3 6

Figure 2.5.5

# % $ '$&

Summary

, ( /# )

# 0 - 2 3 4 . 5 , 6 $ = $ Table 2.5.1

Using Invariants to Construct an Isomorphism

#

'+ $ -4. = # 0 0: + % -42 = = ' = = '

( ; 0) ' #

Section 2.5

93


'

Figure 2.5.6

(

4 ,5 ' -4 $ /# -44

& &,

-4

-4

&% &

-4 -4'

94

Chapter 2


-4)

-4+

-4-

-4/

-4 2

-4

-4 8 /

$ ( ) $ ( ) &, &' !

-4 -40 -4 ' % -43 -4 + -4.

-4 -4- -4 ) % -44 -4 - -45

-4 / % -40 &$ &&

-42

Section 2.6

Matrix Representations

95

-4

-4 -4

-4 -4'

2.6

MATRIX REPRESENTATIONS

A , Adjacency Matrices

2 # # ' 2 L M 7 0: ) $ -.0 ' # # : 0 0 : 0 : 0 : 2 7

0 0 : 0

: : 0 : Figure 2.6.1 "

& # # ' # 2 # #

96

Chapter 2


( ) 2 2 L M ! 2 ' ) !

(

! 7 0 ' # ! 0

# ( /# ) ' # ! % 2 # # 2 L M 7 0:

) ' # $ -.- #

Figure 2.6.2

: 0 : 0 : 0 2 7

0 : : : : : "

: : 0 :

( ) % 7 2 ( !#&

( ) % 2 2 L M ! 2 ' ) !

!#&

8 # ' # ( ) 2 L M # # ( # # ) Brute-Force Graph-Isomorphism Testing

# '

Section 2.6


97

) $ -.2 ' : 0 0 0 0 : 0 : 2 7

0 0 : 0

0 : 0 :

Figure 2.6.3

: 0 0 0 0 : 0 : 2 7

0 0 : 0

0 : 0 : & "

!&$ 8 / 8 A P/ C" , 7 A C" , A C" $# # 1 ' # 2 $ # 3 1 2 3 , 2 ( 3 ) 7 2 A P/ A C" Algorithm 2.6.1:

$ #

Æ

-0

Incidence Matrices for Undirected Graphs

# 4 # : 4 L M 7 0 -

98

Chapter 2


) $ -.3 #

4

7

Figure 2.6.4

:

: :

.

: : :

0 0 : :

: 0 : 0

: 0 : 0

: 0 : 0

: : 0 0

0 : 0 :

" 4 # #

( )

!#&

( )' +

!#&

/ , # # Incidence Matrices for Digraphs

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111

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&&$# # ' 8 # ' 8 ' ' ' () ( ) ' & %8 % % & 8 8 8 7 &8 ?@ F & # 8 8 4 ( ) # %8 (%) # & %8 (%) # 78 & 7 ( ) 7 ( )8 # ' # $ $8 ' F #$& 8 #$ & 8 # #$ & 8 # # 8 # , , # #$# 8 # $ ,8 ' ; # #8 8

Chapter

3 TREES

3.1 Characterizations and Properties of Trees 3.2 Rooted Trees, Ordered Trees, and Binary Trees 3.3 Binary-Tree Traversals 3.4 Binary-Search Trees 3.5 Huffman Trees and Optimal Prefix Codes 3.6 Priority Trees 3.7 Counting Labeled Trees: Prüfer Encoding 3.8 Counting Binary Trees: Catalan Recursion

INTRODUCTION

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$ & D $ ' $ & $ - '

( & , $

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158

Chapter 3 TREES

& /0 /0 &* & / 0 /0 &* & /0 &* &

C!

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3.9


@ F @ A

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@ $ (

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@ @ @ @ @

159


2 # F @ 2 : 2 " A G < ; 5 ) ;0

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@ " F 5! * 5! @ @# F 5! 5! @

Figure 3.9.3

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Figure 3.9.5

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160

Chapter 3 TREES

GLOSSARY

; 8

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6 ; $ $ 8 9 6' ; + $ ;

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161

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162

Chapter 3 TREES

&, 6; +

1 % 3,&; 3& ; ; & ;

,

; + 63 ; 0; +

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; +

+

; ,&; ; 2 , / 0 2 ,&; ; , / 0 2 2 ,&; ,&; ; 2 2 , / 0 + &*;

Chapter

4 SPANNING TREES

4.1 Tree Growing 4.2 Depth-First and Breadth-First Search 4.3 Minimum Spanning Trees and Shortest Paths 4.4 Applications of Depth-First Search 4.5 Cycles, Edge-Cuts, and Spanning Trees 4.6 Graphs and Vector Spaces 4.7 Matroids and the Greedy Algorithm

INTRODUCTION

!

" # $ %&' ( !

)

163

164

4.1

Chapter 4

SPANNING TREES

TREE-GROWING

! * )

$

Frontier Edges

*

$ $ +,,

Figure 4.1.1

- $ +,,

(

.

$ /+ Choosing a Frontier Edge

*

0

Section 4.1

165

Tree-Growing

.

- 1

2 3

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$ +,/ 4 " ' $ +,/ " ' 5 " '

Figure 4.1.2 Algorithm 4.1.1:

.

.

3 2 3 2 6 4 0 4 " ' 0 * " ' 7

. 8 !

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166

Chapter 4

SPANNING TREES

Discovery Order of the Vertices

0 9

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! "#

$ -

9 ( 9 " +; +<' 3 %, $ +,% ) 3 +; +< Two Kinds of Non-Tree Edges

$ +,%

Section 4.1

167

Tree-Growing

$ 9

1

$ $ $ +,% $ Depth-First and Breadth-First Search as Tree-Growing

6 ! 3

. . Resolving Ties When Choosing the Next Frontier Edge

3 2

$ " /<'

! 9 " ' = 3 Tree-Growing Applied to a Non-Connected Graph

"'

$

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168

Chapter 4

SPANNING TREES

Tree-Growing in a Digraph

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169

Tree-Growing

Algorithm 4.1.2:

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( $ +,; " ' 5

Figure 4.1.5

Some Implementation Considerations

. * 9 "* +,,'

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A " B*>?C%D BAC%D B CCD'

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170

Chapter 4

SPANNING TREES

8

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Section 4.2

Depth-First and Breadth-First Search

171

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4.2

DEPTH-FIRST AND BREADTH-FIRST SEARCH

" ++ ;+' " ,/;' " &<'1 F F 2 " ,%/'

" ' " ' G H * * +, Depth-First Search

6

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Algorithm 4.2.1:

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1 ' +

.

.

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.

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,/

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173


Recursive Depth-First Search

* $ "'. 6 0 $ "' 7

6 +,

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Depth-First Search in a Digraph

* +/,

*

% $ &

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Breadth-First Search

3 G H

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174

Chapter 4

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SPANNING TREES

1 2 " 2

Algorithm 4.2.2:

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.

.

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$ $ +/% 8 +/,

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Figure 4.2.3

3 2

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Section 4.2


175

6 "03 $ -'

"$ 3 $ -'

) %*+ + + / ! / 0# & & ' / ! & &# $ ) * + , *+

+/ +/ +/$ +/ +/( +/, +/-

E E E E E E E

/ % / !# & # $ # & ' / / & ( $ ) * + , *+

+/. +// +/ 0 +/ +/ +/ $ +/

E E E E E E E

+/ ( # 2

+/ , 6 ) I +/ - E +/ . 3 # % I J +/ / +// +/ 0 +/% +/ +/+

176

Chapter 4

SPANNING TREES

+/ B. D 3 " /<' #

8 +/, +/& +/ $ B. D 3 " /<' #

8 +/C +/,+ +/ B. D 3 * +/, " /<' #

8 +/, +/& +/ ( B. D 3 * +/, " /<' #

8 +/C +/,+ +/ , B. D 3 * +// ) # 8 +/, +/& +/ - B. D 3 * +// ) # 8 +/C +/,+ +/ . B. D ( * +/,

4.3 MINIMUM SPANNING TREES AND SHORTEST PATHS +,

*

Finding the Minimum Spanning Tree: Prim’s Algorithm

1 ( 2 . 0 $ 6 ,< A # " %&' !

7 # B ;&D G H ! 9 ! " '

Section 4.3 Minimum Spanning Trees and Shortest Paths

177

0

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.

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$ $ +%, .

Figure 4.3.1

5 "# 4

)

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178

Chapter 4

SPANNING TREES

"" @ ,' "' " ' " ' K 4 @ K " ' $ " K' 4 " ' @ "' " ' " ' K

Figure 4.3.2

% # $ % ()

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3 )

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Figure 4.3.3

.

+ # 4

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179

Finding the Shortest Path: Dijkstra’s Algorithm

( . 0

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3 2 3 2 6 4 0 .4 $%" " ' 0 * " ' 7

$ BD

$$ $ +%+ .

180

Chapter 4

SPANNING TREES

3 BD

Figure 4.3.4

"# 16 4

Calculating Distances as the Dijkstra Tree Grows

$ "' 5 $ +%+ . BD 4 BD @ "' $%" BD @ "'

$%" Æ 3

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3

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Figure 4.3.5

181

16 4

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8 ! 3 "

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+%

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182

Chapter 4

SPANNING TREES

" 4 " %# *' " /+ $ & # $ $ " " & # $ # $ $

+%( +%-

8 +%, 8 +%%

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8 +%/ 8 +%+

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4.4

APPLICATIONS OF DEPTH-FIRST SEARCH

3 The Finish Order of a Depth-First Search

6

Section 4.4

Applications of Depth-First Search

183

.

3 +/ "8 +/,&' 7 3 " %%' " 8 ' Growing a DFS-Path

*

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$ $ $

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Finding the Cut-Edges of a Connected Graph

* " ' *

0 , " ' * "$%' , " /+' 0 - - - " 8

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184

Chapter 4

Figure 4.4.1

SPANNING TREES

% - &

!

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Figure 4.4.2

% " &

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' %

.

.

3 2 - 6 + , 9 3 " ' 4 , ( - .4 - 8 =M M= 0 - 7

$ Æ

Section 4.4

185


Topological Sorting by Depth-First Search

* +/ (

* *

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+

.

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3 2 - 3 2 " .4 6 4 " ' 4 " - .4 - " .4 " , 7

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186

Chapter 4

SPANNING TREES

Finding the Cut-Vertices of a Connected Graph

2

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5 "' 3 3 (Æ "' 3 !

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Figure 4.4.3

1 2 7

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Section 4.4


187

++L

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$

*

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(Æ "'

Figure 4.4.5

188

Chapter 4

SPANNING TREES

Characterizing Cut-Vertices in Terms of the

A +/, ++,, "'

0 '"' "' ++,, "* ++%'

% # $ '"' "'

Algorithm 4.4.3:

' % 8

.

. *

3 2 * # 8 0 3 * * $ # '"' $ 3 '"' "' * * 7 *

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Figure 4.4.6

' & 2

Section 4.4

189


3 '"' . "' " ' '"'

' 3 ' '"' "$ B*>?C%D' 6 Escaping From a Maze: Tarry’s Algorithm

2 *

N 2 8 9

2 ,CL; &' . $ $# $ Æ $

A

. 3 -? 3 35 3 35 3 -?

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190

Chapter 4

SPANNING TREES

) %*+ + + ! 0# * + $ ' $ $ &# $ 6 $ , *+ $ 7 $ . !/ * +

++ ++ ++$ ++ ++( ++, ++-

E E E E E E E

++.

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4.5

CYCLES, EDGE-CUTS, AND SPANNING TREES

" # +,+' >

Section 4.5

Cycles, Edge-Cuts, and Spanning Trees

191

( ! 3

3 #

( !

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5 "' *

(Æ "' 0 -

# ½ 4

½ 4

3 - " 2 %,' - - - 4 - 3 ) - - Partition-Cuts and Minimal Edge-Cuts

* 3 & 8& # ,%

0 / / / /

/ /

($ / / ,

/ / / /

/ / / / ! / / 0 / / / / A / / /

/ / / / /

192

Chapter 4

SPANNING TREES

( / / 4 / /

# / / 3 / / / /

. +;+

2 # /+%

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/ / / / ) *

/ / " )' 8 / / " . . . ' 4 Fundamental Cycles and Fundamental Edge-Cuts

0 "' # %,+

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Section 4.5


193

.

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194

Chapter 4

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SPANNING TREES

' # #

Relationship Between Cycles and Edge-Cuts

+&

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A +;,

(-

A +;/

(. !

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Section 4.5


195

(/

£ £ £ 0 . . . £ , " 8 " ' A +;C

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( . % $ +;/ - %

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196

Chapter 4

SPANNING TREES

"/ %' 3 % - 4

½ $ / 3 3

4

½

¾ - 4

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Section 4.6

197

Graphs and Vector Spaces

+; , , +; - +; . , +; / 0 ! @ "> . * +;< ' +; 0 +; +;,: +; 9 +;C

4.6

GRAPHS AND VECTOR SPACES

? Vector Space of Edge Subsets

$ 1 "' 1 "'

4 " ' "

'

1 "' "/' , 4 : 4 1 "' 3 " "/' * *+'

,

1 "' 0'

1 "'

198

Chapter 4

SPANNING TREES

,

4 . . . . . . 1 "' 1 "' 0'

3 - 4 ½ ¾ . . . 1 "' - 4 ½ ¾ . . . # . . .

0 . . . ) 4 . . . % , : * $

1 "' 2 3 4 . . . ( $ 1 "' 9$"/' ) $ $ 3 / @ (" ' 4 (" ' @ (" '

. 8

) 4 " ' 1 "' * , 4 : 4 " ' 1 "' $ G H

1 "' The Cycle Space of a Graph

1 "' 1 "' " '

, $ +<, 1 "'

Figure 4.6.1

# 1 "'

Section 4.6


199

5 1 "' 1 "' 1 "' 2 " +;,,' 1 "'

,$ 1 "' 1 "'

3 Æ 1 "'

1 "' 4 A +;,, ¿ "' # / 4 , / % / / / / 2 4 / 4 / / A / 4 / @ / // / / The Edge-Cut Subspace of a Graph

1 "' 1 "'

, $ +</ ,; 1 "'

Figure 4.6.2

1 "'

, 1 "' 1 "'

A +;% Æ 4 / / 4 / / 4 / / 4 , / % 4 % +

200

Chapter 4

SPANNING TREES

4 4 4 4

>

4

A

4

,$ +<+ 1 "' $ +</ $ 4 1 4 1 4 4 Bases for the Cycle and Edge-Cut Spaces

,(

1 "'

A

> 1 "' - 1 "' 5 . . . - @ 4 , . . . ) - 4 . . . ) , 4 - . . . , , A , 1 "' " 1 "' ' 1 "'

, . $ +<% $ +<, 8 1 "' / 3

4 4 4 4

Section 4.6

201


Figure 4.6.3

' # " 1 "'

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Figure 4.6.5

- 0 O !

"O#0' O ! "OE0' . - 0 $

F 4 O#0

2 OE0 2

202

Chapter 4

SPANNING TREES

! F

* . $ +<<

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@ @ @

4: 4: 4: 4: 4:

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Figure 4.6.7

203

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4 ,/1 4 4 +1 4 4 +1 4 4 C Dimension of the Cycle Space and of the Edge-Cut Space

3 8 +;, +; +<; +<< $

% # ,-

1 "' & 0 "' 4 @ "' 1 "' & "'

+<; +<<

Relationship Between the Cycle and Edge-Cut Spaces

2 2

,. ! - 1 "' 1 "'

5 "' 8 +;C (Æ "' 3

- 0 . . . - 4 . . . @ 4 , . . . *

+<; - -

204

Chapter 4

SPANNING TREES

- 5 - A 1 "' - - 3 - - - 4 > -

,/ ! -

Orthogonality of the Cycle and Edge-Cut Spaces

1 "' ) : "/'

, 0 1 "' 1 "' 1 "'

0 - * A +<C +<L - *

, 1 "' 1 "' 1 "' 1 "' 1 "' 4

3 1 "'1 "' 1 "' 1 "'

"1 "' 1 "'' 4 "1 "'' @ "1 "'' "1 "' 1 "'' A # +<& "1 "''@"1 "'' 4 "1 "'1 "'' 4 4 "1 "'' "1 "' 1 "'' 4 : Vector Space of Vertex Subsets

. 2 2 6 2 "/' / "* * *+' 0 . . .

. . .

3 " % '

3 B % D 4

: ,

Section 4.6

205


* 1 "'

"/' 1 "' 8 1 "' "/' 3 3 1 "' 1 "'

, #

$ +<C , : , , 3 4 : , : : Figure 4.6.8

3

: : , ,

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, , : :

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3 $ 8 +<+ 3 4 " 2 1 "'' / 3 2 3

$ $

3

, $ ' + .

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3

206

Chapter 4

SPANNING TREES

, : : , , , : : : , , : : : , ,

% # , $ ! $ * ' .

8 3 "/' 3 "/' 2 ? 2 ) # +<,%

, ! $

"/'

2

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4 $

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$ 3 "/' +<,+ +<,;

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Section 4.7

Matroids and the Greedy Algorithm

207

; ; ;# * + (& & 1 "' *+ (& &

1 "'

+< 8 +<, +<( 8 +</ +<, 8 +<% +< ( 1 "' $ !

+<. 8 ,; 1 "' $ +</ / $ +<+ +</ $ * +<,

; !- ; !/# $ . ; !" ; !; $ &

+< 0 +< +<

+< $ +<, +< +<< +< ( +<L +< , # +<,% +< - $ +</ 1 "' +< . 8 $ +</

4.7

MATROIDS AND THE GREEDY ALGORITHM

* " ' 4 "

' " ¼ ¼ '

208

Chapter 4

SPANNING TREES

- 0 $ ", : :' ": , :' ": : ,' 4 " ' " ' - 0 4 " ' .

$ 3 - -

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.

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#

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3 2 3 .4 3 2 .4

6 4 #

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.

G

H GH

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Section 4.7


209

-$ # $ +&,

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3 $ +&/

Figure 4.7.2

#

.

(' J A O BO ;
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Figure 4.7.3

; ,

210

Chapter 4

SPANNING TREES

For Which Problems Will the Greedy Algorithm Perform Optimally?

* 2 4 "

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3 4 3 4 0 - 3 4 - 3 " +;/' 4 4 3 A - 4 $ 3 @

.

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-( $ +&% 3 4 4 4 2

-

4 "

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5 "' 3 4 3 4 4 3 3 @ 4 4 3 @ , 0 3 4

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@ / 3 1 @ , 4 3 1 :

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Section 4.7

211


(Æ "' 0 3 4 . . .

4 4 . . . * ' 3 5 3 4 . . . 4 4 . . . " 4 , . . . ' Æ

" ' 4. "3 ' "4 '

" 4 , . . . '

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A "3 ' "4 ' , " ' A 4 3 3 @ " ' "' * "' "'

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.

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8 +%, 8 +%%

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&

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212

Chapter 4

SPANNING TREES

+&.

4 +&/ " , " +& 0 "* +&,' " " "

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' 2 +& 0 - 4 " ' - +& $ 0 2 4 "

' ! 2 +& 0 . ",' 3 1 "/' 3 ! +& ( 2 "' +& , 0 2 4 "

' G 2 H 2 O " ' +& *

) * " ) 0 "

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3 O 8 +%, +%+ +%

4.8


+C < *

213

Glossary

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Figure 4.8.1

GLOSSARY

#. " 9 . 3 " . " . " 2 . 9 " 2 . " . " 3% , . , " /+' & "'. " - & . " 8 "

214

Chapter 4

SPANNING TREES

9

.

.

.

9 .

. # . 1 ) # . 1 "' . 2 . 2 . 9

" . 1 "' . . 16 . * +%/ & # " 9 . : & # 9 . " ' . . . ) 1 . 1 "' 1 "' . "/' . . . 2 . 2 & .

Glossary

215

9

. . . . ! # . ) . ) # # . # . # . 1 * +&, # " # 4 "

'. " ¼ ¼ ' 4 "

'. . . 2 4 "

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6

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6.3

(9!-

HAMILTONIAN PATHS AND CYCLES

; =

1

(5? " 0

"

%

5 ;2>G(<=

"

3 !

% 3

1 (72

268


Figure 6.3.1

# )

"

&

%

&'"

I% DJ

"

Æ

Showing That a Graph Is Not Hamiltonian

? 2 %

9

9 F

7 %

%

1 (79

Figure 6.3.2

0

Section 6.3

269


'? 2 ! 7

%

2 #

7" 9

0 ;

= 9 %!

1 (77

Figure 6.3.3

0

'? 2 2 7

%

7

#

2

#

Sufficient Conditions for a Graph to be Hamiltonian

3 F 2C<9 ;*F<9+=

! ! ;*!(G+= % "6> ./$ . ! 7 ;= B ; =

#

"

0 "0 4 "0 ; 7= Æ ;= B ; = 2 B

4 A A ;

1 (7)=

270


0

Figure 6.3.4

1 A 9 2

"0

;

1 (7<= ; =

0

B 2 A 9 9

Figure 6.3.5

;= B ; = A

B

A B B B

A 2 B B

B 2

A 9 B ; B =

9B7A 2

; %! "# > .+!$ . ! 7 ;=

! F

Æ " * 69+

%% . ! * ;=B ;= #/012% ; %& . ! ;= ;=

Section 6.3

271


' ( )* * * % 6 6 ! " #

(7

(7!

