Graphs, Algorithms, and Optimization

305 Pages·2016·7.43 MB·English

by  Kocay W.

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Cover
title: Graphs, Algorithms, and Optimization Discrete Mathematics
and Its Applications
author: Kocay, William.; Kreher, Donald L.
publisher: CRC Press
isbn10 | asin: 0203489055
print isbn13: 9780203620892
ebook isbn13: 9780203489055
language: English
subject Graph algorithms.
publication date: 2005
lcc: QA166.245.K63 2005eb
ddc: 511/.5
subject: Graph algorithms.
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Page i
DISCRETE MATHEMATICS AND ITS APPLICATIONS
Series Editor KENNETH H.ROSEN
GRAPHS, ALGORITHMS, AND OPTIMIZATION
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Page ii
DISCRETE MATHEMATICS AND ITS APPLICATIONS
Series Editor
Kenneth H.Rosen, Ph.D.
Charles J.Colbourn and Jeffrey H.Dinitz, The CRC Handbook of Combinatorial Designs
Charalambos A.Charalambides, Enumerative Combinatorics
Steven Furino, Ying Miao, and Jianxing Yin, Frames and Resolvable Designs: Uses, Constructions, and
Existence
Randy Goldberg and Lance Riek, A Practical Handbook of Speech Coders
Jacob E.Goodman and Joseph O’Rourke, Handbook of Discrete and Computational Geometry, Second Edition
Jonathan Gross and Jay Yellen, Graph Theory and Its Applications
Jonathan 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
David M.Jackson and Terry I.Visentin, An Atlas of Smaller Maps in Orientable and Nonorientable Surfaces
Richard E.Klima, Ernest Stitzinger, and Neil P.Sigmon, Abstract Algebra Applications with Maple
Patrick Knupp and Kambiz Salari, Verification of Computer Codes in Computational Science and Engineering
Donald L.Kreher and Douglas R.Stinson, Combinatorial Algorithms: Generation Enumeration and Search
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Charles C.Lindner and Christopher A.Rodgers, Design Theory
Alfred J.Menezes, Paul C.van Oorschot, and Scott A.Vanstone, Handbook of Applied Cryptography
Richard A.Mollin, Algebraic Number Theory
Richard A.Mollin, Fundamental Number Theory with Applications
Richard A.Mollin, An Introduction to Cryptography
Richard A.Mollin, Quadratics
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Continued Titles
Richard A.Mollin, RSA and Public-Key Cryptography
Kenneth H.Rosen, Handbook of Discrete and Combinatorial Mathematics
Douglas R.Shier and K.T.Wallenius, Applied Mathematical Modeling: A Multidisciplinary Approach
Douglas R.Stinson, Cryptography: Theory and Practice, Second Edition
Roberto Togneri and Christopher J.deSilva, Fundamentals of Information Theory and Coding Design
Lawrence C.Washington, Elliptic Curves: Number Theory and Cryptography
Kun-Mao Chao and Bang Ye Wu, Spanning Trees and Optimization