(7%

"

(7&

(7+ D 6 6'! "

(7 >" >

(7, >" >

(7- >" >

(7. >"

! 8

(7/ (" 2G (7 '

!

(7! ' 66 60! !

(7%

(7&

(7+

(7

272


(7,

(7-

(7.

' (7!/ !

% 3 ;

1 (72= 1

#:'&$ (7! :

"

"

" #

(7!!

8

5 '

8

(7!%

F8 ;: (7= I ;= J ; I ;= J=

(7!&

Æ ! 8

(9 "

(7!+ :

2 9 1

H H

1

2 9 1

(7! '

("

(" (7!,

"0 ;5?

2 9 = (7!- 2C

% H

5

D ;5?

= (7!. 7 7 7

96 2 2 2 %

D

Section 6.4

6.4

273

Gray Codes and Traveling Salesman Problems

GRAY CODES AND TRAVELING SALESMAN PROBLEMS

%

I J #

; =

I J

9 "

;

= H &

GGG 2GG 22G G2G G22 222 2G2 GG2 3 7

"

"

H 3

1 ()2

Figure 6.4.1

0 ; ;

3

; B 2="

" ? 0 G

"

0 2

0

(5?

96

1 ()2

1 ()9

G 2

GGG

K

;G22=K

GG2K

GGG

274


Figure 6.4.2

; % ; !

& ! 9 )"

9

G G G 2 2 2 G

' * ?

3

I J

1

3 . ()2

2 > G22 ;A <=

; = ) (

0

&

Traveling Salesman Problem

!

$ % ;$%=

1 ()7

K

"

"

Figure 6.4.3

7 *

Section 6.4


275

'

' 26
.

, 8 ' ;

. = !

H M ;2662= ' , ;2><(= 5 ;2><(=

"

' 5 2C7) $ 1 :

2C<)

3# F F 1 $ L ;*F1 L<)+=

0 )C ; F:

)> =

2C>G

: ' 72>" ;*:'>G+=

8 2G

2 2G

: "' ( "" # / +

Heuristics and Approximate Algorithms for the TSP

5

Æ " %

H 8

!

" % H

' "

"H

*44 , ><+

'

% "

?

" "

276


Algorithm 6.4.1:

: :

? ? % );= A G % A G

?A B 2

0 );= A ?A

"

K

1 ()7

5 ;=

1 ())

Figure 6.4.4

;

"

' &! & *' A ( #/* 3142%

5 '

/

#

"

()7 4 2 9

'

B ' &% * 2 *' !

! ! ! #/(*.112%

Section 6.4

277


Two Heuristic Algorithms That Have Performance Guarantees

"

# "

Algorithm 6.4.2:

#

? ? 1

: + : + : ? 1

0

:

0

&!

1 ()<;=

6"

1 ()<;=

1 ()<;=

"

Figure 6.4.5

) 0 &!

&& *' !

! 45 4 %

()9 %

" "

"

9 ; =

0

278


9

;% = 9 ; = # %

9 ; = 9 ;% = !

()9

& :

8 5

. L

: ' ' ; (9= "

"

: 8

'

Algorithm 6.4.3:

8

? ? 1

4 ,

"

1 " , : +

: + ()9 : ()9

: 8

" *44 , ><+ &+ *' !

! , 6

(5? '

"

/

"

TSP’s in Disguise

'

8Æ

5

' ' 2 9

' ? 1 " ? 1 G

A A G ' A 2 9 ;

1 ()(=

G

Section 6.4

279


Figure 6.4.6

? ! 1 "

!?

'

" A ' A A

Figure 6.4.7

!

? % 1 "

%?

=

' ;

Figure 6.4.8

%

280


0

&! 7! * * + ?

0

0 ' # 0 5

0

D F "

0K

'

0 " 0

&% , " ?

#

1

; =

"

0

&& * ) ( ? .

-

' ' A 2 5

D

A 2 - 4

-

)

' #

"

"

-

8 &

' " #

1

8

" 1

%

-

½

¾

½

¾

' ( )* * * & () E

()2 3

)

3 7 . ()2

Section 6.4


281

, -! # % 6 ! / " # #

()!

()%

()&

()+

$! # # % 6 " & # / " # #

()

. ()9 (),

. ()7 ()-

. ()) ().

. ()< ()/ ' ? %

" ; 29=

0

! # # % 6 " # & # , / " # #

()

. ()9 ()%

. ())

()!

. ()7 ()&

. ()<

- ! # # % 6 " " ) & # 6 / " # #

()+

. ()9

()

. ()7

5 0! # # " #

F

2 9 7

3

"

;

= ()2 ()9

;=

"

282


3

"

;

= ()2 ()9

;=

" (), ()-

(). 0 -

. 0 0

0 !

0 ! ! B 2 ; *" " " =

0

0 ; B 2=" ' ;5? : 0

2 0 ' 0 = ()!/ * (+ % ()2

. ()G ()< 6.5


(< F "

1 (<2;=

(<! F "

1 (<2;=

Figure 6.5.1

(<% :

(<& 3

"

" "

"

283

Glossary

(<+ # $ % '

% (< F

0 9 B (<, F

0 7 B (<- F

(<. 3 0 " (</ F + 0 + "

" .

" (<

1 (<9;= " (<!

1 (<9;= "

Figure 6.5.2

(<% F

1 (<7;= (<& F

1 (<7;=

Figure 6.5.3

(<+ "0 + 0 + 3

" (< 3

"

"

"

"

GLOSSARY

?

284


23 ?

9 H 2 9

K

;9 =" # 0 23 4 9? 9

9

K ;9 =" # 0 ? ? " ? ? ? ; ?

9 " H ? @ ; A ? ; =

K

?

?

5=

?

?

; = "

" K ; = ! ; = A ); = ; = ?

0

?

?

? ? " 1 > ! ? 0 4; 2 9 ?

" B '

Chapter

7

PLANARITY AND KURATOWSKI’S THEOREM 7.1 Planar Drawings and Some Basic Surfaces 7.2 Subdivision and Homeomorphism 7.3 Extending Planar Drawings 7.4 Kuratowski’s Theorem 7.5 Algebraic Tests for Planarity 7.6 Planarity Algorithm 7.7 Crossing Numbers and Thickness

INTRODUCTION

Æ ! " # $ % Æ & !

'

( ) * *&

285

286

7.1

Chapter 7

PLANARITY AND KURATOWSKI’S THEOREM

PLANAR DRAWINGS AND SOME BASIC SURFACES

+ ,

( ) " Planar Drawings

( "

,

- **

Figure 7.1.1

! .. / .. /

0

1 2 / 3

4 ,

, , 3 ,

" , - *#

Figure 7.1.2

Section 7.1

Planar Drawings and Some Basic Surfaces

287

1 3 3

, ' 5 6

Three Basic Surfaces

7 8

! 7 8 Ê

9 9 :

Ê % ",

* #; ! , 8 - *8

Figure 7.1.3

- *< : ; ; ' ! : * - *< ' ! : *

Figure 7.1.4

!

288

Chapter 7


5 6 5 6

1 3 )# " " % Riemann Stereographic Projection

= %

Ý ; ; ; 7 8 ; * ; > = / 5 6 ! 5 6 - *

Figure 7.1.5

" #"

1 ' , 5 " 6

? = / "

, @ A .

$ $

% = /

Ý

Section 7.1

289


Jordan Separation Property

/ "

@

" . B 7 Ê

7 / C; *D ; : * : + .

", 7 C; *D ; : * / - 5! 6 % - *&

Figure 7.1.6

& " "' ( "

) 3

7 Ý , 7 7 , * - *

3

Figure 7.1.7

" "'

' ! B !

Ý

290

Chapter 7


7

+ ' Þ , ' . E -

/ - *< E ( % Æ F E *)) "

G *H;

"

( = ! " #$ Applying The Jordan Curve Theorem to the Nonplanarity of

and

3

E ( 3

" 8

% $

0 ; < B E ( * # 8 < * ( , ; 5 6 B E ( / , ; * # 8 < - *) @ 8 ; * # ; ; # 8 ; ; 8 < ; ; < * ; #< , * # 8 < *

Figure 7.1.8

-

! "#$%& Þ

Section 7.1

291


3 #< <; ;# * 8 E (

*8

)

$

0 ; # < * 8 B E ( # 8 < #

, ; 5 6 B E ( / , ; 8 ; 8 # ; ; 8 < ;

- *H

Figure 7.1.9

-

1 , # 8 < #

3 , * , , * ,

, * "

Figure 7.1.10

.

- ***

3 ( )

292

Chapter 7

Figure 7.1.11

0 1


" /

"

! " # I 1 '

* ;Ô *

* *%

Ô

Ô

$ % " ! " I 1 > ,

Ô *) ; ; * * ; *

7.2

** *2

8 ; < ; ;

SUBDIVISION AND HOMEOMORPHISM

- #*

! ! ! 5 6 , (

3

, 5

6

Figure 7.2.1

!'

! " ! % 3 "

Section 7.2

293

Subdivision and Homeomorphism

Graph Subdivision

0 ,

¼ ¼¼

Figure 7.2.2

!'

# , ¼

, ¼ ¼¼ / ¼ ¼¼ 0 ¼¼

, '

Figure 7.2.3

9 * - #< 3

, 9 * * 3

, * : %

!' /"," , %/","

Figure 7.2.4

%

9

* *

$ $

'

Barycentric Subdivision

/ "

! J

294

Chapter 7


,

,

- #

Figure 7.2.5

!," " !'

$ % $

0 ¼ + ¼ 5 6 , 56

$ & $

B

##

$ % & $

(

3

56 , 5 6 "

*

$ ) % $

B

#8 "

#< - #&

Figure 7.2.6

" !," " !'

Graph Homeomorphism

Section 7.2

295


% ? - # 8 I

Figure 7.2.7

/ " "

1 K 7, #<

$ * ' % $

#* $ $

B

## + Subgraph Homeomorphism Problem

$ ) - #) - ;*#8;

# !

Figure 7.2.8

" " ,

K - #) ,

, ; * # , 8 - #H % / /

; * # B

296

Chapter 7

Figure 7.2.9


! , "

* - #*; " / / 2

?

Figure 7.2.10

0 1

" " ,

"

& ' '( (

#

#

#

#%

#) ' % ! % L I 1

#*

& ' ')(

# #3

#2 # 4

Section 7.3

297

Extending Planar Drawings

& ' '( *+,

# #

7.3

# # %

<

EXTENDING PLANAR DRAWINGS

' # " , "

#

' 1 7

! 1

1 ' # I 1

1

Planar Extensions of a Planar Subgraph

, , ! J "

$ $

$ , Æ

@ ,

Æ

298

Chapter 7

Figure 7.3.1


/

/ ' ,

,

$

( ( $

(

@ ,

Figure 7.3.2

!' ' ,

/

$ % ' 1 ' % 1 $

3 1

' - 88 + G ,

Figure 7.3.3

5 !,

Section 7.3

299


Amalgamating Planar Graphs

' #

$ ) ' % % $

( = / $ * ' ) ' ¼ " ) ¼ % " # : : $

" : 8 - 8<

Figure 7.3.4

- ! K, " ! ¼ "

8 - 8

"

Figure 7.3.5

!

'

, % " A

K ! "

" ! - 8&

" "

" # : : :

300

Chapter 7

Figure 7.3.6


$ !, "

, ' " ' $ ¼ " % " #$ : $

'

, %

A

$ ¼ / $ 1 % ,

- - 8 ( 8

Figure 7.3.7

/'

1

- 8) !

Figure 7.3.8

%/'

Section 7.3

301


Appendages to a Subgraph

3

" ! 0 , !

1 % F A,

0

%

0 /

0 ,

3 - 8H ! ,

, K

!

Figure 7.3.9

! 6 6

3

,

1 7 / 7

302

Chapter 7


Overlapping Appendages

56

3 0 1

B

- 8*;

Figure 7.3.10

, " '

$ 2 ' % $

+

( *8 E ( 0

= 3 - 8*; 0 1

"

& - -( #

. #

8

8

Section 7.3

303


8

8%

& -$ -%( #

#

8)

8*

8

82

& -/ -'( ( " "

83

8 4

8

8

& -- -0( 1

2 "

( #

8

8 %

304

8 )

Chapter 7


8 *

8 ? 1 1

$

7, 8 2 £ £ . ,

£

88

8* ¼

7.4

KURATOWSKI’S THEOREM

+ ! ! J .

%

! *+,-./ $

*< *

#&

!

Æ 3 ! Æ , ! 3 , , , , ,

1 3 1 , 8 3 ' 1 , 3 - 1 " !

Section 7.4

305

Kuratowski’s Theorem

Step 1: A Minimum Counterexample would be Simple and 3-Connected

' 0 % $

B ( 8#

!

** ' 0 % ( $

3 , B ( 8

*# ' 0 ' ( % %

% $

, , %

*# / & # - <*

Figure 7.4.1

/ & %

: & % ! ! 0 &

% 9 ! % 9 @ % 9 , & , % 9 3 % 9 $ % 9

%

8<

*8

% ' 0 % ( (

306

Chapter 7


$

%

, (

B

*8

- <#

Figure 7.4.2

-" '"

B

8&

*<

' 0 % - $

.

* Step 2: Finding a Cycle with Three Mutually Overlapping Appendages

0 ! 0 8 * * # *

- <8 5 6

,

Figure 7.4.3

," !

1 $

+

#*

Section 7.4


307

' % $

3

8

## ' %

$

& & >

56 &

- << @ B

##

#8

Figure 7.4.4

7 , " ","

% % $

% +

@ + B

#8

#< ' 0 % $

# %

3 #

#< ,

308

Chapter 7


B

#8

# Step 3: Analyzing the Cycle-and-Appendages Configuration

! J "

, # ! ' ' % 0 $

0

- < +

Figure 7.4.5

8

"," "

2 B " Æ

3 , 3 ¼ ,

!" !" ( ¼ : - <& 3

& & ( * Figure 7.4.6

Section 7.4

309


!" ! !" ! ( :

¼ ¼ : - < 3 & ¼ 0 , & & ¼ & & ¼ , " ¼

/ - < : ¼ : ( * G G &

Figure 7.4.7

Figure 7.4.8

!"

!" & , - <) " & " " ! ( #

Figure 7.4.9

!" !

!" ! & , 0 & & - <H & ¼ & & & & ¼ ¼¼

310

Chapter 7


& ¼¼ ( # <* ( ,

, % ' " " %

2 $

B <* ! ! 8

!

! Finding

or

in Small Nonplanar Graphs

" ¼

< & & ¼ & 5

6

@M '% - <*;

Figure 7.4.10

8

! !, ' "

"

! < / / - <*;

% / ! - <**

Figure 7.4.11

8

6

% / ! - <*#

Figure 7.4.12

8

6

Section 7.5

311

Algebraic Tests for Planarity

% / " ! - <*8

Figure 7.4.13

0 1

8

6

" %

4 # ( # 5

<

<

<

<%

<)

<*

< <3 9 < < 9

7.5

<2 & < 4 & < < %

ALGEBRAIC TESTS FOR PLANARITY

! 3

312

Chapter 7


, 7 % ) . % About Faces

1 ( 3 ( (

) - * +

Figure 7.5.1

!

1

!

1 G ! + . J ! / ! " " ! . * #

) 3 - * * !

# ! ) 8 ! ) 7 " & !

$ ) ' 1 '

% $

(

0

- #

Section 7.5

313


Figure 7.5.2

"' "

E

3

,

) ' - * - 8 + ** ! ) ) **

Figure 7.5.3

' !

Face-Size Equation

) #" $ 3 ' 1 % # : *

¾

$

7 2 !

! " .

314

Chapter 7


)% - < ,

% # : *# %

¾

* : * 9 * 9 * 9 *

Figure 7.5.4

: * 9 # 9 8 9 & : *#

!

Edge-Face Inequality

!

$ ) $

3 +), : * ! 3 +), : # ! : ¼ : ! 3 ¼ : ! * + ! # K

+), 8 3 ! , ! + , ! ! ,

! $ )% ' 1 + % $

!

! (

@ : - (

, , !

,

Section 7.5

315


Figure 7.5.5

" " 9 ! ,

, - , * ' "

$ )) % 1

)*!, $

( * 3 ! !

8 ( # 3 !

)*!, : * *

*

. ( 8 B

< ! *

3 ¼ 1 ¼ ¼ * ¼ 5 !6

* * ¼

)*!,¼ * : )*!,

)* # %$ 3 ' 1 % # )*!, ( $

3 - . 7% # :

¾

*

)*!, , (

316

Chapter 7


Euler Polyhedral Equation

' < & < < & 9 < : # ' ) *# & ) *# 9 & : # 1 = ' C' ;
3 *; 7 9 ( : #

? "

*H< 0 0

*)** 9( : ; N .

*)H*H;<

) & $ 3 ' 1 % 9 ( : # $

3 ! - . - % +

3 ( : *

9 ( : * 9 ( 8*8 : * 9 * : #

! K,

7 7% - : ; ! - : 9 * 0 . - : (

* : # : 9 * - &

/

Figure 7.5.6

B

8*) . 9 % B E 2

8

( : (

9*

Section 7.5

317


9 ( : : : :

9 (

* 9 * 9 ( # 9 * 9 ( 9 * 8 9 ( : #

" 7 ! -

Figure 7.5.7

/" " : ! /"

)) # - & & ' 8 7 ' , ' -

Poincaré Duality

- I N C*H;;D 1 £ 1 £ £ ' " ( )

' 1

" @ ? 1 & '

1 1 3 , £ - ) £ ( £

Figure 7.5.8

1 '" !

318

Chapter 7


£ - H / , , 3 £

, £ £ £ 1

Figure 7.5.9

1 " !

3 N £ ,

£

£

3 N

.

£

£

1 £

3 N , £ - , £ £ ( + ,

$ )2 4 5 3 ' 1 % £ : ( £ : ( £ : $

% " N

% ,

!

£ , - *;

Figure 7.5.10

" '

Section 7.5

319


)* 2

- **

2 - H % * #

- ** % * 8 <

;

Figure 7.5.11

1 N

/ .

5 6 1 7- 3% & 7J **# Algebraic Proofs of Nonplanarity

7 % 9 ( : # .

E > 7 % %

% !

E Æ

#; O !

)3 ' 1 % 8 & $

7 7% * 9 ( : # 7 7% # )*!, ( # 7- 3% 8 8( # 8 )*!, # < ( 8 # 9 8 #

< * & 8 #

)

8 8

# &

320

Chapter 7


) 4 % $

8 & : H : *; % / 8 & " H 1 E ( 7 7% 8 & 3 '

% H " 2 ,

% $ ) ' 6( *H % $

8 & : 8 ) & : *) *H / 8 & B H 1 * CI&HD )8) ), *H

**

% !

A More Powerful Nonplanarity Condition for Bipartite Graphs

- H % %

)

& H 8 &

) ' 1 % # < $

, ! * 9 ( : # 7 7% # )*!, ( # 7- 3% 8 <( # < )*!, # < ( < #

< * 9 < # & # #

)

# #

# <

Section 7.5

321


) % $

# < : ) : H % / # < " *#

Figure 7.5.12

.

Nonplanar Subgraphs

H *# ) % " $

7

)2 : , / : : #* & : * * : 8 & : * I ! - *8 I *<

Figure 7.5.13

"

$ ) ) 7 $

322

Chapter 7


$ ) * 8 % $

% H *#

F

, &

<" ) % 0 )3 ),

*8 - *< +

!

Figure 7.5.14

0 1

, " )

& $ $( # 6

# 7 " 8 3 # 8 9 #

%

Section 7.5

323


& $$ $%( # 6

! 3 #

9 #

)

* 2

7, * 7, 8

7, # 7, <

& $/ $'( # 6

(

! ( (

(

3 # 9 9 : #

3

4

7, * 7, 8

7, # 7, <

& $- $0( # : ;

)

% *

7, * 7, 8

7, # 7, <

& $ $')( # # ( 9 < =

3

2 4

7, * 7, 8

7, # 7, <

N £ & $' $'0( # " (

%

) 9

0 * 8

& $' $-'(

4

<

2

8 9

3

#

& $-- $)( #

& *8 % & *8 ) ** * ** *<< &*8 2 *#) ;# 3 8& &) %4 8& *;# % $ * *) " # <

324

Chapter 7


& $' $$( #

9 8 &(

% %%

7.6

9

% %)

PLANARITY ALGORITHM

C$@&
3

"

Blocked and Forced Appendages of a Subgraph

,

0 3

0 3

* 3 - &* ! , ! !

Figure 7.6.1

! ! "

$ * ' 1 % ( 1 $

0 " /

2 B E

Section 7.6

325

Planarity Algorithm

0 3

* 3 - &# ! , !

! "

Figure 7.6.2

Algorithmic Preliminaries

B ( 8 !

! <

# 3 # ' # " 3 Selecting the Next Path Addition

B

, + , , ! 3 3 , !

3

/

3 " 3

, 3 >

, ,

,

, ,

326

Chapter 7


The Algorithm

K , % , % 1 ,

, $ ,/ / " = 4 6 # + 6 -07 4 8 - ' : , 3 ! -07 3 1: 1: 1: $ ( ,

7

= Algorithm 7.6.1:

Outline of Correctness of Planarity Algorithm

, 2 " " 2 ¼ - &8

Figure 7.6.3

2 ¼

- H) CB @&D

&8 C';*D I / CI
?

=

C?= )*D '" C?= HD

Section 7.7

0 1

327

Crossing Numbers and Thickness

" *

& 0 0'( > 0

& 7, <* & 7, <# & 7, <8 &% 7, << &) 7, < &* 7, <& & & 2 &3 9 & 4 & & & & & C( /D 3 &*

7.7

CROSSING NUMBERS AND THICKNESS

% " +

5 6 Crossing Numbers

Æ

,

8 )8

0 3

3

3 ,

3

$ 3 : * $

B *<

3 * - *

3 *

Figure 7.7.1

, "

328

Chapter 7


$ 3 : * $

B * 3 * #

3 *

Figure 7.7.2

-

, "

Lower Bounds for Crossing Numbers

* ' % 3 8 9 & $

0 , H 8 & 8 &

: ,

3 8 9 &

$ % 3 : 8 $

: * : & 8 3 * 8 & 9 & : 8 - 8

3 8

Figure 7.7.3

"

) ' % 3 # 9 < $

0 , *# # < # < : ,

Section 7.7

329


3 # 9 <

$ * 3 $

Figure 7.7.4

: #

: *#

:

3

*# # 9 < : #

- <

3

#

"

%

-

Figure 7.7.5

/"

Thickness

" 7 4 7 ' 3 + % . ! @ . . @

%

.

4

- &

330

Chapter 7

Figure 7.7.6


!

' %

4

8 &

$

B H

8 & $ 2 4 : # $

B 4 # - &

4 # 3 ' %

4 # < $

B *#

# <

Straight-Line Drawings of Graphs

3

' C'8&D - C-<)D ! -J

C ); )*D

! J -J

4 ' % 0 1 " $

3 8# < $

3 5 # #

Section 7.8

%

331


$ #

$ H

$ % 9

) 3 3 9

* 2

3 3 9

3 4 4 4 : # 4 : 8 " 6 " 5 6 ", -

Figure 7.7.7

/, /

' $ 9

" #

% )

7.8


) ( , 8 & > 7 7% 7- 3% , 8 > E / ) $ % 8 8 8 8 < < < ) $ % 8 8 8 8 < < <

332

Chapter 7


)% 6 7 3 , / ,

,

I L - : )) $ < H, )*

% < < ) < <

& )2 ( 8 8 L 7 ,

)3 8 < 7 7% 7- 3% ) 4 0 < / $ 7 ,

) $ &, ) $ H, . 9 . ) $

) % $

) ) $

Section 7.8


333

) * B

) $

) 2 > *! 1 * # - ! ) 3 > *! H 1 * 8 - ! )4 =

) (

) (

) 3

I 1 F P

)%

I 1 , )) 4 )* ( 4 ) = - & - & 1 -

*&* 3 , %

334

Chapter 7


GLOSSARY

! 1 , ! "1 " 1

' 1 1 ! 1 !," " !' 6 9 1 , 1 " * ! , ! 1

! , " 1

! " ! 1 1 % ! " ! 1 " " " 1 " " ! 1 , " ! 3 1

)8 "'6 " 7 1 1 # 1 , 1 "1 ! 1 , , , / , , '"1

335

Glossary

!, ! 1 ! , !, ! 1 " 1 7 Ê " "1 7 , 1 ,

" 1 1

"/ ! 1 1 ( ( ( , "/>1

!

. " 1

" " 1 ! 1

( ) + ,1 7

. 1 . ! 1

1

! , 5 6 1

! 5 6 1 * 7 1 C; *D ; : * : / C; *D + . ", " 1 ; : * : - 5! 6 1 ; : * : 1

1 ,

336

Chapter 7


7 8 Ê1 9 9 : $ "? 1 ! 1 N

'"6 6 " 1

1 ! ! 1 1 7 " #" 1 ; ; ; 7 8 ; * ; / 1

N £ " 1 > " 1 .

/' ' 1 2 / 1 Ê % ", 1 ! * #; , 8 1 * #; , 8 3 / 1 !' 1 ¼ ¼¼ ¼ ¼¼ , !' 1 % " 4 1

Chapter

8

DRAWING GRAPHS AND MAPS 8.1 The Topology of Low Dimensions 8.2 Higher-Order Surfaces 8.3 Mathematical Model for Drawing Graphs 8.4 Regular Maps on a Sphere 8.5 Imbeddings on Higher-Order Surfaces 8.6 Geometric Drawings of Graphs

INTRODUCTION

!

"

#

$

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

337

338

8.1

Chapter 8

DRAWING GRAPHS AND MAPS

THE TOPOLOGY OF LOW DIMENSIONS

* . / . / 0 +1

Some Subsets of Euclidean 2-Space and 3-Space

Ê & . /

Æ .. / . // 2

.

/

3 3 .

/

* 1& Ê .*

Ê / *

. / 3 *

/ 3

.

*

/ 4 3

.

+ & * & &5 &

Ê

Figure 8.1.1

* 6&

/ 3 3 2

.

* 6& . / 4 3 2

Figure 8.1.2

Section 8.1

339

The Topology of Low Dimensions

Topological Equivalences of Euclidean Sets

7 8 7 8 & 5

9 & & 9

9 %

* : . / % % %

.. // 2

* % ;

% ( . /

<5 & & +6 =

* %

Figure 8.1.3

! ! " "

7 8 7 > 8 5 & +, * >

% *

5

340

Chapter 8

Figure 8.1.4


# " ! " ! "

%

$ * 5 6&

*

5& 5 +- 6& 5

*

Figure 8.1.5

!

Topological Model of a Graph

* 5

6& 9 ?4 @ Ê 6& .4/ 2 ./ 2 & & 5

2 *

. / ?4 @ * . / .4 / . / 2 . / 5 Ê *

5

*

7 8 * 4& ?4 @ * & ?4 @

Section 8.2

Higher-Order Surfaces

341

9

&

. / . /

& +A

Figure 8.1.6

% "

9 *

&

4& & & 4& * &

& '() ) ) + 0 .4/

% + 0 . / 3 1 % + " % & B + # . % / C

8.2

HIGHER-ORDER SURFACES

* '

4 &

% &

342

Chapter 8


Torus and Möbius Band

D

* .14/ & &5

6&

% &

*

. +1/

Figure 8.2.1

: > > * +11

Figure 8.2.2

" "

* +11 78 * * 5 +16

Figure 8.2.3

( " * "

# . 78/ .

78/ ;

Section 8.2

343


9 ( 78

78 78 & +1, +1-

Figure 8.2.4

" + +

Figure 8.2.5

#, * *

* . +1A/

Figure 8.2.6

#, * *

-

* EF

9 * ! EF

EF $ EF < Bounded and Boundaryless Surfaces

&

%

344

Chapter 8


*

*

* $ *

* . * EF

Closed Surfaces

5& 9 ./ * ./ *

./ *

%

/ * *

. / 2 4

0 * % * EF

' 7 8 & # 7 8 &

% EF 9 0 % EF 6& 0 % 5 . / * % * 7 & 8 Classification of Surfaces

* 0 * % +1

Section 8.2


Figure 8.2.7

345

*

! "# $%&' * % * 7 8

EF * EF % 5

9 *

EF ,& 6& ' ,& *

7 8

EF 9 * ,& .

6& / *

* 5 0

0

Figure 8.2.8

*

() )

! "# $%&' *

!

% * . / EF

346

Chapter 8


& '() ) ) ! !"#

+1 +1 +1 +1 +1$

* .44/ * .44/ * .4/ * 6& * 6&

!$ !%# &

+1. * .44/ +1/ * .44/ +1 * .4/ +10 * 6& +1 % * 6& +1 $ EF ! EF * +1 +1 C > & C

B +1 C +1 C

8.3

MATHEMATICAL MODEL FOR DRAWING GRAPHS

*

?4 @ .4/ 2 ./ 2 .D '

5 / & & C 5

Section 8.3

Mathematical Model for Drawing Graphs

347

"

5 *

+6

./ * 5 ./ & ./ * 5

Figure 8.3.1

* " " + "

# Normalized Drawings and Imbeddings

" 9 ./ * ./ * #

. / ./ * +61 9 ./

G ./

G ./

&5

Figure 8.3.2

* + "

" H

5 5

' .; 5 .//

348

Chapter 8


&

G % 9 E & & &

Ê D 9 "

*

Eliminating Edge-Crossings

% &

0 # +66 &

Figure 8.3.3

" " " * ! " " "

&

+6, *

D &

Figure 8.3.4

" " " * "

9 6&

5 +61 C# # #

Section 8.4

349

Regular Maps on a Sphere

* + 9

0

; &

& '() ) ) !' !'$# & & &

+6

+6

+6

+6

+6$

+6.

8.4

REGULAR MAPS ON A SPHERE

* # & & * ;

5 ;

5 ; &

& Map Theory

; &

&

* # $!

5 5 5 5 5

350

Chapter 8


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

Figure 8.4.1

+ 12 12 "

* +,1 * 6& 6& G * 6& 5 ,& G . 5 / *

Figure 8.4.2

+ "

9 *

- , 9 -

./

9 *

E 5 0 5 % ./

* * 9

Section 8.4

351


9

%

* +,1 &

Degrees and Face-Sizes of Regular Maps

*

Æ ./

- 9 Æ ./ A -

./ !"# $./ % 1 & % .1/ .6/

.,/ .-/ .A/ ./ .+/ .I/

6 !"# $./ 6% 1 1 % 6 3 % 2 1

./ .1/

1 A 1 1 1 A 1 1 A 1

.,/ .-/

.4/ Æ ./ 2

6

Ú ¾Î

!./

1

./

Æ ./ 2

.1/

Æ ./ A

.6/

%

1

Æ ./ A

* 1 .I/ ./ .1/ 1 4

' .

/0 10 2 -

* +,1

- ' ) 3

/0 10 2 -

$ +,6

352

Chapter 8


Construction of the Regular Simple Maps

$ +,6 +,, 6 , - & ' 6 , - * * +,-

$ 4 #

) 30 0

2

,# 1. 3 #/ #

2

1# 1. 3 #/ #

% 2

, 1. 3 #/ #

-

0 % 7 8 %

# 3 % 2 1 % 1 2 #% & ' % 1 2 & %

*

& ' # * * +,- & ' 5 #

6 6 6 , , , -

6 , 6 , 6 , -

< 6 , A , * 1 + 1 A $ 14 64 1 C 1 A 1 + D 4

1 4 14 +

1 64 14 1 + 14 4

, 4 ,

,#

1#

,

& 6& J &

&

&

Section 8.4

353


*

+,6

Figure 8.4.3

3! "

9 * +,6 * 5

9 0 %

* +,-

&

&

& '() ) ) ( !) !))# & & &

+, +, +, +,

: 2 : 2 1 & ' 2 , : 2 , & ' 2 1 : 2 1 & ' 2 6

+,$ & 6 +,. C . +,1/

+,/ ( 2 1 & ' +, ( & ' 2 1

354

8.5

Chapter 8


IMBEDDINGS ON HIGHER-ORDER SURFACES

*

> 0 % ' Cellular Imbeddings

;

9 % .*

71&8/

9

$ +- % & & & +-./ 78 .

! / +-./ %

Figure 8.5.1

+ * "

&

=

& +-./ % & E & +-./ % &

- $

-

! 0

*

Section 8.5

Imbeddings on Higher-Order Surfaces

355

- $ -

* ( +- 0 >

$ +-1./ +-1./ $ ./ ,& * 7 8

D ./ 5 5 * 0 5 5

Figure 8.5.2

+ * " 12 4 1*2

Flat Polygon Drawings

% % $ +-6 +1 &

>

Figure 8.5.3

" "

356

Chapter 8


> ,! .

?EA@/ %

$ +-,

'(' (

))

Figure 8.5.4

" * "

0 & > 1* . ?EA@/

% +-- 5

Figure 8.5.5

" "

>

+-A 5 D

'

(

Figure 8.5.6

* "

Section 8.5


357

Surgery on Imbeddings

* > . / K . / % % 5

5

$$ C > .* L/ +-

Figure 8.5.7

* " + + "

*

& <5 +- *

* 5 +-+ . / *

Figure 8.5.8

5 " 3 + !

Euler Polyhedral Equations for All Closed Surfaces

; & '( 1 2

"+. /

9 0 & ) 1 !"# $./ %

358

Chapter 8


. / & E 9 ( ) ./ 2 % ./ 2 ./ % 2 " % & % % &

6 $ - 0 5 .

. . -

0 5

C

# #

$ - 3 % 2 1 1! -

-

3 % % 1 *

% ; C

= +-6

<5 , +-I

= +-6

* & = +-6 3 %

Figure 8.5.9

" * "

Section 8.5


359

,

% * 5 3 % +-I % 5 & % 3 % 2 3 % 3 1 3 % 2 1 1! 3 % 2 1! 2 1 1.! 3 /

$$ - ) 3 % 2 1 * -

*

* +-, 5 EF

$. * +-A 3 % A I 3 6 2 4 2 1 1! $$ * +- +-+ 3 %

I 14 3 I 2 1 2 1 1! $/ * +-4 I * 3 % 2 A - 3 I 2 4 2 1 *

Figure 8.5.10

* " + 0 "

* % 3 %

). / *

). / 2 1 1! 1 * &

J & & -./

$ I *

-. / *

)#./ )#./ 5 ). /

360

Chapter 8


Average Degree and General Surfaces

* ' * +,1

$. - 9 ). / Æ ./ A -

*

* +,1 5 1 ./ % 6 .1/ 3 % 2 ). / .6/ .,/ .-/

. & %/ . %/

). / 6 1 A A ). /

Æ ./ 2

.A/ Æ ./ A

. ./ .1// . .6/ /

1

. * 1/

A ). /

. .,/ .-//

$ 0 ). / 2 &

A * $0 * 3 A ). / 2 4 * +-A

3

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

+- +- +-$ +-/ +-0

. 3 . /

+- +- +-. +- +- %

3 , 1

!" !")# *

+- +-

+- +-

!"" !"!# *

+- $ +- /

+- . +-

Section 8.6

Geometric Drawings of Graphs

361

( !"+ !"$# &

+- 0 +- +- +-$ +-/ +- +-0

8.6

+-% & +- +- +-. . ( 3 ( , ( & 6

GEOMETRIC DRAWINGS OF GRAPHS

J H 5

& $ & *

&

J " & .0 +6/ Some Types of Geometric Graph Drawings

( & 5 * & " "

. +A.//

"

"

" ' . +A./ /

362

Chapter 8

Figure 8.6.1


7+ " 12 " 4 1*2 " "

Some Properties of Graph Drawings

' ' D

5' ' * " & 5 '

" * "

*

5

* "

Minimizing Total Edge Length

! Ê

! Ê

5

' .*

& / * '* & Ê . +A1/

Figure 8.6.2

8 "

Section 8.6

Geometric Drawings of Graphs

363

9 Ê

/. / ( M

Ê &

* Ê 5

Ê &

5 +A6

Ê " '

Figure 8.6.3

" "

9 .?=*4,@/ <(& Minimizing Area

* C

. +A,

Figure 8.6.4

) " " + "

. +A- &

Figure 8.6.5

) " " " + "

364

Chapter 8


Representability

; '

N D:*"DJD< = C: ;
)"#). 9 1 6/ +AA

Figure 8.6.6

+ " + " )"#). 9 1 6/

* & N E<EHE 0( <<

0

0 0 " " 5

0 9 5 5 -

& 5 A .?E 0I1@/ <(& 5 % A .? ;IA@/ < 5 1 Ê 5 I Ê & .?=CI-@/ Voronoi Diagrams and Delaunay Triangulations

O C &

&

. ' 3 3 ' ( * + 2 Ê Ê 5 . / . / . / * , 2

* & . / O . / . /

Section 8.7


365

* C

O +A O C

Figure 8.6.7

9 " 7 "

D & C

, Ê &

, " C

5

9 * O /. /& /. /& .0 ?$44@/

9 ' C .0 =*4,@/

9 Ê

P. / ( M .? ;++@/

8.7


+ C EF + $ 9 6& ,&

+ C

+ +$

C 5 +6 C &

366

Chapter 8


+. & 5 ." 9 H ( %

& / - 2 . / ' 1&

2 . /

+/ C 1& 5 & + ; 1& 5

GLOSSARY

* + " " 9

9 ./ * ./ * #

. / ./

" 9

5

" + "9 5

' 9

* + "9 * 9 * + 9

* 9 9 & * " 9 9 & % "9 " 9 %

"9

367

Glossary

#, * * 9 &

9

9 . / 3 2 . /9 ' 1& 2 . / , &

! Ê 9 5 ! 9 &

.

' 3 3 '

(

' ! Ê 9 &

9 EF " 9 & EF

* * 9 % EF

7 +* "9 -

7 " 2 9

5 & . / . . / . / 7 " Ê9 5 9

& & .41/ + &5

+ " " 2 . / 9 & & 5

" + "9

9 " 9 '

" 9

368

Chapter 8


" * "9

5 5 5 5 5 " " 9 # "9 . / % 9 4& ?4 @ 9 & ?4 @ 9 & Æ .. / . // 2 . / 3 3 . / #) *9 Ê 9 9 5 3 9 5

9 Ê 9 3 % G -. / " 5 ). / * 1 ). / 2 1 1*! &

: " 9

% * " 9 +6

./ * 5 ./ & ./ * 5 " * 9 % 7 8 " " Ê 9 & ' " " Ê 9

5 9 . / 4 3

Glossary

369

9

" 9 & %

78 * " " 9 &

!9 5 9 &

%& ; * 9 &

9 "9

9 % 9

" 9 +" +* 9 5 Ê

0 #, * * 9 & & "*

9

+ +

9

4 * " 9 & * 9 EF * 3 9 & % 5 * + " " 9 ./ #

G ./ G ./

'

9 . / 3

* 9 EF

* 3 9

% 9 ?4 @ .4/ 2 ./ 2 ?4 @ .D ' 5 /

370

Chapter 8


9 .4/ 2 ./ 2 .

7 8 / 9 .4/ 2 ./ 2 " 9 9 1& " 9 &

9 &

9 & 6& 9 & <! 9 &

+ "9

" " + " 9 & & & 2 . /9 2 . / ! *+ + 9

& & ?4 @ 6& 4 ?4 @ " 9 ! 9 9 % * 9 9

% & * 9 " ! 9 9 & & 9

" " 2 . /9 6& 5 6& *

9 % " " + "9 " Ê 9 & 5 +"9 ' 9 . / 4 3 2 9 . / 3 3 2 9 " 2 Ê 9 Ê 5 . / . / . /

Chapter

9 GRAPH COLORINGS

9.1 Vertex-Colorings 9.2 Map-Colorings 9.3 Edge-Colorings 9.4 Factorization

INTRODUCTION

! " # $ % & ' # # # ( ) * ) # +# #, # * - # # Æ # * . # - # # #, # * % ' # / ## Æ 012* 3 # # # * ( 4 ## #5 #5 *

371

372

9.1

Chapter 9 GRAPH COLORINGS

VERTEX-COLORINGS

6 # # * ( ## # ## 75*

* 7 # 7 # # Æ 8 9 Æ # 7* The Minimization Problem for Vertex-Colorings

# # - # # # # # * : 75 ; # # * ( # * $ 75 - #5 5 # 7 *

75 * 75 # # # # < * : = 7 # #5 ## * # 7 5 * 75 ( 0** # 7 >5 *

Figure 9.1.1

# # 8 9 # < - # 75 * 8 9 ; * 8 9 ; 5 8 95 *

Section 9.1

373

Vertex-Colorings

?5 ( 0**

( 0** ?5 8 9 ?* ' #, # 5 * ?5 *

Figure 9.1.2

>5 ( 0**?

8 9 > # 5 8# #9 8 9 >* ' 8 9 ; >*

Figure 9.1.3

!

: # 75 # * 5 ## # 5 #, * #, < # #, * Modeling Applications as Vertex-Coloring Problems

% ## 75 # # # @ * $ # * 6 # #5 # #, # # # - * A#,B # # # < - * &# - 5 5 * 6 # 8 9 - - - # # # * 8$ *?*2*9 $ # < # ## 5 * ( 7 # # # # *

374


# < - # 7 7 # * $ * 6 # #, # # # * ! # # ## * 75 8 9 # # # @ *

6 # A B 7 # * # ## A B * # A B # 7 # # # * C # # # < * # 7 # #, # * - ## # * Sequential Vertex-Coloring Algorithm

85 9 # # 7# 57 5 * D 5 # E+$ * F # # ;

E+$ # 7 * ' 7 * 0** - # G # * 5 35# H)D 10I* 6 # # ?5 35 * Algorithm 9.1.1:

: :

"# $%

7 * 75 : *

( ; " 8 9 :; # 5 # * & 75 *

% - 75 # ( 0**> >5 *

Section 9.1

375

Vertex-Colorings

Figure 9.1.4

#

' 7 7 # 7 ?5 #

( 0***

Figure 9.1.5

#

: 3 ( 0** ##

#

? - # ## *

8 # 9 ## # 7 # # - 5 * 57 7 # 8 J # 9 0** # * Basic Principles for Calculating Chromatic Numbers

5 * # # # +7 0** # 0**?* : $ 8 9 7 5 * : $ 8 9 # - * # 8 9G # # - 8 9* F K # *

376


& 8 9 Æ 8 9 L & - Æ 8 9 L

## 7 Æ 8 9 *

& 8 9

& # #, # *

7 #,* 8 9

- *

% ' 8 9 8 9

8 9 ## 7 # *

& 8

9 8 9

& $ 8 9 # < 8 9 *

! # ; 81 : 9 ( 0**2* 3

5#, #5 ; 81 : ?9 15 ?5 * 8 9 ; # - 0**> 8 9 8 9 ; 1 ; > >5 M ? > # 2*

Figure 9.1.6

81 : 9

& ( 8 9 8 9 & % # # *

Section 9.1

377

Vertex-Colorings

L # # ## # 7 # 7 *

& )

8 L 9 ; 8 9 L 8 9 & 6 , L # # 7 #, 7 * $ 8 9 - # # 8 9 - # 8 L 9 8 9 L 8 9* D 8 9 L # 8 9 # < * Chromatic Numbers for Common Graph Families

F # 4# 0*** Table 9.1.1

% ! 8 9 ## ##

? ? >

& * 8 9 ; & # # # # < *

& + 8 9 ; ! & 5 # 7

# 7 * 6 # 8 9 0***

% ' 89 ; ! & * 0*** : D # # *

378


# ## # *

% ' , 8 9 ; ! & * % ' 89 ; & * % ' 89 ; & *

& 8 9 ; ? & ! 8 9 ; * 0**>

8 9 888 999 ; L ; ?

## # *

; L #

& ! 8 9 ; ? & ; L 0**2 # ! 0** 8 9 ; 8 9 L 8 9 ; L ; ?*

& ( ! 8 9 ; >! & , L 5

0**2 # 0**? 8 9 ; 8 9 L 8 9 ; ? L ; > *

& ) " 89 ; & F 0** 89 * 5 *

Chromatically Critical Subgraphs

( # 8 9 #* ( # #

7# #* * 8 9 ; # #5# 8 95 # 8 ** #5 5 9* #

Section 9.1

379

Vertex-Colorings

( ## ?5 # # #

# ? * ) ## >5 0** # ( 0**1 5# 5# # ?* >5 * ## >5 8 +7 9*

Figure 9.1.7

- ! '

& * " ! # #"

8 9"

- + " #

& F # 7 # * ! # 8 95 75# 7 0**1* # # # 8 95 7 *

# # #

# * # 8 95 # * Obstructions to -Chromaticity

8 9

* * 5 * 8 0***9 & $ 8 L 9" " " & 6 # ## 8 L 95 # 5 *

8 L 95

8 L 95

*

+ *

5

H! 4 I #

## *

380


5 4

; ## 5 * , ## >5 # ?5 >5 * ( 0** # ## ## 0**> >5 *

Figure 9.1.8

.

Brooks’s Theorem

7 #

7 57 * 7 57 * HI 57 * HI " # # # * " #

# 57 * 7 #, 7 * / , "! " %" & ? # & & " & " 7 * ( 5 #* 7 # 7 8 7 # 5 9* 6 7 # # 7 # #* 8 9 ; * # 8 7 5 HI 9* $ 5 7 #, 7 57 # #, 7 57 * F 5 HI 7 #, 7 * ' # *

- 01 2 "!

& # # Æ 8 9 ? 8 9 Æ 8 9 & 6 H" 1I - # * $ ; * # * ( 7 7 # Æ 8 9* 37

Section 9.1

381

Vertex-Colorings

8 >*9 # # #* 6 # 7 7 5 #7# * ' 7 Æ 8 9 5 #7# * 6 - Æ 8 9 * $ # 57 * " 75# # ## 75 ; * # 7 Æ 8 9* F ! 7 Æ 8 9 * F 7 # Æ 8 9 * % # 5 #* F " 0**M 7 5#, # #* ) # # #* ! 7 7 Æ 8 9 5 #7# * 6 - Æ 8 9 7 * - # # * Æ 8 9 * Heuristics for Vertex-Coloring

75 # 7 # # Æ # H !2 F10I # - # #

# * # # #* * # * # # < # # #, * Algorithm 9.1.2:

: :

$% / 3 4

57 * 75 *

% # # 7 # 7 7 # #* 7 : 8 9 :; * & 75 *

# # # ( 0**0 8 9

5#5 # ?5 ( 0**0 8 9*

382


Figure 9.1.9

5 #

Coloring the Vertices of an Edge-Weighted Graph

( & ' H/ E0I

$ ## ! ! ! * # # # # # # < * # 75 # # # # # ## # < * # & 8 ** # ## # < 9* , # # @ 4#* @ # # # # #* # #, #5 # # #* ( # # - # # # # # # # * $ 89 # # # # # # # * ( 5 8 9 #5 # # # * , # 5 * H/ E0I # # A # ÆB A # ÆB #*

6 %7"" " 0** - # - * !

- *

Section 9.1

383

Vertex-Colorings

0** ! # < # - ## # *? MM *

' (#% (#() " "" ) " ""

0**

0**

0**(

0**)

0***

0**+

0**

384


' (##* (##+) ,

0** , +7 0**?* 0** +7 0**>* 0** +7 0*** 0** +7 0**2* 0** +7 0**1* 0** ( +7 0*** 0** ) +7 0**0* 0** * - 75 *

0** + ) # ( 0**1 - 75 # ?5 * 0** - 75 # # * 0**, ## # * 0** # 7 # * 0** - 75 # # #7*

0** 5#5 +7 0*** 0** 8*9 - - * 8' : - 75 *9 0**( ) 0**1 0*** 0**) ) 4 7 # # - * 0*** * 0**+ ## >5 * 0** >5 * 0**, , *

Section 9.1

385

Vertex-Colorings

' (#%# (#%-) " .

0** 0** 0** 0** 0**(

+7 0**?* +7 0**>* +7 0*** +7 0**2* +7 0**1*

0**) # ## 8 9 ; " # 8 9 ; " # * # # #$8 9 #

% 7 % 7 % * 8 M**9 '

(#%/ (#0#) " " .

0*** 0**+ 0**

0**, 0** 0** 0** 0**

+7 0**?* +7 0**>* +7 0*** +7 0**2* +7 0**1*

! # # * ! # # * # #

8 9 ; # # #$8 9 ; *

8 9 - - 8 9*

8*?9

0**( * 0**) #5 * 0*** H" 1I 6 0** # 0** # +7 0**? 0**0* ' (#02 (#0() , $ . . " " " 3

0**+

0**

386


0**(, H" 1I # 7 5 # 4 # # * # # * #5 # # +7 0**> # 0**>0 # 75 # #* 0**( ) $# $ 7 ## * 6 # # 7 # # - * 7 # # 7 * ( # 7 # 4 *

> 2 > > > 2 2 > > >

9.2

MAP-COLORINGS

## *

( # % &8% 9 Æ % # &8% 9 #* # ## Æ # * # # M ( 5! 8# 9 # ' # # * 6 &8% 9 # # # * ' # ! 2* Dualizing Map-Colorings into Vertex-Colorings

## ' :

% : ( 5 ( ; # # * ( # * 5 #

# # # < * ' : &8'9 ## *

%

( 0** >5 * C5

#* - *

Section 9.2

387

Map-Colorings

Figure 9.2.1

# # K # * 6 # 7

K # * : # * ( 0** # 8 9* C # #*

Figure 9.2.2

.

" 4# # # . # 8 1*9 75 # * # ( 0** ( 0**? # 7 # ; * 5#, # 4 5#, 7 *

Figure 9.2.3

- 4

&

Geographic Maps

% # # #* : 6 5 * ( ( # $ # 5 # # C #

388


# $ # ( 0**>* $ $ 4# # " * 6# # ! #, # 3 # F* #, # < *

Figure 9.2.4

3! 8' 4 "

: 7 # * # ( ! $ ( 0** # *

Figure 9.2.5

4 % 9"

$ - >* 6 # ( 0**2 5 F # ?5 * 0** $ - * % +7 >5 $ 8 +7 9*

Figure 9.2.6

- "

# $ * $ ## # ( 0**1 * % N # 3# # # / # *

Section 9.2

389

Map-Colorings

Figure 9.2.7

- . 9"

: # - 4 # ! # # # ( ! * , L # * Five-Color Theorem for Planar Graphs and Maps

( ! * F* / H10I ## # ' # H0MI # Æ Æ * ' #K *

75 # ## # *

75

5*

( ( 0** / 5? * # / ##*

Figure 9.2.8

. . :

- 0;. + ,2

( & $ # # 5 #* 6 Æ 5 * $ Æ 8 9 2 8 *>*9 7 # * 0** 8 9 2* 75# 5 0**1* 37 # 5 * 6 # 7 5 # 7## # *

390


# 7 * # ( 0**0* ! # >5 ( 0**0

## # # / 5> $ 7 8 ** # 9* # $ / # >5 7 * # > # / ( 0**0* 5 # * 5 7# 5 # 7 *

Figure 9.2.9

". :

$ / >5 # 5 7 * / 5 >5 # # ( 0**M* # # 7 # 7# # # ( 0**M* F D # ! 81*9 * $

Figure 9.2.10

: '

$ ?5 # 5 # < #

/ ?5 ) 5 ?5 * / ) # 7 5 * - 0 ; *)2 $ )"

: # ' H'12I ( ! 4# # * ( # ' # ' ## # # 8 0MM9 # * C MM #* + # # & $# $ # H& $$ 01I*

Section 9.2

Map-Colorings

391

6 %7"" " 0** 7# 7 # *

0** +7 0** # 7 # * 0** 7# 7 # *

0** +7 0**? # 7 # * 0**( 6 >5 7 ## ?5 7##O +7 * 0**) # +7 0**? # 75 * 0*** 7# 7 # *

0**+

( # / 5> *

392

0**


( # / 5? *

4 "

1 # # # * 0** , ! ( ! # A B # , *

0** ) >5 8 ( 0**29 $ # * 0** ! + # # 5 : ( F 3# "7 ) # $ 4#* % O 0** ! 3 * 0** ! * 0** ( ! * 0** ) ! *

0** * ! *

Section 9.3

9.3

393

Edge-Colorings

EDGE-COLORINGS

( # #5 # 75 * 75 #5 #5 * #5 75 #5 * The Minimization Problem for Edge-Colorings

# *

# < #

:

#5 ; # # * ( # *

#5

# *

#5 #,

# # # < *

: % 5# 75 < #5 # #* ) 5 7## # * # # 5 *

# 5 ( 0*?**

Figure 9.3.1

. (

8 9 5 # < - # #5 * 8 9 ; * 8 9 ; # 5 # 8 95 * #5 ( 0*?* ( 0*?* * # ?5 #, #* 8 9 ; >* # # ¼

¼

¼

¼

394


Figure 9.3.2

4

Modeling Applications as Edge-Coloring Problems

* : $ #

#* # # # < # #* - # #5 # * : ! ! * * * 6 ! * * * : ! # - # # * : ( ! ! # * * * # , ! # * # ( 0*?*?*

Figure 9.3.3

! !' !

4# #* 6 #5 # #5 * #5 ( 0*?*> #*

Figure 9.3.4

4

Sequential Edge-Coloring Algorithm

- #5 - 75 0**

Section 9.3

395

Edge-Colorings

#

# *

Algorithm 9.3.1:

"# %

*

: # : #5

( ; " 8 9 :; # 5 # * & #5 *

Basic Principles for Calculating Edge-Chromatic Numbers

+#5 5 # # 5 75 5 * : $ 8 9 7 # 5 * : $ 8 9 * 7 # #5 * # - # * & 8 9 % ' ! 8 9 Æ 8 9 & 8 9 8 9 ¼

¼

¼

¼

¼

# 5

¼

( 0*?*

8 9 # 7 5 7 8 9 ! 0*?** ' 8 9 ; * ¼

¼

¼

Figure 9.3.5

( !

Matchings

" # ## 7 * 6 # # #5 * 8 9 # 5#,* 7 #*

396


# 7 ## 8 9 *

# # 8 9 ¼

: 6 # # #5 * # # #5 # 4 7 * & ! 8 9 8 9 ¼

¼

+$#,

#5 # 7 * Algorithm 9.3.2:

% !' < <

: * : #

5 * 6 4 :; M* % ; :; L

( # 7 + * ( # + 89 :; :; + 8#5# 9 & #5 * 5 # 5 # 7 # D +# #* F # # ?*> # # *

Edge-Chromatic Numbers for Common Graph Families

6 # #5 4# 0*?* 75 # # 0** 7

75 # 7 *

& ( 8 9 ; Æ 8 9 ; ¼

& ) 89 ; ! ? & * 89 ; & + 8 9 ; ? & 8 9 ; Æ 8 9! & , 89 ; ¼

¼

¼

¼

¼

+$#, +$#, +$#, +$#, +$#, +$#,

Section 9.3

397

Edge-Colorings

&

8 9 ; ! ?

!

¼

+$#,

( 0*?*2 # 5 *

Figure 9.3.6

( .

& " 89 ; ? & : ¼

5 # M # 5 8 # ( 0*?*1 ; 19* C # , M # # # 8 # 9* $ # 8# # # # ( 0*?*19 #, # *

Figure 9.3.7

-.

$ # # ? 8 # 9 # 8 ## # ( 0*?*19 # ?* 6 % % % % # # 8 # 9* # % # # 5 #* : 4 7 ## * $ ; # 0*?*> 8 9 *

¼

% ' " 89 ; & , ## ¼

7 * 0*?* # 5 #5 7 7* #5 7## #5 7 # , 7 7 8 ( 0*?* 9*

0*?* 4 #5

## # # *

398


Figure 9.3.8

Table 9.3.1

! 8 9 Æ 8 9 ; ? ##

? ##

¼

?

Æ 8 9 Æ 8 9

Chromatic Incidence

7 # #5 , * ( #5

7 # # # * C 7 *

#5 # < #5 # 7 * 6 # # 8 9*

# # 8 9 # * #5

8 9 ;

8 9

¾

( #5 ( 0*?*0 # <

#5 # ## # # # # * # # * ( #5 # ? # #5 , # *

Section 9.3

399

Edge-Colorings

Figure 9.3.9

-

# - # *

& " ! 8 9 #,8 9

% ' ( "

8 9

¾

#,8 9

¾

& ) " # !

8 9 ; #,8 9

% ' * "

8 9 ;

¾

¾

#,8 9

Edge-Coloring of Bipartite Graphs

#5 - 75 * 3 #: 8 9 ; Æ 8 9* 4 ## # * # # # * #5 # * * ¼

# # * *

/ + & & " %" # % & #: * # 5 # #5 # - *

400


$:

* ! # 8# #9 7 # >* # ## # - # 5 #* # 7 7 # ##5 # # 5 #* 7 # > 7* # 7*

Figure 9.3.10

% ' %

%:

* ! 7 , 7 ##5# 7 7 5# 8 ( 0*?*M 9* F +K #5 - 8*9 ##5# 7 #* 7 5 4 8>*9* 37 # 5 # ! * #5 # # #5 # 7 # * £

£

£

£

6 8 9 #5 ## 5 # # 5 #

#*

( # >5 ( 0*?* # / 5 #5 ## #*

Figure 9.3.11

. . :

/ " &

# & & & - " " # & F " 0*?* / 5 #5 # 7 ## # # # # # 7* #5

Section 9.3

401

Edge-Colorings

# # # 7 # 55 - # 7 * # # #5 7 #*

- , 0:= )2 8 9 ; Æ 8 9 & " P ; Æ 8 9 # # 8 9 ; P* ¼

¼

! 0*?* P 8 9* 37 # P5

# 8 9 7 * $ #5 7 8 9 #,8 9 8 0*?*29* # # * F P # P # # # 7 * 6 " 0*?*0 ## # * ¼

Vizing’s Theorem

! # 8 9 Æ 8 9 ## ! 0*?* N 4 K # #

8 9 * ¼

¼

# # 5 % # * # # # # # #, * 7## # # % * "

) 6 ( 0*?* # 5 # # # # #*

Figure 9.3.12

( !

/ & "

- " " + , & + 7 / # 8 #5 9 # # # *

- 0$> ) )(2 0 ))2 # " Æ 8 9 L & #5 # # 8** 0*?*9 # * 6 # # * C # # - # #* 6

402


# # # 7## # * # # #* $ 7# Æ 8 9 7 * " 7 - # 7 * ! 7 - # # # #* 8( 7 *9 $ 5# # 7 - # # * 37 * 6 7 - # # # # ( 0*?*?* 7 # # # 7* $ # *

Figure 9.3.13

!'

6 # 7 - 5# # 7 -

# # 7 * ! 5# # 7 - # # 7 * " . 7 # 7 - 8 . 7 9* #5 ! 7 # 7 -* 6 .* .: ; . #

* # # # * # ( 0*?*>* 3 #5 # # *

Figure 9.3.14

% .

! ; .* 6 6 .* " / 5 #5 # 7 * F # # 5# # 5# # 7 8 # .9* F " 0*?* / # # 7

* # # # $5

Section 9.3

403

Edge-Colorings

* 6 # ! #*

Figure 9.3.15

% .

7 * # * 7 -* ! # * # # ( 0*?* * $

Figure 9.3.16

% ! .

7 * # * 7 * # # ( 0*?*2 * # # * $

Figure 9.3.17

% .

7 7 * $ # 7 - # - # # 7 -* # 7 * 3 ! # ( 0*?*1 * $

404


% ' 8 9 ; Æ 8 9 8 9 ; ¼

Æ 8 9 L

¼

& # N 4 K # ! 0*?**

5 8 9 ; Æ 8 9*

8 9 ; Æ 8 9 L * ¼

¼

# 35

H' I*

/8 9

, *

7 #

: N 4 # Æ 8 9 8 9 Æ 8 9L /8 9* #5 # N 4 K 7 A B # ( 0*?* , # #* ¼

Figure 9.3.18

' ? @ # Æ

L /

Line Graphs

# #5 75 *

)8 9 # , # # #, # # # 7 *

*

#

# ( 0*?*0*

Figure 9.3.19

-

Section 9.3

405

Edge-Colorings

& " )8 9

#"

& # # *

: F HF2I # # # ( 0*?*M ## * $ 35 # # #5 # 75 *

Figure 9.3.20

- ! !

6 %7"" " ' (%# (%#$) " "" " ""

0*?*

0*?*

0*?*

0*?*

0*?*(

0*?*)

0*?**

0*?*+

406


0*?*

0*?* ,

0*?*

0*?*

0*?* 0*?*>* " 8 9 4 7 * 8 9 8 9 * ¼

¼

¼

0*?* 0*?** 8 9 ; # Æ 8 9 ; * 0*?* ( 0*?*2* 8 9 ; ?* 0*?* ) 0*?*1* 89 ; * 0*?* * 0*?** 8 9 ; ?* 0*?* + 0*?*0* 8 9 ; Æ 8 9 * 0*?* 0*?*M* 8 9 ; * 0*?*, 0*?** 8 9 ; ?* 0*?* ## # #5 * 0*?* $ # * 0*?* $ #5 )8 9 * 0*?* +7 # # #

#5 - Æ 8 9* 0*?*( ) 7 # # # # #5 * 0*?*) " Æ 8 9 8 9* * 0*?** " ## * * 0*?*+ " ?5 5#* * 0*?* ) 7 0*?* # # #5 * ¼

¼ ¼ ¼ ¼

¼ ¼

¼

0*?*, H" 1I 6 0*?* # 0*?* # +7 0*?* 0*?**

Section 9.4

9.4

407

Factorization

FACTORIZATION

% #5 #5 #, # # # ## # #5 # . 3 * 6

4 #5 # * % %* * ' 4 # # @ # ! ?* Factors

*

#5 *

#5

( 0*>* 4 *

Figure 9.4.1

4>

5 *

4 5 *

( 0*>* 5 4 *

4 89 * 6 4 89 # # # ?5 # >5 *

Figure 9.4.2

-. >

5 # * # 5 #

# 5 *

408


Tutte’s 1-Factor Theorem

* HM>I 4 K 5 @ # 5 * # # # K * ## * !" % ## % # 7# % * / . & " K # # # ### #, * L K # % * 6 # % 8 L 9 % 7 % * 6 # % ## #* # # , % * 6 # % # ## # ## - % 8 # 9* 6 # ## ## # 8 9*

/ & ! & # /! # - 0! % 0" & F 2** + -* 6 # # # # 7 # - 8 #9* % 7 7 - # 7 # * 8 # # # #*9 6 # 5 *

" 0*>* 7 # ( 0*>*?* 6 # # # # * # # # 5 # " 0*>**

Figure 9.4.3

-. ' .

" + # 0 * # + 0 #5 8 9 8 9*

Section 9.4

Factorization

409

K 5( # " .4 H" 1I*

- 0!" $ !2 0"

% ! % # % & 89 $ 5 * ## % 7 # 7 # 7 # 7 % * 6 % % * 89 F # 7 K # 5 * F ## # # 5 K # 8 " 0*>*9 # # 7 5 * % # # 5 * " 7 ; #, 7

% $ *

7 7 ## * + 7 #, 8 # 9* K # ## * ' ## 7 * 5

7 8 #9 # * #, 8 # 9 # # * $ # Æ * K # % ; ## * % $ * 6 5#, - # * 7 #, * F # ## # 5 * 6 + # 0 5 L - # L * % # < + 0 5 - # 5 * $ 7 # + # # 0 7 # M + 0 + 0 # # * + # 0 5 + 5# # 0 5# * " + 0 # - * 8C 5 + - # 5 *9 6 # # 0 5# + 5# 5 * 6 # # # 1 ; 8 L - L - 9 - # * $ ( 0*>*>*

410


% ! 1

Figure 9.4.4

$ # # # 1 * 1 7 # ? 7 # # # * 6

" 0*>* 1 5 * F 5 1 # + 5 * Petersen’s 1-Factor Theorem

##

# # $8 9*

- 0% " $ !2 $ " " /"

0" & F K 5( Æ K 5 # * " % # # % # ## % * %

? %

7 % # ? ?5* ( ## % # , %

;

7 # ? * F +K 5$ 8*9 # * $ ## ? ##* $ 5# 2

? # ?$8

%9

;

?$8 % 9 ? %

K # $8 % 9 % *

% ' ( $ " " /" %" & F K 5( 5 * $ ?5 #5 5 5 *

Section 9.5

411


: (* FQ 8HFQ?I9 # 8 95#5 # 5 5 # 5#5 # 8 L 95 5 5 * : # ! ?* 89 " . # ! ? # @ * & " $ ! H/Q 2I + 5

2 M 5 * % " $ ! H0I* + # 5 * 6 0*>* 0*>* 0*>*

%7"" "

5 *

?5 5 5 5 5 5 * # 5 ,

0*>* # # 5 # 5 5 * 0*>*( 5 # 5 * 0*>*) 5 * ' : # * 0*>** 5 # # 5 # # # * 0*>*+ 5 #5 5 * 0*>* $ 5 # #* 5 O 0*>* , 5 # # 5 * 0*>* 5 # # 5 *

9.5


(-# (-+ #/

0** 0** 0** 0** 0**(

% .

2

% 5 * ?5 % # ?5 * >5 % # >5 * 5 % # 5 * 25 % # 25 *

412


0**) % 15 * 0*** # 8 L 9 ; 8 9 L 8 9* 0**+ 257 8 9 L 8 9 ; #5 * 0** 257 8 9 L 8 9 ; 1* 0** , ! 5 2 * 0** 75 # #5 3 ( 0** # * 0** 75 # #5 ( 0** # *

Figure 9.5.1

0** ! 7 # # * 6 7 O 6 # O

Figure 9.5.2

0** >5 057 * 8' : # # ?5 >5 057 *9 0** ( ! 7 # ?5 * F * 0** ) ! 8? : 9* 0** * * * * #

# ?5 *

Figure 9.5.3

0** + ! *

413

Glossary

0** ) >5 ' " * % ? # < # O % ? # < # O 8 ## 7*9

Figure 9.5.4

<' A

0**, ! # * 0** # ## > # # # # * 0** # - > # #* % * 0** ! ## * 0** ! - 80 : ? >9* 0**( ! ## # 3 ( 0*** 0**) ! ## # ( 0***

Figure 9.5.5

0*** ?5 ?5#5 * 0**+ ## # < 5 5 5 *

GLOSSARY

8 : # < # # * ! : 7 # 7 57 *

: 7 * #5 7 : # < #5 # # 8 9*

414


!: : # < - #

# # 8 9* : ## " * % : 7 % - 8 5 9 ### % # # &8% 9* ' : # # # ##*

: 5 #5

- 7 #* .: #5 7 #* # : 7 #, * # ! : -* 75 : * ! : 7 5 * : 75 * : 75 7 # < * : 5 7 # < * : # * ! ': 8 L 95 8 L 95 * ! #$8 9 : # % 7 % 7 % * : 8 9 ; * ! 8 9 : # < - # #5 * ! : # 5 * : # A B * : #5 7 # < * ! : # # 5 # # # #, # * ! # 8 9 : 7 # ## #* : * : # # * . : ; L * ¼

¼

Glossary

415

: * : # * > : #5 #5 *

> : 4 5 *

! : 7 # ##

# # 8 9* : 5#, #* : 5#, * 8 . : L # ## # 7 # 7 * : : ## # * : : ## 5 # # 5 # #* ! : 7 57 * : )8 9 # # # #, )8 9* : ## * ## :

## * : # 5#,* : 7 #* : # #5 # # # 5 * ' /8 9 : 7 # , * ! : # # # * ! ': 7# * : ## * . : ; L ##* : 5#5 5 # 5#5 * : " * : ## 8 9 - - 8 9* : #5 # # # # < * : # # # < *

416


: # # # # < *

:

# * -A : # % ## % # 7# % * ! : 7 5 * : 75 # * : 75 7 # < * BC : 8 9 ; * BC ! # # 8 9: # < - # 75 *

Chapter

10

MEASUREMENT AND MAPPINGS 10.1 Distance in Graphs 10.2 Domination in Graphs 10.3 Bandwidth 10.4 Intersection Graphs 10.5 Linear Graph Mappings 10.6 Modeling Network Emulation

INTRODUCTION

!

"

417

418

Chapter 10

10.1

MEASUREMENT AND MAPPINGS

DISTANCE IN GRAPHS

# $

% # $ & '

#

# $$ (

# $ # $ & & # $ & # $ # $ ) # $ # $

*

( ( Eccentricity, Diameter, and Radius

# $ # $ & # $

#$

* #$ & #$ & # $

#$

#$ & #$ +

# $ & #$ #$

, & # $ +

#$ &

, #$ & - #$ # $

Section 10.1

419

Distance in Graphs

Figure 10.1.1

#$ & - #$ &

#$ &

#$ #$ #$

* *

* . / * 0

# $ * * #1 " $

!"#

1 #$

(

2

3

# # $ & # $$ # $ # $ # $ & # $ + Periphery

4 5

+

# $ &

#$

#$

, ,

#$ & -

$%&'&("# !

# $ #$ &

+

& ) 6 # -$

1 #$

#$ & 7 # $ & # $ &

420

Chapter 10


#$ 1 & # $ # $ & 1 # $ 1 # $ # $ & # $ # $ & # $ #$ Girth and Circumference

#$

#$

Figure 10.1.2

, 8 9

#$ & 8 #$ & 9

Convexity

+

,

: ;

,

: ;

: ;

$ , , 8 : ; & # $ &

Figure 10.1.3

+

: ; &

Section 10.1

421

Distance in Graphs

$ )* , , 8

# $

Steiner Distance

3

+

# $

1

!

, - 1

& # $ & <

Figure 10.1.4

'

. -8

+ 1 *

1

Total Distance and Medians

# $

# $ &

# $

+ , <

1 # $ & =

Figure 10.1.5

, ) # $ & =

422

Chapter 10


! " # $# ) $

# $

6

> & ,

# $ &

&

? & & & # $

& #

$ & # $# ) $ # $

@ ) 1 + &

, ! # $ &

&

Figure 10.1.6

# $ # $ ) # $ # $ # $# ) $ & # $# ) $ # ) $

) ) ) & & # $ & @ 6

# $

& ) &

# $

# $

Section 10.1

Distance in Graphs

423

# $# ) $ & ' 6

2

, :/A= ; # $ & ) ) ) # $ ) # $# ) $ &

+

#$

+ )* B

, < < > #$ & !

+ # 3 & , .

& 6

6 # $ & # " $

- ./' ' ')

. + - <

$ ! + 0 (

? ? 8 ? A ! ? ! C ? " ? " C 0 ? # . $ # $ ? # . $

!

1 , 0 0 . #$ & "

0 8 ? -

424

Chapter 10


$ ? 8 ! ? 8 + 0 0 A ! ( ? " - # $ 1 / * 0 "

2

,

#$ * #$ * * #$ * # $ * * '

0 ? * 3 0 #$ 3 # 1

# 0 ## $ & # $ $ 3 6 0 2 ! 3 #$ 8 0 #$ + 3 ? #A. $ 0 3 2

0 #$ #$ ) # $ ( ?

0 !

10.2

DOMINATION IN GRAPHS

+

(

? / :/!; > > :> !; , *(

:AA -; , :AA1=D; 3 $ + $

# $

$ 6 $

Section 10.2

425

Domination in Graphs

+

& #$$

%#$

#

, #

$

2 # $

8

Figure 10.2.1

,

, <

%# $ & * Three Applications of Domination

A

) :/98; $ .

(

) # % $. 1 (

+ 6

) & . ?

* * * *

* *

!- 6 * #1 " $ Private Neighbors

/ ,

426

Chapter 10


' # $

' : ; ' : ; & ' # $

3 $

$

' :; $ &

$ + $

. /

6 E *

$ ' $ $ ( $ ) $

> :> !;

:> !; $ $ $

#$ /

$

$ $ # $ 1 $ 3 $

$ $ :> !; $ $ ' $ $ + $ 0 / 0 #$ & $ $ & 6 $ # 0 #$$ 6

$ .& $

%#$ ( #

Section 10.2

427


Bounds on the Domination Number

, ! :'+19=;

, +

) Æ #$

%#$

#

#

*+,-. %#$ Æ #$

.& 0

*/$,0. %#$ )#$

)#$ 1 !

)#$ Æ #$

Independent Domination

+

6

*#$

+

%#$

, %#$ & 8

Figure 10.2.2

(

*

% %#$ %#$

1 # 3 $

$

$ 0

428

Chapter 10


# .& % %#$ *#$ 3

# & *#$$ 0

%#$

.& % %#$ *#$ .& # *

0 =

Connected Domination

+

$ $

%#$

$

, 8 %#$ & -

Figure 10.2.3

* ))

Distance-+ Domination

'

+ F + $ +

$ $ # $ + +

%#$

+

> %#$ & %#$ 7 %#$ %#$

!

, - %#$ & @ %#$ & -

Figure 10.2.4

* )2

Section 10.2

429


- ./' ' ') ,

"" "#

$

%#$

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ANALYTIC GRAPH THEORY 11.1 Ramsey Graph Theory 11.2 Extremal Graph Theory 11.3 Random Graphs

INTRODUCTION

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RAMSEY THEORY

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Section 11.1

473

Ramsey Theory

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Ramsey Numbers for Arbitrary Subgraphs

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Chapter 11

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$$$,

475

Extremal Graph Theory

9 )& -

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11.2

EXTREMAL GRAPH THEORY

2

2

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2

C D 2

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Figure 11.2.1

* / % % &

476

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() , () , ()

() , () ,

!

, () ()

$ - ! ./.. , ,

Section 13.1

535

Flows and Cuts in Networks

Figure 13.1.1

*

0 1 2 ( % )

Feasible Flows

+

*

$ ( )

() *

. ( 3

) () () ( ) () , ()

" 3 ! & () &

! ./.3 % ./.. 4 5

+

Figure 13.1.2

( ) ( )

,

()

()

536

Chapter 13


$ ( ) ( )

Cuts in - Networks 6 ( )

( ) ( 7-) ( -/)

+

* 4 ( ) ()

! ././ ( ) ()

( ) , () ,

Figure 13.1.3

( ) ()

$

, ,

Figure 13.1.4

! ./.7

!

!

+ , , , 0

( ! ./.-)

Section 13.1

537


Figure 13.1.5

Relationship Between Flows and Cuts 0 ( )

( ) 1 2 ! ( ) ( )

,

()

()

"

() ,

Figure 13.1.6

() ,

! () ! " #

!

( )

,

6 ( )

()

( ) ,

() ,

$

!

,

()

()

()

()

()

() , 8

() ,

()

()

538

Chapter 13

6 + ./.3

()

()

,

,


()

()

9

9

()

()

! ./.: " ././

Figure 13.1.7

5 , ( ) ,

()

() , : .

# $

$ ( )

!

,

()

()

$ " ././ () ,

,

()

$

%

! ./.: ./

.8

The Maximum-Flow Problem and the Minimum-Cut Problem -/

Section 13.1

539


!

$

( )

!

" ././

( )

,

() () ( " ././) () () ( ) () ( )

,

( () )

-/

# $ % &' ( $ ) *

!

$ ( ) ( )

" ./.-

# $ + &#, - $ ) *

$ ( ) , ( ) $ $

!

+ ; " ./.-

(;) ( ) , ( ) ( % )

+

! ./.< .8 6 = ./.:

Figure 13.1.8

540

Chapter 13

# $ .

$


() ,

() 8

$

% "

/ #011 1

! " # ! $

./.

./.

./.

./.

% & !

./.

+

%

./.

./. 0

.

./. 0 8 8 >

2

./. > ? ? = @ = ./.:

Section 13.1


541

3

./. ? % ./.A ? = @ = ./.:

./. ? % ./.A ? = @ = ./.:

./. ! ? = @ = ./.:

./. 0 > $ () $ = @ = ./.:

./. + 0 B

./. % ./.

= = @ = ./.: " = ./.<

542

Chapter 13

13.2


SOLVING THE MAXIMUM-FLOW PROBLEM

! ! C! !53D % $ 6

Using Directed Paths to Augment Flow 0

, ( ) ( ) , . ! ( ) ( ) 6 E E "( ) ( )

! ./3. 5 = , E , 3 <

Figure 13.2.1

45 % .

? A ! ./33 $

Figure 13.2.2

# 6 5

# @

Section 13.2

543

Solving the Maximum-Flow Problem

F

* 4 A .8 $

-Augmenting Paths

$

, ,

!

# , , ,

* =

0

'

@ ! ./3/ $

Figure 13.2.3

7 6

+ $ # #

()

()

() % 8

! E E ,

() () ()

E $

544

Chapter 13


* = # E E , E

4 E E #

A

Figure 13.2.4

! ! ./37 # , E , .

45 55 # 6$ E , .

' 6

1G2

! &) 5*

$ E $ ; #

() 9 E ;() , ()( )E

# #

(;) , ( ) 9 E = 8 ;() () E

# ; #

!

! # # ! ./3-

Figure 13.2.5

6 7

Section 13.2

545


E

# ;( ) , ( ) 9 E ( ) E

(;)

,

;()

;()

,

E 9 ( )

;( ) ,

# $

$ $ $ Max-Flow Min-Cut Theorem

&#8 9 ) * $ $ !

( () 0 " ./3. )Æ ()

0 = * ( ) ( ) +

0 + ,

() ,

() 8

# = ./.<

&9) 9#* $ & !

./3/ ( ) " ./3. ./3/

Finding an -Augmenting Path " ./3. ! ! ( 7.) #

546

Chapter 13

Algorithm 13.2.1:


- 9 )

** "* C*+D

! () *, 8 C ,D H ! # + E , E

! # () *, () 9 E % C D () *, () E I

-

$ ! ( ) E % 8

$ &

()

()

() % 8

Figure 13.2.6

) - ( )

( )

( ) % 8

* ! 6 Æ $ ./38 1 2 ! ! ( ) 6 Æ !

Section 13.2

547


0

! ! ./3: . ' 3'

Figure 13.2.7

7 $ 3' 5

% J ! ! C%J:3D - ! $ ./33

Algorithm 13.2.2:

)5 55 !

** "* # *,

( )

H 8 *, . H +

+ 0 $!(() *, H *, *, 9 . % I I # I #

$ ./3/ $ ./38 ./33 1!!%J2 ! ! % J

548

Chapter 13

Algorithm 13.2.3:


)): 9 )

** "* C*+D

! () *, 8 C ,D I $ $ ./33 # + E , E

! # () *, () 9 E % C D () *, () E ? I

$ ./3/ % ( )

Figure 13.2.8

0 3

( ) , 8

; 55 # E , 3

Figure 13.2.9

0

( ) , 3

; 55 # E , 3

Section 13.2

549


Figure 13.2.10

0 ( ) , 7; 55 # E , 3

Figure 13.2.11

) ( ) , 5 , ) * $

4 & ) * $ ./3/

% J C%J:3D $ ./3/

0 ( ) $ ./3/ ( ) Æ ( C:
/ #011 1 ./3 0 A % ./3/ F F

.. .& , . ! $

./3

./3

550

Chapter 13


./3

./3

%

./3

+

./3 $ -88 -88 A88 :88 588 588

= K K

.

./3 L " 0 0 H H " 0 $ ! ! * . M 3 M / M 7 !

2

./3 0 7 - 7 / 3.

5 !

>

3

./3 + " * M

M

./3 ! $ ./3/ ( % ./.A)

./3 ! $ ./3/

Section 13.3

551

Flows and Connectivity

./3 >

./3 C / 0D ! $ ./3/ % ./33 ./35

13.3

FLOWS AND CONNECTIVITY

N -/ N 1 8.

"

( +,( ) % +,( ) +,() % ( +,() ( +,() , +,()

!

+ - * * - * ( +,(* ) , ( +,(* )) 6

- - - -

-

!

( +,( ) +,( ) , " , +,() ( +,()

( +,() , +,()

,

" '

!

" , . & 5./ 6 .-3 & 6 " , ! ! . " , ! 9 . + .//. ; " , ! ; 6 ! & ! ! 9 . &

552

Chapter 13


Two Basic Properties of 0-1 Networks

$

8 .

!

() , . $ &

'

!

+ , & = () , 8 () , .

() , () , ( +,()

() , () , +,()

( )

( +,( ) +,( ) , ( ) , +,() ( +,()

( +,() , +,()

,

6 " .//3 ( ) & ( ) , & *

() ,

. 8

( ) , , ( % ) ( ) ,

Separating Sets and Cuts = -)

(

+ . $ . . .

+ / $ / / / /

* !

Section 13.3

553


!

() , . &

!

+ , 0 0 ( ) 0 1 0 1 1 1

( )

1

,

0

,

( ) ( .) (

1 )

Arc and Edge Versions of Menger’s Theorem Revisited

& ) 95< * / ' / & / !

+ / " ./// .//7

! = ( ./37)

* = N " ./// .//7 ! = ( % ) N .

. .

%

%

. . . . . ( . ' . ' % "

" +

1 / ) / , ) $ , ) 1

!

0 + , ) ( ) / 1 , ) ,

554

Chapter 13


) /

1 $ , ) ( 1 ) , / $ ( ) 1

.

. . . . & .

!

+ " 1 . "

. + 1 .

1 1

. 3" + 1

. 1 + .//: 1 1 " " 4

& . & . " "

2 &5 ) 95< * . ' . & . !

+ " '

& . " ' &

.

2 & . $ .//5 2 . &

' , 2 , 2 ' 6 $ .//< .//-

" ,' ' " , "

" " &

Determining Edge-Connectivity Using Network Flows

.

3 (.)

3 ( ) . .

Section 13.3

555


+ -/- -/

! 3

. &

3 (.) ,

3 (

)

% "

" .//.8 .//A 3 (.) . .

. ( $ .//5 .//<) 6

+ . . .

!

. 3 (.) & 3( )

!

0

) 3 (.) , 3 ( )) ( " .//.8) Æ 3 ( ) 3 ( ))

+ . .

. .½ ¾

) . +

.½ ¾ ( )

.½ ¾

4 ) ) . , ( )

3 ( )

,

( )

(" .//7 $ .//<)

( .½ ¾ )

,

( )

,

3 ( ))

,

,

( ) ( .) ( ) (! = ./37) (" ./// .//-)

556

Chapter 13

# $


. 3 (.) ,

!

3 (

)

+ (

, ) 6 " .//.. 3 (.) , 3 ( ) 3 . Æ

Algorithm 13.3.1:

5#$ #

** . "*

3 .

= . + . 3 *, !

$ $ ./3/ . ( ) 3 3 *, ( ) I 3

$ .//. () $ ./3/ ( ) $ .//. ( ) Æ $ ./3/

Using Network Flows to Prove the Vertex Forms of Menger’s Theorem N ( .//- .//A)

# (5 ) (5 /*

+ & / / * % % /

% / % / ) )

Section 13.3


Figure 13.3.1

557

5 5 5 /

+ & . ( /) $ . ( /) . ( /) ( )

( ) Relationships Between Digraphs D and N / %

/ % "

/ '

' % "

' & ' % " /

%

/ & % "

+ &= ) 95 (5*

' / ' / & / !

$ .//./ .//.5 N ( .//-)

558

Chapter 13


. &= ) 95 4 >*

' .

' . & .

!

+ . . .//.: $ .//5 .//<

Determining Vertex-Connectivity Using Network Flows N

N

& 3 ( ) +

+ -/-* + . & 3 (.) 3 ( ) &

*

( ) &

Algorithm 13.3.2:

=#$ #

** . , "* 3 .

= . 3 *, ! *, 8 H ! 3 ! *, ! 9 . ! 4 , ! 9 . & =

$

I 3

$ $ ./3/ ( ) 3 3 *, ( )

Section 13.3

559


$ $ .//. O C%:AD 3 $ ./// . Æ

/ #011 1 ! - - ! $

.//

.//

.//

.//

% 2 , !

.//% .//+ .//. .//

% .//. % .//3 % ./// % .//7

3 . ! - - ! $

2

.//

.//

3

.//

.//

& , . !

.// .// .// % .//

% .//A % .//.8 % .//.. % .//.3

560

Chapter 13

+ .// . .//


0 + .//.

0 , & . 0 *

() ,

. 8

( ) , ,

2 .//3 .//

.// .// .// .//

.//

" $ .//5 " " .//.8 " $ .//./ " $ .//.7 " $ .//.- " $ .//.5

" ! = ./37 $ ! N .//- " ./// .//7

13.4

MATCHINGS, TRANSVERSALS, AND VERTEX COVERS

N

Matchings in a Graph

$ ' . . '

) ' ' ) ) '

$

* $ . . =

= . ! ./7. $ . . $ .

Section 13.4

561

Matchings, Transversals, and Vertex Covers

Figure 13.4.1

5 .

Maximum Matching in a Bipartite Graph

$ . .

) * ) + 0 F & & $ & & & K ! ./73 & % & &

Figure 13.4.2

5 5 6

Converting a Maximum-Matching Problem into a Maximum-Flow Problem + . . . . *

(.) . .

(3) % . ) . )

(/) ! .

(7) ! ) . )

(-) % . () , .

562

Chapter 13


! ./7/ $ ./7. .

Figure 13.4.3

6 5 . 5 .

Relationship Between Matchings in . and Flows in .

!

. .

. $ . .

!

+ . () , 8 . . + '

. . . ' . ( % ) = ' . . 5 . .

() ,

. . 8

, 5 ' & 5 '

. ( % )

# $

' . $ . ( ) , ' $ '

! " ./7. @ ! ./77

/ . .

$ ./7. . . " ./7.

Section 13.4

563


Figure 13.4.4

5 5

Algorithm 13.4.1:

9 ? 95

** . "* ' . ' *, = . . $ $ ./3/ . ! 5 . (5 ) , . ' *, ' I '

$ 5 . .

*

@

% C%5-D %

' '

) , - ( * 0 H K $

Transversals for Families of Subsets

+ , 1 1 1 ( ) ( ) & , , 1 , . ,

564

Chapter 13

+ , $ , 1 1 1

1 , $M


1 , $ M

1 , M

1 , M

1 , $

& , $

) . / * 0 , ! 1 , 1 1 1 Relationship Between Bipartite Matchings and Transversals

+ . ' . '

' ' 4

! ./7-

Figure 13.4.5

5 5

6

Finding a Transversal by Finding an

/-Saturating Matching

+ , , 1 1 1 ( ) = . .

, , ) ) )

Section 13.4

565


1 ) $ ) 1 .

%

! ./75 & , $ % ./77

Figure 13.4.6

& , $ 5 5

Hall’s Theorem 45 Æ

+ . - (- ) & - (- ) - -

$

" - (- ) -

!

&@ < ? >A 2* . . - - (- ) !

( () 0 ' . - - ) ' % ) (- ) - - (- ) )Æ ()

0 #N = - (- ) - = . . ( " ./7.) $ . . . ! = ( ./37) Æ . ( )

566

Chapter 13


+ . - , & ( ! ./7:)

, -

Figure 13.4.7

, - -

, - 9 - 9 , - 9 - 9

( , .)

( . )

- 9 (- ) 9 ( (- )) , - 9 (- ) (- ) 9 ( ! ./7:) - 9 (- ) 9 ( (- ) ) , - 9 (- ) - 9 - ( #N = ) ,

# $ &@ < *

0 , 1 1 1 ! 1 ! . ! ,"

!

#N 6 B

* @ #N # $* 6

- 7 (0 % )

Section 13.4

567


Two Graph Factorization Theorems

$ $ % . $ ! . ! . $ ! % !

&&'!

*

( % )

. , % 8

123 4 % ,

!

0 - - ,- 0 , , ( )

(- )

,- ,

, -

6 #N 6 B . 0 , . . . . (, .)

$ $ $

*+ " % * $ . .

% &$ !

*

,( % )

. 7

1)564 %

!

H . + . 3, . %B = H 6

7 ,

- ,

& B 6 , . , 0 6 ./7- 6 . 2 6 6 2

.

568

Chapter 13


! , . . 2

2 , 2

. 2 3 . . 3 (3, 3)

Maximum Matchings and Minimum Vertex Covers = -

+ . . . .

$

+

! ! ./7< ( ) ( ) -

Figure 13.4.8

5

! + &' ( $ 95*

. . '

'

% "

# $ . &#, - $ 95*

' . . ' , ' % "

* = ./7< ( % )M

- . (- ) ( (- ))

2 &:B5A 2 *

.

!

.

.

&

+ . . & , , ! ./7A

Section 13.4

569


Figure 13.4.9

9 ,

= . . + - - % (- ) ½ - ½ (- ) , ½ (- ) 6 ) , . ( . #N = #N ' ' , ( ./7/) .

4 . . ' ' , ' , ' ' 9 , = .375 ' . ' ,

0-1 Matrices and the König-Egerváry Theorem $ 8 .

1

3 &:B55C$A 2 * 8 9 & 9 !

+ . & . ./7A ( % )

: ) * 0 > 7 @. @3 @/ @7 @-

. 7 / 7 3

3 5 7 < 5

/ / 3 / 5

7 5 / 3 3 7

7 7 : -

570

Chapter 13


+ ' - - 4 . 7 4 8

. 8 ' , .

8 8 . 8 8

. .

. . 8 . 8

8 . . . .

. 8 . 8 8

.N 7

Summary of Equivalences Among Theorems in Chapter 13 N ( .//.<) ! = ( ./37) JP N ( ./7A) #N ( ./7/) JP %Q ( ./7.8) 6 ! = , ( .//- .//A .//.: .//.<) ! = , # ( ./7/) # , JP ( ./7A) JP , JP %Q ( ./7.8) JP %Q , # (% ./7/3) , # (% ./7/.) , ! = (% .//3-)

/ #011 1 8 , ! -

./7

./7

Section 13.4


571

./7

./7

./7 ! J B # 6 ! = # L > ! B %

B K

%

./7 $ # > F

K =

"

B R I I 6 % =

0 H L " 0 0 O L 0

# 1 45 - - * ! 9

+ ./7. ./72 ./7 3 ./7

. 3 7 3 7 3 / . 3 / . 3 - . - . 3 3 - . 3 / . 3 7 . / 7 . 3 / 7 3 / 7 . 3 3 / - . / 7 - 7 -

!

./7

./7

./7

./7

572

Chapter 13

./7


+ . . (

" ./7.) " . ' . . .

%

./7 + ' . ! . 5 . ( " ./7.) =

() ,

. . 8

, 5 ' & 5 '

. 0 .

+

./7 + . Æ E " Æ E .

.

./7 0 & % ! & ! . & ! " & F &

2

./7 + . , , " ! ! . " .

3

./7 + , 1 1 1 . 3 ,

" " . "

./7 + . - " - . ( 3/) - .

.. . :; < 1 1

./7

./7

./7

. 8 . .

. . 8 .

8 . . 8

8 . 8 8

. . 8 .

8

8 8 . 8

. . . 8

8 . 8 8

. . 8 .

8 .

8 8 . 8 .

. . . . 8

8 . 8 . 8

. . 8 8 .

8 . 8 . . 8

Section 13.5

573


./7 0 & & & & F

& K

H. H3 H/ H7 H-

L . / 5 / 5

L 3 : / < -

L / 3 < 5 :

L 7 < / 5 7 /

./7

%

" (" ./7:)

./7

+

" (= ./7<)

.

./7 0 ./7A JP

2

./7 % #N / - ( = ./7<)

3

./7 ! JP %Q ./7.8 8.

./7 ./7

13.5

> #N ./7/ N ./7.< > #N ./7/ JP %Q ./7.8


./- + . , , " ! ! . " ' ' ' ' . (#* % ./7.A)

./- $ & .8 .8 & % & &

6 & & " /8 38

./- + ' 8N .N .N . " .N

./- " * $ .

574

Chapter 13


./- + . + . . " 9 , (#* % ./73.)

$ " " 8 , (" ) .

$

%

./- % - #& * ? #N " 8 " " (#* + , . 3 , 1 1 1 1 $ 8 8 (" 9 .) #N = )

+

./- ? #N (#* M N 3 ) $

Figure 13.5.1

.

./- > . 3 / 7 - 5 : <

K , - 5 : , 7 - 9 , 3 / 5 : < , . 7 8 , . / 7 # , 3 - 5 < & , . - 7 , / - #*

GLOSSARY

55 # *

6

# , , , *

6 5*

.

*

()

$ * $ * ,

()

M

* () , ()

575

Glossary

*

5$ 3(.) .* . 3 ( ) * .

5* 5 .* . 5 .* . 5 .* . !* ! . 8 5 .*

. ! D6 E * ! *

* () * . ( ) () () 3 ( ) () , ()

* 55 * ( ) # , , , *

* @ < # . * - (- ) -

()*

() ,

.* & .

() ,

-

$ F ( ) () *

( )

5

* " 8 , (" ) .

"

7* 5$ 3( ) .* .

$ 3 ( ) & .*

.

) ' *

& )

'

5 .*

.

*

576

Chapter 13


* 5 * * * * =

* ' 3 * 8 . ()*

() ,

.*

() ,

.

7 *

, ,

5 ' . *

' '

5 /*

/M /

5 5 .*

.M

.

5 . ( /) . ( /)M ( )

5 G 5 * * = E #* () ()

()

* = $ * , 1 1 1 ( ) * & , , 1 , . ,

( )* ( ) ,

()

.*

()

M

. .

*

Chapter

14

GRAPHICAL ENUMERATION 14.1 Automorphisms of Simple Graphs 14.2 Graph Colorings and Symmetry 14.3 Burnside’s Lemma 14.4 Cycle-Index Polynomial of a Permutation Group 14.5 More Counting, Including Simple Graphs ˝ 14.6 Polya-Burnside Enumeration

INTRODUCTION

577

578

Chapter 14

14.1

GRAPHICAL ENUMERATION

AUTOMORPHISMS OF SIMPLE GRAPHS

!! " !

# $ % # $ $ $ &' &' $ # $ % $ % &' ( &' &' % " # $ % # $ $ # $ $

&' # &' ( ) &' )

&'

# &' &' The Sets &' and &' are Permutation Groups

$

* " $

+ % ,

* + % ,

$ $ # &' (

-

Section 14.1

579

Automorphisms of Simple Graphs

&' .

# # & #' &' &'

%

& !!' /

# 0

# # & '

&' ( &'&'&'&' &'& ' &'& ' &'&'& ' &'&'& ' &'&'& '

&' ( &'&'&' & ' & ' &'& ' &'& ' &'& '

Figure 14.1.1

# !! # Automorphism Groups of Some Other Simple Graphs

& '

& ' 1 # $ # "

Figure 14.1.2

23 4 !0

580

Chapter 14


5 23 # 1 5 #

& '

23 2024 ( 6 5 7! 3 #

& ' & ' 5 # #

&'& '& ' & '& '&' . 5 ! Æ

Figure 14.1.3

& ' 3

# $ # ( 6 2020 5 7! 3 #

& '& ' & '& ' Æ

5 #

# & '&'& '& ' & '& '&'& '& '& ' 5 ! # ( 4 .

( # !0

Figure 14.1.4

!

& ' * # ( ( & ' $ . &' ( & ' ( $ & &' ( & ' ( '

Section 14.1


581

. ( &' ( & ' (

( ! " ! - " -

&! "'&' ( !"&&'' ! - " - # ! " . ( & ' ( - -

& ' ( 11 . ( & ' - - # $ 8 &1' # $ # $ # & ' ( !&1'

Figure 14.1.5

" 4161 ( 7!9

# $%&& &

202

202

202

202

202

202'

202(

202)

582

Chapter 14


!

202*

202+

202

202

202 : &';

202 )

# :

; : # ;

202 )

: ; : # ;

202' )

$ 6 : ; : # ;

202( : & ' !# 4# 0# ; : # ; 202) : & # ' # # & 2!'; : # ;

Section 14.2

14.2

583

Graph Colorings and Symmetry

GRAPH COLORINGS AND SYMMETRY

# . # &'

3 204 206 Coloring a Set Subject to the Action of a Permutation Group

#

$

&'

2 ! % % % $ $

&$' $

2 ! % % % $ &$' &' & '

&' & ' ( $

< / &

# 4'

* ( + % ,

&$' $ ( $ &' & &''

&' & ' &' & ' &' & ' &$' (

* ( # 4

20!2 ( ( & ' & ' # 9 2!9 !09 3 &' & ' &!' 20!2 &!' % 2 ! & 2' 3 & !'

4

# &' & ' + &' & ', ( 222 !22 2!2 22! 2!! !2! !!2 !!! Æ

Æ

Æ

584

Chapter 14

Figure 14.2.1


,-

% . !22 2!2 22!

( Induced Permutation Actions

. # 20!2 #&!' 8

$ $

( + % ,

&' & ' $ &' & ' $ $ &'

$ ( + % ,

% &' & ' &' & ' & ' ( &' & '

&' & '

$

% &' & ' &' & ' " & 20!4' &' & ' &' & ' &' & ' = 3

( + % ,

( + % &' & ', &' & '

*

=

&' & '

Section 14.2

585


. 20!! $ &' & ' ( 222 22! 2!2 2!! !22 !2! !!2 !!!

Figure 14.2.2

$

> & '

Equivalent Colorings of a Graph G under Aut(G)

. # ? $ $ # $ & / 20!2' : 3 &' & ' &$' &' & ' &$'

* &'

# &' &'

# &' &' &' & ' &' & ' &' & ' ) &' & ' &' &' & '

# 20!4 2@9 5 # Æ

Figure 14.2.3

! ,- -/

586

Chapter 14


20!0 <5 #

Figure 14.2.4

! ,- /

Counting Vertex- and Edge-Coloring Orbits One by One

# A 3 # " 3

"#

! "# 4 & ' #

) &' &' &'&'&'&' &'&'&'&'&' 5% % # &'&'& ' &'& '& ' 5% % # &'&'& ' &'& '& ' 2@9 & '& ' &'& '& ' Æ

20!6 &' &' & & '

Figure 14.2.5

'

2B #&!'

& '/" &' &

'

. ) "

! ( 4! 20!B 20 & ' &' & ' ># &!' "

& ' ) )

Section 14.2

587


Figure 14.2.6

- 0 " &' &

'

Elementary Application of Symmetries in Itemizing Colorings

: # 3

&' & ' #&!' #&!' 3 3 # # #

3 20!7

Figure 14.2.7

-/ ! ! "

C 3 #

3 3 " 3 !9 & ' . !9 #&!'

'

&!' 29 %

2 2 29 ( ! !

3 3 3 0 3

588

Chapter 14


(

&' &4' &!' = 4 20!@ # 3 4

Figure 14.2.8

& /,- //

) 2@ 4 ) ! # # 20!B !0 &4' # # &4' 2@ 06 ( !0 # 4 . @2 ( 4

# $%&& & ! !" #!$ &' % !&

20! 20! 20! 20! 20! 20!' 20!(

D #

! ! &' #4$ ' !"

20!) 20!* 20!+ # 20!4

Section 14.3

Burnside’s Lemma

589

20! 20! 20! D 20! # !& !! &' #!$ ' !(

20! 20!' 20!( 20!) 20!* 20!+ 20!

# 20!4 D #

!!! !! &' #4$ ' !"

20! 20! 20! 20! 20!' 20!( 20!)

14.3

BURNSIDE’S LEMMA

# 20!4 D #

3 . CE " CE = " &' :

CE & 206' &' &' & ' &' & ' Stabilizer of an Object ( + % , &' ( &' (

*

590

Chapter 14


$ $ 2 &'

& ' ) &' ( ( &'&'&'&' 5% % # ( &'&'& ' 5% % # ( &'&'& ' 2@9 ( & '& ' Æ

. & ' % &' ( ( &' ( ( &' ( ( &' ( (

$ & ' ) &' ( ( &'&'&'&'&' 5% % # ( &'& '& ' 5% % # ( &'& '& ' 2@9 ( &'& '& ' Æ

. & ' % &' ( &' ( &' ( &' ( ( &' ( ( Fixed-Point Set of a Permutation

* ( + % ,

*& ' ( &' (

* & ' $

2 $ "#

# & '

& ' "#

"#

* &'&'&'&' ( * &'&'& ' ( * &'&'& ' ( * & '& ' (

Section 14.3

591

Burnside’s Lemma

* &'&'&'&'&' * &'& '& ' * &'& '& ' * &'& '& '

( ( ( (

# CE Relationship Between Stabilizers and Fixed-Point Sets

1 ( + % ,

&' (

* & '

# # $

# 2 &' ( 9 & 2E' #

. & ' & # 2042'

&' ( ! F ! F ! F ! ( @

"## & # 2044' *& ' ( 0 F ! F ! F 9 ( @

' . & ' & # 204!'

&' ( 2 F 2 F 2 F 2 F 0 ( @?

"# & # 2040' *& ' ( 6 F 2 F 2 F 2 ( @

Relationship Between Stabilizers and Orbits

1 ( + % ,

&' ( &+

&'

)

&+ &' ( ( % % % ( 2 % % % $ $ % % % ( &'

592

Chapter 14


( 2 % % % &' ( Æ & ' $ ( ( &' ( 2 % % % ) % % % &' &' ( C ( &+ &'

( & ' ) 5% % # 5% % # 2@9 Æ

&' ( ( &'&'&'&' ( &'&'& ' ( &'&'& ' ( & '& '

& ' ! ! # 2042 & ' 0 # &' ( 3 ! (

) & ' ) &' ( ( &'&'&'&'&' 5% % # ( &'& '& ' 5% % # ( &'& '& ' 2@9 ( &'& '& ' Æ

& ' &' ( 2 ( 0 (

% # #

1 ( + % ,

2

(

&+ &'

)

% % % ( .

2

&+ &'

(

2

&+ &'

2 2 2 ( 2 ( ( 2 ( (

Section 14.3

593

Burnside’s Lemma

*

* 2044 2042 = $

2 * 2044

Figure 14.3.1

0

Proof of Burnside’s Lemma

234 5 ( + % ,

2 *& '

(

* 2042 204! 2044 CE 2 *& ' ( 2 &'

2

& * 2042'

& * 204!'

&+ &' 2 ( 2

&+ &' 2 ( & * 2044' ( &+ &'

(

Direct Application of Burnside’s Lemma

CE #

# :

CE ( + % , % 2 * & ' ? !

#

#

+

& ' "# # 2046

594

Chapter 14


*& ' ( 0 F ! F ! F 9 ( @

) & ' ( 0 CE ! ( @G0

& ' "# # 204B *& ' ( 6 F 2 F 2 F 2 ( @

) & ' ( 0 CE ! ( @G0

# $%&& & ) )(

* ' * + * , + - ) , + - )! , + - )) , + . /

204

204

204

204

204

204'

204(

204)

204*

204+ 204 204 204 F 204 204 204' F

# 2022 # 202! # 2024 # 2020 # 2026 # 202B # 2027 # 202@ # 202H # 20229 # 20222 # 20220 # 20226

Section 14.4

595

Cycle-Index Polynomial of a Permutation Group

)" ))!

* ' * + * , + - ) , + - )! , + - )) , + . /

204( 204) 204* 204+ 204 204 204 204 204 204' 204( 204) 204* F 204+ 204 204 F

# 2022 # 202! # 2024 # 2020 # 2026 # 202B # 2027 # 202@ # 202H # 20229 # 20222 # 20220 # 20226

14.4 CYCLE-INDEX POLYNOMIAL OF A PERMUTATION GROUP

=

$ #

&$' # #

Cycle-Structure Monomial of a Permutation

$

596

Chapter 14


*

( + % , $

, & ' (

(

+ $ $

( &2 7 H 4'&! 0 @ B'&6' , & ' ( 2 0

# & ' ) &' &'&'&'&' 5% % # &'&'& ' 5% % # &'&'& ' 2@9 & '& ' Æ

& ' ) &' &'&'&'&'&' 5% % # &'& '& ' 5% % # &'& '& ' 2@9 &'& '& ' Æ

Cycle-Index Polynomial

# &' # &' &' & ' &' & '

*

( + % , $

% % % ' ( 2

&

, & '

, & '

# 200! # & '

' ( 02 & F ! F '

&

Section 14.4


597

) !

!' ( 20 &! F ! ! ! F ! ' ( H

&!

20!6 & ' # &!'

# 2004 # & '

' ( 02 & F 4 '

&

!

!' ( 20 &! F 4 ! ! ' ( 20

&!

20!B & ' &!'

# Correctness of Substituting Into the Cycle-Index Polynomial

= CE & 2040' $ # &' &$' * ( + % ,

% &' & ' &' & ' " & ' ( &' & ' &' & ' &' & ' &' & ' &' & ' = 3 ( + % &' & ', &' & ' & ' &' & ' &' & ' = &' & '

598

Chapter 14


. -& % % % ' -&$ % % % $'

$

1 ( + % , &' & ' $

* & ' ( , & '&$ % % % $'

&$' "#

$ $ *& ' ( $ C $ , & '&$ % % % $'

' < 20!!

& ' &' & '

* 2002

( & ' & '

, & ' *& ' @ &222'&!22 2!2 22!'&2!! !2! !!2'&!!!' ! &222'&!22 22! 2!2'&2!! !!2 !2!'&!!!' !

(

( + % , &' & ' ( &$ % % % $'

CE

&' & '

( ( ( (

2

2

2

*& '

, & '&$ % % % $'

& * 2002'

, & '&$ % % % $'

$ % % % $'

&

(

& ' 5 # & '

' ( 2! & F '

&

) !

!' ( !2 &! F ! ! ' ( !9

&!

#&!' # 20!6

Section 14.4


599

) # &' 2 & ' ( & F ' ! 2 &! !' ( &! F ! ' ( 29 ! &!' # 20!B 8 4 2 &4 4' ( &4 F 4 ' ( 06 ! # 20!7 &4'

# $%&& & (

, &'

! , &'

!

0

1 2 #!$

3

# $ 0

1 2 #!$

3

#$

200

200

200

200

200

200'

200(

200)

200*

200+ 200 200 200 F 200 200 200' F

# 2022 # 202! # 2024 # 2020 # 2026 # 202B # 2027 # 202@ # 202H # 20229 # 20222 # 20220 # 20226

600

Chapter 14


" ! 3 ! &' + # $

&' & '

200( 200) 200* 200+ 200

# 2022 # 202! # 2024 # 2020 # 2026

!! !( 3 ! &' + # $

&' & '

200 200 200 200 200'

14.5

# 2022 # 202! # 2024 # 2020 # 2026

MORE COUNTING, INCLUDING SIMPLE GRAPHS

.

CE & 200!'

&!' Counting Necklaces

0 3

3 #&!' & & ' &' & '' 2062 3

Figure 14.5.1

/,- -/&!'/

Section 14.5

601

More Counting, Including Simple Graphs

. # & ' 2 & ' ( F ! F 4 F ! @

200! 2 ! F ! ! ! F 4 ! F ! ! ( B &! ! ! !' ( @ > #&4' C 200! #&4' % 2 4 F ! 4 4 F 4 4 F ! 4 ( !2 &4 4 4 4' ( @ I ) # # 4 0 ( 2! # 206! # #

Figure 14.5.2

-/

!

4 F 2! F B ( !2 #&4' 200!

"

Counting Vertex-Colorings of a Bowtie

2064 ./ #

Figure 14.5.3

"! ./

./ #

&./ ' &+'&0'& '&'&' &+ 0'& '&'&' &+'&0'& '& ' &+ 0'& '& ' &+ '& '&0 ' &+ 0 '& ' &+ 0 '& ' &+ '& '&0 '

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

602

Chapter 14


# &./ ' 2 & ' ( F ! F 4 F ! @ # &./ ' 2 F ! F F ! F ! & ' ( @ C 200! #&!' ./ 2 &! ! ! !' ( ! F ! ! ! F 4 ! ! F ! ! ! ( 2! @ 2060 " ) # 3 # 3 #

Figure 14.5.4

6/" -/ ./ ! ! "

. ) 200! &!' ./ %

! ! !' ( @2 ! F ! ! ! F ! ! F ! ! F ! ! ! ( !2

&!

" # Counting Simple Graphs

200! &' # & & '' &!'

&!' &' & ' .

! &!'

! #

/ 2062 # 0 6

Section 14.5

603


General Strategy for Counting n-Vertex Simple Graphs

& %

& '

) 3 # Æ & %

& '

# & ' # & ' & % !""#

" ! #

Counting Simple Graphs on 4 Vertices

& '

2 F B F @ F 4 F B ' ( !0

&

!0 # & ' " % % 2 J % B

( B ! % @

( 0 4 &4 2'1 ( ! % 4

$ " % B

&0 2'1 ( B $

& '

2 F H F @ F B ' ( !0

&

& ' , & ' , & ' # , . , & ' ( , & ' ( 4 + % . 2 # "# .

604

Chapter 14


. , & ' ( , & ' ( 4 + %

! 2 "# 2 "# !

# ! . ! , !

. , & ' ( , & ' ( 4 + % 4 4 2 4 4 , )

. , & ' ( , & ' ( 4 + % ! "# . ! ! ! ! ,

. , & ' ( , & ' ( 4 + % 0 ! 0 ! , &

$ !! "

A / 2064

2 ! F H ! ! F @ ! F B ! ! ( 22 ! ! !' ( !0

&!

22 2066

Figure 14.5.5

/-

% &!' B0 & #

' 22 2066 # #

22 B0 & #'

Section 14.5


605

Simple Graphs with 5 Vertices

$" %

C

0#

2 % % % ' ( 2!9 F 29 F 26 F !9 F 49 F !9 F !0

&

2

% % % ' ( 2!9 F29 F26 F!9 F49 F!9 F!0

&

% % % !' (

&!

2 ! F 29 ! F 26 ! F !9 ! F 49 ! F !9 ! F !0 ! ( 40 2!9 %

# $%&& & & &" 5

0 6 !

+

7 0 6 !

+

206 206 206 206 206 206' 206(

& & 5

0 6 !

+

7 0 6 !

+

206) 206* 206+

606

Chapter 14


206 206 206 206 206 . ./

I 3 3 I 3 # 3 &' &' # 2064 206' 2066 206(

206) I 206* I 4# 206+ ) 4 = # 2062H; 206 I #

14.6

˝ POLYA-BURNSIDE ENUMERATION

/KE # 204 200 206 = # $

# Inventory of the Coloring Classes ( + % , $

&' & '

*

' 20!6 #&!' 2 0 9 3 ! 4 2 3 4 ! ! 3 ! 2 4 3 2 9 0 3

Section 14.6

607

Polya-Burnside ˝ Enumeration

' 20!B &!' 2 6 9 3 ! 0 2 3 0 4 ! 3 0 ! 4 3 ! 2 0 3 2 9 6 3 ˝ Polya Substitution

L /K &+/47,'

200! $ & $' #

* & % % % ' #

( + % , $ # $ F F F

$

* & % % % ' #

( + % , $ $ # /K $ . & F F '

˝ Polya’s Enumeration Theorem

89 6 " # +C99, +C90, ' 7 8 ( + % , & & F F ' '(

$ &' & ' Æ & F F ' ( 2 ! % % % $

' < # 2000 # & '

' ( 02 & F ! F '

&

/K !

& F ' ( 20 & F ' F !& F ' & F ' F & F '

( 20 F 0 F B F 0 F F ! F 0 F 0 F 0 F ! F F ! F ( F ! F 4 F ! F

608

Chapter 14


. /K 3 Æ # # 20B2

' < # 2006 # & '

' ( 02 & F 4 '

/K ! 2

& F ' ( & F ' F 4& F '& F ' 0

( 2 F 6 F 29 F 29 F 6 F 0 F 4 F 4 F B F B F 4 F 4 ( F ! F 0 F 0 F ! F &

# 20B! Inventory of n-Vertex Simple Graphs

/K ! & ' #

" $

' / 2064 # & '

2 F H F @ F B !0

& F ' 2

& F ' F H& F ' & F ' F @& F ' F B& F '& F ' !0

( 2 F B F 26 F !9 F 26 F B F !0 F H F 2@ F !7 F 4B F !7 F 2@ F H F @ F 2B F @ F B F B F B F B ( F F ! F 4 F ! F F

(

. 2066 0# % 2 9 2 2 ! ! 4 4 ! 0 2 6 2 B

Section 14.7

609


# $%&& & ' 20B /K 4# 20B I 6# 20B I 6# 20B I 6# " 20B # 20B0 # 20B4 # 20B! = 40; 20B' /K 6# 20B( /K B# 20B) ) /K 4 # 206!9

14.7


" ": 5

- ; 0 .

<

, 89 #$ * < #$ -

;

0 .

<

, 89 #$ * < #$

207 207 207 207 207 207'

$ 6

#

8M 1#

610

Chapter 14

207(

207)

207*

207+


207 = # &A ' # #

207 * #

# A

N

207 . " # 4 / # ) % * #

GLOSSARY

%

&' %

611

Glossary

%

$ %

2 ! % % % $

&' &'/ % $ &$' &' & ' % $ 3 / 2 5 % 2 & % % % ' ( , & '

% $

/ % , & ' (

+ $ $ / &' % $ /,- ( + % ,% ( $ &' & &'' &' & ' &' & ' ,- / % &' ,- -/ % # &' 9/ % % *& ' ( &' (

- % % ( + % , &' & ' % ( + % , &' & ' - % $ " -% ( % " ": % &' + % ,% " $ ; / % /K #

612

Chapter 14


; " $ # % $ F F F # #

& % % % ' "0 ": % ( + % , &' ( &' ( - % -/ &' % $ #

Chapter

15

ALGEBRAIC SPECIFICATION OF GRAPHS 15.1 Cyclic Voltages 15.2 Cayley Graphs and Regular Voltages 15.3 Permutation Voltages 15.4 Symmetric Graphs and Parallel Architectures 15.5 Interconnection-Network Performance

INTRODUCTION

Æ

!

"

#

$

613

614

15.1

Chapter 15

ALGEBRAIC SPECIFICATION OF GRAPHS

CYCLIC VOLTAGES

! %

Symmetry and Specification Reduction

!

&

!

!

' ( )*))

+ ,

'

!

- ( )*))

Figure 15.1.1

- ( )*))

' ,

'

) . +/ 0 -

1 . +/ ) . +/ (

)*)-

Figure 15.1.2

' )*))

' 2 .)/

'

3

+ .-/ !

3

.+/

'

Section 15.1

615

Cyclic Voltages

' ! '

Voltage Assignments and Their Meaning

4

. 5 /

)*))

. /

2 ! 2

! - ( )*)- ' ( )*))

!

( )*)+./ ,

4 6 ./ ./

Figure 15.1.3

! 2

7 . /

7 7

616

Chapter 15


2

(

!

! ( )*)+

7 7

8 .))/ ( )*)+

Vertex Fibers and Edge Fibers

!

0

9

0 2 (

: 0 2 (

:

# ( )*)-

$

+ !

7 7

; ( )*)$

Figure 15.1.4

"

!

: 7 : 7

Section 15.1

617

Cyclic Voltages

: 7 : 7 -

*

)1

4

#

#$ !

:

:

# :

#$ (

(

Artificiality of the Arc Directions

5

<

. = >/ !

( )*)*./

( )*)+./

- . +/

) . +/ ( )*)*./ ,

( )*)+./

Figure 15.1.5

618

Chapter 15


#

. / . / ./ 7 ./

#$ !

?

?

5 ! 7 1 ) . /

7 5

!

# !

#$ 0 . / !

5

!

%&2

!

Voltage Graphs with Self-Loops

!

./

' ( )*)' ) . +/

( )*)+

Figure 15.1.6

$

#

4

Section 15.1

Cyclic Voltages

619

" ( )*)@ 7

.* 2 ) -/ % ,

) . */

*

) . */ #

- . */

*

- . */

Figure 15.1.7

( 7 .* 2 ) -/

)

7 .@ 2 ) -/ ! @

) . @/ # @

- . @/

Figure 15.1.8

(

7 .@ 2 ) -/

* 4 )*)A

.' 2 ) -/ ! ' + 7

) . '/ <

+

. '/

Figure 15.1.9

( + 7 .' 2 ) -/

620

Chapter 15


# . /

#$ !

)*))1

Circulant Graphs as Coverings of Voltage Graphs

% !

5

5

2

+

. 2 /

#$ ! %

( )*))1

Figure 15.1.10

.)+ 2 ) */ $

!

9

!

5

B

B

B !

Section 15.1

621

Cyclic Voltages

4 ?

7 .- 2 ) - / % ( )*)))

Figure 15.1.11

) -

(

%&2

. /

Designing Specifications of Given Graphs

!

6 )*))

Figure 15.1.12

# ,

! 6

( )*))-./ )1

)* 4 .)1 )*/ 7 * C 6 )*))

C 6 )*)-

# 6 )*)$ 6

!

+ -

( )*))-./

622

Chapter 15


! A

+'

% ) - + $ . A/ !

(

!

5

7 ) - + $

! ! - B )

.- B )/

! % ) - . - B )/ !

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)*)

)*)

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Section 15.2

Cayley Graphs and Regular Voltages

623

)*)! (

)*)-

)*) (

)*)+

)*) (

)*)$

)*)' &

)*)" D 6 )*)* )*)) 6

1 )*)* < E

15.2

CAYLEY GRAPHS AND REGULAR VOLTAGES

9

9 .);@;/

!

& .)A))/

%

5 9

Cayley Graphs

! 9

2 7

D

5 <

2 7

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(

) !

%

?

( $ @

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624

Chapter 15


0 7

! . /

7 7

5

.C

- )*)/

0 7

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( )*-) 9

)

9 .* 2 )/

Figure 15.2.1

+ . . )/ . . )/

, 1 4

( )*--

.-$ 2 ' A/

9

9

Figure 15.2.2

+ 1 .-$ 2 ' A/

0 7

!

9 . / 9 . / : 7 ! 9 . /

?

:

Section 15.2


625

( )*-+ 9

- + !

Figure 15.2.3

$ . . - +/

9

%&2 ( )*-+

9

;

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( )*-$

Figure 15.2.4

. - +/

%&2 0

9

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( -1

.- 1/ !

( )*-* +$ 9

Figure 15.2.5

+ . . 1) )1/

626

Chapter 15


! 9

) - ( )*-'

Figure 15.2.6

+ . . ) - +/

- 9 9

9 0

- !

5

9

!

!

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F

! ! ! ! ! ( )*-@ 9

!

Figure 15.2.7

. $ 1

Regular-Voltage Graphs

!

!

9 7 . /

./ ! ./ 0

Section 15.2

627


2

7 . /

! 2

, GD !@@ D !;@H

= >

!

" ! " ( )*-;

Figure 15.2.8

. / 1(

%&2 ! " 1 )

- ( )*-;

)*-@ " 9 (

9 I6 ?

9 Æ ! 2 7 . /

7 7 5

5

!

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628

Chapter 15


2 8

) ! ( )*-A

,

;

Figure 15.2.9

. /

Graphic Conventions for Clarifying Voltage-Graph Drawings

!

!

%

!

%

* ( )*-)1 ( )*-A !

Figure 15.2.10

0 $ $ 2 1 )

Section 15.2


629

Voltages on a Bouquet to Specify a Cayley Graph

!

9 %

#

. /

#$ !

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. /

#

( . /

6 )*-) ( )*-)) ./ ./ 9

Figure 15.2.11

$

.

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# 5 % % % !

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#

Figure 15.2.12

$ 1

630

Chapter 15


Multiple Components

)*-;

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"

Figure 15.2.13

( 1(

! !

"

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The Natural Transformation and Vertex-Transitivity

2 # !

&

0

%&2 4

& 4

# 2 # & 2

#$ 0 !

!

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!

& . / 7 & ./ 7 ! &

# &

& ½ &

Section 15.2


631

! & ( )*-)$ . ( )*)@/

&

Figure 15.2.14

$

. 2

: #$ 0 : ! & ½

! &

)*-)+

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)*- !

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632

Chapter 15


)*-' . / ! K

) - + $

.) - + $/ .) -/ )*-" . / !

) - + $

.) - +/ .) - $/ "

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# $

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%

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15.3

PERMUTATION VOLTAGES

5

Permutation-Voltage Specifications

!

Section 15.3

633

Permutation Voltages

0 7 G' 2 ( H

) '

7 . /

7 G' 2 ( H

7 G' 2 ( H

!

7 G' 2 ( H

K 5 ) -

5

( )*+) ./

./

Figure 15.3.1

3

,

5

.) - +/ ,

5

.) - +/ 5 ) 5 - # 5

.) - +/ 5 -

5 + 5

.) - +/ 5 +

5 ) !

2 7 G' 2 ( H 7 . /

7 ( 7 ( 5

5

!

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634

Chapter 15


2 6

%&2 !

5 ./. / 5

./ )*+)

%&2

%

Drawing Graphs Specified by Permutation Voltages

!

5

( )*+-./ K % (

)*+-./

( )*+-./

5

Figure 15.3.2

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692

Chapter A APPENDIX

& ( ( ( ( "$( ( % ! "$( % ) "$( % "$( ( %

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ALGORITHMIC COMPLEXITY

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Algorithmic Complexity

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A.6

SUPPLEMENTARY READING

- 0' ,:41 08 C&1 08 4(1 0,:(1 0 :=1 - 0F:41 0@.;B1 - 0# JC(1 0.CC1 0,CC1 0C;1

BIBLIOGRAPHY B.1 General Reading B.2 References

B.1

GENERAL READING

ALGEBRA and ENUMERATION

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

! " ./ &) 0123 ! 0 1 4 0,

&23 ALGORITHMS and COMPUTATION

% 5 67 8 " % # 5

, ./ &67 167 ! 1 . / &67 "3 ! " $ , ! ( &3 3 0 * # ! / % 9, &3 3 , &3 695

696

BIBLIOGRAPHY

+32 " + + ! "# % *. / &32 !66 * ! ./ &66 /62 % ! / $ (.%

&62 COMBINATORIAL MATHEMATICS

1 ' ( 1 " , ( : 1) * 1 9 " ( %

:) 0* & 0 ' % * ;"< 0.%

&& ( ! ( , % ! 1 =$ & * 6) 9 ! * (.%

&6) * 7 ' % * 9 " 0.%

:7 !7 % ! % 1 =$ &7 & 9 " / > ! :& GRAPH THEORY

16? $ 1 @ .%

&6? 1+ /62 @ + 1 " ' + * / &'()&*() - &62 1 6 1 1

A ! &6 1 032 1 5 ! * 0 + , " &32 $+) $ + + , 9 " $*$ ( :) $B? $ ( B 0 %

:? # * # !.8 ! " : 6? $, 5 ( &6?

B.2

References

697

6 0 $ , , ( &6 66 * 1C,=$,, &66 D) + D

-# $*$ ( :) %2 9 % ./ &2 !: ' , 0 @ ! !, / > ! &: /& # 1 / ! " (.%

:& /2 * / / + , &2 // * / / / > ! & SURFACES and TOPOLOGICAL GRAPH THEORY

63 + / # :& ;9 " / .E &63< 023 / ! 0 % &23 *3) * ! &3) /6) / . * " @ .%

&6)

B.2

REFERENCES

E , -# , ,, , , Chapter 1: Introduction to Graph Models

* 6) 9 ! * (.%

&6) /) * / - Ü&7 -# + D

$*$ ( :) // * / / / > ! &

698

BIBLIOGRAPHY

Chapter 2: Structure and Representation

033 1 # 0' $ , ,

/0 & ;&33< :6&F:67 @33 @C @ G , 0 0 0 :) ;&33< .: Chapter 3: Trees

% 5 67 8 " % # 5

, ./ &67 %?: # %4, , ,,,. 0 12 ) ;&?:< &6F&&& Chapter 4: Spanning Trees

% 5 67 8 " % # 5

, ./ &67 167 ! 1 . / &67 #? " / #C , # & ;&?< :2F:3& 9 2: * / 9 , 3H 0 ?H2 ;&2:< 7)? '?2 1 ' - , , 3 & ;&?2< +32 " + + ! "# % *. / &32 -: - - &: (?3 * $ (, ! , I 3 / 72 ;&?3< !66 * ! ./ &66 !: ' , 0 @ ! !, / > ! &: /32 # / , ( &32 /2 * / / + , &2

B.2

References

699

Chapter 5: Connectivity

% 5 67 8 " % # 5

, ./ &67 167 ! 1 . / &67 $%26 ("J ' A, 'A 1 &26 2&F27 9 9?2 + * 9 # * 9 0, K 0 /0 0 6 ;&?2< 7F)) %2: 9 % ,,, 0 0

0 0. 400 )6 ;&2:< &&):F&&)2 !: ' , 0 @ ! !, / > ! &: 2& / 7. 0 0 :7 ;&2&< ))&F)?? Chapter 6: Optimal Graph Traversals

$(6 % $ 0 / ( ! . ,, . , , , :2 ;&6< )?F? #9 ?) 1 #I # * 9 ! 0 ! . , , 0 1 0 : ;&?)< 77F)& #"?& @ 1C " $ - % :6 ;&?&< :7F:&3 #?: # ! , , 0 5 0 0 : ;&?:< 2F6& "2? ", $ , , 0 1 0 &7.& ;&2?< 737 " 37 ", " + 0 " $ . , 0 ? ;&37< 66F&:) 9 % 9 2 1 . &. 60 & # 0 @ .%

,, & 9 & % 9 2 1 . &. 60 7 # 0 @ .%

,, && 9 ) % 9 2 Ü): -# + D

$*$ ( 1 * 9 :)

700

BIBLIOGRAPHY

93 @ 9 , , , , , / :2 ;&3< ?76F??) 2: 0 ,, 0 & ;&2:< :37F:33 %2 % " , G , . 30 0 3 0 7& ;&2< ?)&F?2: %/3? ( % % ! / - " . C /0 0 &6 ;&3?< 6F63 ++'!6? " + + ' + % * ' # !, % ! / &6? -2 - - @ %, 0 0 23 ;&2< ?? (32 $ % (, - , / :7 ;&32< ?))F??) (!6: $ % (, ' ! I ! $ (.%

&6: * 6& " 0 * * " C - , , /0 & ;&6&< 232F26& * % * G 0 .1 & E. $ $ , ! * !+33 # * I * " ! ( 0 +

, , /0 0 2 ;&33< ?27F?6& ! 32 ! ! I I (. , , , / :7 ;&32< ???F?2? !, )& $ 1 !, / - ) 0 0 )6 ;&)&< :77F:73 1 67 $ + 1 , , . / 9 + 9 ! * * 0

;"< 7 !.8 ;&67< 32F&&& / 3: # * /

!Æ 0 5 0 0 :) ;&3:< 37F3?? Chapter 7: Planarity and Kuratowski’s Theorem

1 032 1 5 ! * 0 + , " &32

B.2

References

701

#0(2) #, D 0 * ( H . G 1%0 1 6 ;&2)< 77F)3 9)6 E 9 - .

0 0 && ;&)6< ::F:77 * 3 + * % * ., , :. ,. /0 0 0 0 : ;&3< 2&&F2&3 * 6& + * % * , I :. , 80 0 )) ;&6&< :)&F:)3 %2 9 % ./ &2 % 3) % * " C "Æ ( / :& ;&3)< ?)F?26 @?) 0 % @, 2 $, 5 ( &?) 6 $ , ( /0 0 0 3 : ;&6< :))F:3& 6& $ , ' L , /0 ? ;&6&< ::?F:)& /72 ' / 1, I, 8 , /0 0 0 60 )2 ;&72< :&F:: /& # 1 / ! " (.%

:& /) * / - Ü&7 -# + D

$*$ ( :) Chapter 8: Drawing Graphs and Maps

/66 / $ , , 5 1 7 &66 $ $ , $ , $ , , Ü&7? - # ; ' % * < $*$ ( : $ 0 3: % ! 0 $ / - 0 1 " !.8 &3: "/2 ( " ! / I , " ,,, @(. &2 ;&2< 2F6: ;! # # 1 * ,<

702

BIBLIOGRAPHY

63 + / # :& ;9 " / .E &63< +#? + # 1 $ , , 7., :7F:? ( ) / , # ! 5 0 0 ?? !.8 &? + ) + * , " Ü&7 -# + D

$*$ ( :) 023 / ! 0 !.8 &6 ;9 " H % 1 > / &23< 0 !: $ 0 , ! ! , ,,, 0 0 6 ;&:< :2?F:7 Chapter 9: Graph Colorings

%32 ' E / % " , . 30 0 0 0 6: ;&32< 3&&F3&: 1J76 9 1J 5J B J ! , - 0 0 -%0 & ;&76< :3?F:63 126 + / 1 # 3 ! &26 13 # 1 I @ , 0 :: ;&3< :?&F:?2 $62 0 / $ , , , 0 1 0 7) ;&62< &7F:: 3 0 * # ! / % 9, &3 % 6& E % @(. , . /0 & ;&6&< 3&6F3: 'D: + ' D

/ , 0 1 0 & ;&:< ?F23 'J &2 # 'J 5J #, 0 0 0 33 ;&&2< )?7F)2? + 3? + + AI /0 0 0 3 & ;&3?< :2F:3& 06& 1 0 $ &*9& . &6&

B.2

703

References

(&6& ( # J

0 &? ;&6&< &7F :: ( ) 0 ( ,, ! Ü?) -# + D

$*$ ( :) * !! 3 @ * # ( ! ( !, * , . , /0 0 0 3 3 ;&3< &22F&67 Chapter 10: Measurement and Mappings

++00) $ + + * + ( 0' ! ' 0I @ . , . , 0 0 & ;&)< 3F&3 12: $ 1 0 &2: 137 $ 1 - @ .%

&37 1!67 % 1 00 ! ( 0 0 -0 &6 ;&67< :2F:3? 1) * $ 1, 3" Ü) -# + D

$*$ ( :) 1$8 * $ 1, * $ * ( 8 1 4 0 ?) ;:< 7F:3 1#6? * $ 1, * # # "# &? ;&6?< 37F&3

,

108? * $ 1, 9 * 00 * ( 8 , 5 0 :&3 ;&?< )&F?: 1082 * $ 1, 9 * 00 * ( 8 .. ,. 0 22 ;&2< &&F&6 13) ( 1, 0 ;&3)< :?F:&: 1% 9 1 9 % ./ & $66 9 * ' $ + 7 . , ( +, ! # $ ;&66< &?&F&26 $$#6: ( B $ $A A ' # @ " , ,F / 2 ;&6:< ::7F:?)

704

BIBLIOGRAPHY

$#' 3? $A A ' # @ " * * ' , H

* * :) #, $ , ! 5 / - ;&3?< $- B 6 $ - * -

, ! % 1 B ! : 0 0 &&) ;&6< 7F)& $B) $ ( B Ü& -# + D

$*$ ( :) $ 36 " $ + ( 5 ( &36 #167 * # # * $ 1, I , 0 0 2 ;&67< 7&?F7&3 " ( 22 ( "M / , + (A 0 /0 0 &6 ;&22< &2F&&: 3) 9 / 0 . 0 3 &2 ;&3)< )3F?2 0 6: 0 $ , $ + 0 , I 0 0 7? ;&6:< 7:&F77& 0 6) 0 $ , $ + 0 , / 0 0 ;&6)< &?3F&3 7 0 $ , @ $, 5 ( :7 %2) + % % -, , , / &: ;&2)< &7&F&7? %%) / % 0 % Ü: -# + D

$*$ ( :) %%! 6 / % ! %, ( ! 0 # &6 %@ ?7 9 % * B @ , , %,

0 0 ?6 ;&?7< &7)F&)& %-! 0 % - * -

, % $ ! - ! ! , 0 : ;&< &7F& %-!& 0 % - * -

, % $ ! - ,,, ! . . ; D 9 * ' $ * + , # 9 %< !E 0 ( ;&&< 77F)7

B.2

References

705

00& 0 ! 9 * 00 % 0 0 E . ;D

"< / E : ;&&< 3?F3:7 0!& 0 ! 9 * 00 " * !, /0 &? ;&&< ?37F?33 +/ D + ' /

, , , / 7& ;&< 3?F) +067 * + ! 0 I , , , 0 0 2 ;&67< 7&F7:: 0)? " !I C.0I ! L, 0 0 77 ;&)?< 77F73 00 0' 9 * 00 !E 0 , & 0& B 0

0 ' % * ;" < 0.%

@ D ;&&< 72?F77 -2: - - 0 0 0 0 76 ;&2:< :2F:&: /36 * / * / : ;&36< :2?F:23 / !3 % 1 / 1 # " !,, * , , 0 * E

0*E 5 & ;&3< Chapter 11: Analytic Graph Theory

1 @) 1 1

A 8 @ 2$ Ü6& -# + D

$*$ ( :) 9) * 9 1 Ü67 -# + D

$*$ ( :) * ! * + , 1 + * % ! 1 / > ! & ?? * " 0 $ , , 0 /0 0 3 ;&??< &F3 )& ( A - , , ;%< ; !0 5# )6 ;&)&< )72F)?:

706

BIBLIOGRAPHY

/ ) @ / , 1 Ü6: -# + D

$*$ ( :) Chapter 12: Special Digraph Models

167 ! 1 . / &67 1 6& .$ 1, $ , $ H /0 ? ;&6&< &F)7 1/') " 1 # / ' Ü?2 -# + D

$*$ ( :) $3? " $ (.%

&3? %J7 * %J %, 0 0 0 : ;&7< :?F7: +?7 % + - , , EEE 30 0 3 0 &? ;&?7< &)7F&)6 0) !1 0

Ü7: -# + D

$*$ ( :) 06 ! 0 , 0 0 ?7 ;&6< 23F6 0 26 / 0 % * / &26 *) ' 1 * Ü77 -# + D

$*$ ( :) *2 ' 1 * ,H I 0 &&? ;&2< &3&F:&& !7 % ! % 1 =$ &7 6: $ , ".C %, , 0 5 0 0 )? ;&6:< &?&F&26 !: ' , 0 @ ! !, / > ! &: B6 $ N B " , .C %, /0 < . 0 1 0 ;&6< 3F6& Chapter 13: Network Flows and Related Problems

"2? ", ( K 0 /0 0 &3 ;&2?< ))F)23

B.2

References

707

"'3: ", * 0 ' , , , Æ. K , / & ;&3:< :)6F:2) "3 ! " $ , ! ( &3 9 92: + * 9 # * 9 " "# ( 5 ( &2: 'J &2 # 'J 5J #, 0 0 0 33 ;&&2< )?7F)2? 036 " 0 "# 0 # &36 (&6& ( # J

0 &? ;&6&< &7F :: !: ' , 0 @ ! !, / > ! &: Chapter 14: Graphical Enumeration

1 ' ( 1 " , ( : 1) * 1 9 " ( %

:) %(37 9 % " 0 ( , 2 , ( &37 ( 73 ( ' , I ,, $, 8

26 ;&73< &)?F:?) Chapter 15: Algebraic Specification of Graphs

$2 + $ , , , 0 : ;&2< )?&F)3 33 + /

, , 0 &6 ;&33< :37F:67 +: 9 + 0 ' , &: Chapter 16: Non-Planar Layouts

32 ! * + $ , /0 0 0 3 : ;&32< :67F77 #22 * # , 1 , 0 /0 0 &6 ;&22< 6&3F6::

708

BIBLIOGRAPHY

"2 ", ,

0 0 0 3 ;&2< .2)2 "2? * ", - /0 1 0 0 30 0 3 ;&2?< &:&F&:7 9066 0 9 + + 0 9 ,,,. , / 7? ;&66< ?:7F?7) 3) + 8 0 ;&3)< :7F:)2 37 + ! * 1 , 30

0 0 0 3 ;&37< ):F)? 3) + ! * /0 0 0 3 &3 ;&3)< :&6F:77 33 + /

, , 0 &6 ;&33< :37F:67 63 + / # :& ;9 " / .E &63< 27 / - , $ 30 0 0 0 2 ;&27< :3:F:3? %&6& + %4 5J ( , @ 0 76 ;&6&< )33F?6 @ !/3& " @ 1 0 ! / - ,,, /0 0 0 3 && ;&3&< :?6F:23 @ *!/3: " @ * # * 1 0 ! / ' . , ,,, /0 0 0 3 &: ;&3:< :2F:23 6 $ , , @(. , /0 & ;&6< ?26F?32 Appendix

$ 3& ! $ , , 0 (

= '&> ;&3&< &?&F&?6 "3 ! " $ , ! ( &3 3 0 * # ! / % 9, &3 /62 % ! / $ (.%

&62

SOLUTIONS AND HINTS

Chapter 1 Introduction to Graph Models 1.1

Graphs and Digraphs

!

709

710 1.2

Solutions and Hints


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711

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1.6

Vertex and Edge Attributes: More Applications

(

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+

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5 4

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0 7 . (

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Chapter 2 Structure and Representation 2.1

Graph Isomorphism

712

Solutions and Hints

2.2

A A A

A A A


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8 .

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A

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715

Solutions and Hints

Chapter 3 Trees 3.1

Characterizations and Properties of Trees

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716

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717

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3.7 Counting Labeled Trees: Prüfer Encoding

* * / * /

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1

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718

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Tree Growing

)

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719

Solutions and Hints

4.3

Minimum Spanning Trees and Shortest Paths

. " /

4.4

Applications of Depth-First Search ! 7 . .

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720

Solutions and Hints

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721

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/

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! * * . $

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8 8

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DeBruijn Sequences and Postman Problems

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728

Solutions and Hints

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"

A

729

Solutions and Hints

7.2


& .

.

F

.

0 . . 0

7.3

. . 0 ; .

. 0

.


!

0 JK

! .

6

0

.

7.4

.

.


; 7 .

.

9 0

730 7.5

Solutions and Hints


() & 5 " / C C

()

(

()

" /

"

()

0

" G

" /

0

" * "

( 0 " / / G "

" /

()

C 0

"

G

"

G C / "

& 5 . . .

/

, $ /

; .

$ . 0 ()

0

4 2

/

7 2

7.6

/

1.

" *

" G

Planarity Algorithm

9 8 &G .

; 9

7.7


! G G /

731

Solutions and Hints

Figure Sol7.6.2

, G G

1 ()

1 ()

, G G ?

2( )

*

3 C " 7

7

Chapter 8 Drawing Graphs and Maps 8.1

The Topology of Low Dimensions

()

6

"

*

*

*

2( )

732 8.2

Solutions and Hints


! 7

( )

+ + *

; 7

() ()

;

JK J.K () ()

8.3

Mathematical Model for Drawing Graphs

8.4


8.5


733

Solutions and Hints

F

. LG .

$ $ 3

2.

2

>

3

7 . . > 0

Æ

*3

! 3

Chapter 9 Graph Colorings 9.1

Vertex-Colorings

8.

734

Solutions and Hints

8. .

$ ? * G

7

; .

; . *

;

4 * *

. /

0 . 0

8 . 0 . .

& . . 0 0

*

0

! . 0 27 ! *

. 2 *

6.

2

9.2

Map-Colorings . 0

735

Solutions and Hints

B 1

! & !

. M . @ M . .

9.3

& 0

Edge-Colorings

& / /

* !

! I . . #

736

Solutions and Hints

. .

( . . / . . .

) & .

.

. 0

1.

.

9.4

Factorization

'

C '

$ ? *

Chapter 10 Measurement and Mappings 10.1

Distance in Graphs

( ) 8

"

2

2

4

8

2

8

( ) 2

& .

( )

10.2

() ()

&

() () C *

# 2 .

()


($ )

F /

($ )

"

"

A

( ( ( ( ( (3

($)

($)

* )) " * )) "

* )) " "

"

( ( * ))"*A ( ( * )) " A ( (3 * )) "

/

"

737

Solutions and Hints

10.3

Bandwidth

(. ) "

/

& *

(.

. .

*)

. .

. . *

& * (.

. . *) . .

10.4

. 7

Intersection Graphs

'

($ )

"

2 2 * *

2

7

.

10.5

" 5(/ ) " * ((/ ) / 6 7 " 5( ) * ) + " '

'


+

$

& $ *

.

738 10.6

Solutions and Hints


'

. 7 .

/

6 3

3

7 . 3

@

&

J

K ( C *)

.

/ . / 0 . 7 5 .

Chapter 11 Analytic Graph Theory 11.1

Ramsey Graph Theory , ** *

(/ ) ( ) C (/

- ** *

(/ )

/C

)

* " C *

* " *?

,

739

Solutions and Hints

(/ )

11.2

"

C


11.3

(

) " /

Random Graphs

*C

" *

( )

"

G . ( )

( )

"

G

" *

!

(& $ ** / * )

Chapter 12 Special Digraph Models 12.1


. .

! * * * . . .

$ 7 9 9 7 7

740

Solutions and Hints

C *

* *

. .

! .

@9 8 * * G

/

. <

@

. 9 . .

/

*

/

*

C

*

"

.

G

*

*3G

*3G

/G /G

*

! .

< /

/ * / A

/ *

A

/ *

A

/

,

9 .

"

C

C

C

"

; . @9

.

&. .

. . .

12.2

Digraphs as Models for Relations &

.

741

Solutions and Hints

"

( "

)

1 . . /

2

/ ? 3 G

. 2

12.3

Tournaments

&

< /

. . . .

$

. 7 .

7

/

7 2 7

Æ

Æ

. ? *3

Æ G . 7

8

.

&

* 8 . 2 (. - * / /) . .

742

Solutions and Hints

; . . .

. .

12.4

Project Scheduling and Critical Paths

; !6! .9 . . $

' . $

- ()

*

.- ()

/

*

*

*

*

*3

*3

G

*

*

3

/

/

0 / ; >

9

.- ( ) - (*) ()

"

" *

& 5

.

&

& 7 "

" *

. .

! . .

& . .

)

$

)

9 7 .

9.

)

7

- 0 -

.-

+9

12.5

Finding the Strong Components of a Digraph

. 7 . ;

;

-

- 9 9

743

Solutions and Hints

: $ *

-

/

/

;

-

&

A A

! *

&

Chapter 13 Network Flows and Applications 13.1


. 7 .

. >.

1 - */ * G

7 >.

-

/

9 .9 .

. 8 9

-

. . *

! J K

J 9K

<

.

8( )

/

2

( )

2

& 9

/ - .9 / (/ ) " GA (/ ) " *A ( ) " /A ( ) " A ( ) " /A ( ) " GA () " A ( ) " A ( ) " /A ( ) " A ( ) " /A ( ) " ( - ) " A ( - ) " ?A ( - ) " 3 8 / - >. .9 ! *G

! >.

&

.

.

2

.9 .

.

.

744

Solutions and Hints

.

.9 .

.

13.2

# F */ * / - */ * G


! 2 2 >. . .

& . >.

745

Solutions and Hints

13.3


$

. *

9

.

! >.

/ .

7 >. *

. . 2 /

0

8 $ */ / * . 9.

, ( ) " / , () " /

, ( ) 3

$

"

,(

) " /

" / &

. . *

, ( )

.

! >. /

>. * 0

, ( )

- */ / *

9

" /

. 0

; . $ */ / ?

/ . '

/

, ()

, ( )

"

" /

& >. * . * ( ) " ( ) " & ( ) ( ) . 7 >. ' .9 3 6 >. * >. 7

;

3

3

; .9

N * .9 3

3

.

3 ( )

"

/ * ;

(

746

Solutions and Hints

$ */ * * ) ; >. < . >. Æ 5

13.4


>. .9

. .

. >.

/

*

! . .9

. >.

6 1%

.

'

( *) (/ ) ( ?) (3 *) ( G)

* / 3

9

() 3 ( )

&

2

, 2

3 ( )

Æ

()

() 3 ( ) O 3 ( ) Æ

. . 1% 7

$ */ *G

. ;

"

()

()

"

9

3 ( )

747

Solutions and Hints

$ * PQ $R

*/ / */ ?

N .9 (F : " : . . : ! . '

N */ G) :

; . 9.

. . . 9.

Chapter 14 Graphical Enumeration 14.1


4 &

()( )( )()

5>

()( )(

4 ()( )( ) (

) ( ) ()( ) ( )( ) ( )( )

=

=/=

=/= =

14.2

$


()()( )( )

)

()( )(

$ ()()( ) (

) ) ()( ) ( )( ) ()( ) (

)

748

Solutions and Hints

/?

"

"

*

.

/

*

.

. . 9

*

. 9

*3

14.3

"

. . 9 /

*

.

/

Burnside’s Lemma

()

"

7

( )

"

7

( )

"

7

(

)()()

()

"

7

(

)()()

0 ()()()() 0 ( )()()

" #

/

"

*C*C C " C 8

C C / "

* "

C

8

"

()

"

7 ()( )

( )

"

7 ( )( )

( )

"

7 ( )( )

0 ()()()

"

0 ( )()

"

"

"

"

"

0 ( )()

0 ( )() 0 ( ) 0 ( ) *

C C " /C*C*C*CC 8 "

*

C "

C

" *

" /

" #

"

749

Solutions and Hints

14.4

. $ * / / . $ * /


*

)

&

*

)

&

*

( C

*

( C

"

"

* *

( C

( C

)

)

&

&

G

* 9.

1 * G ( )

750

Solutions and Hints

14.5


"

( )

"

* * * *

C C

(/ /)

C

"

"

*

C

/

* *

/ C */

*

/ C C 3

*

//

C

*

"

C

"

3

/

C *

"

/?

-7 . . * . . * . .

.

/

. *3

! / . ! / . *

14.6

. 9 $ * / * $ * 3

˝ Polya-Burnside Enumeration .

C

C

C

(/ /)

"

"

* *

C

C C

C

C

C

C

C / / / C /

/

G C G C

"

*

751

Solutions and Hints

Chapter 15 Algebraic Specification of Graphs 15.1

Cyclic Voltages

!

9 9

2

%

. *

.

* ( )

752 15.2

Solutions and Hints


& * /

.

- . 9 9 .

JK

: L? 9.

# (

)( )( )( )

(

15.3

)( )( )( )( )( )( )( )


753

Solutions and Hints

( ** **)(* *** *)(**)(*)

(*)

15.4


15.5

Interconnection-Network Performance

(. )

"

(: . ) "

/

( )

(

C

"

) "

(. ) "

(: . ) "

( )

"

( C )

"

754

Solutions and Hints

Chapter 16 Nonplanar Layouts 16.1

Representing Imbedding by Rotations

(

(

(

(

)(

)

(

)(

16.2

G=

(/=)


*?

*

G

3

C*

C *

?C

" /

*

" *

" ?

755

Solutions and Hints

16.3

Voltage-Graph Specification of Graph Layouts

F . .

( P4')

0 " 0

"

"

16.4

&

Non-KVL Imbedded Voltage Graphs

F . .

16.5

. 3 . *

The Heawood Map-Coloring Problem

# * 9 9 8 * * / #

;

8 *

756

Solutions and Hints #

;

#

;

Graph Theory and Its Applications, Second - Jonathan L. Gross.pdf

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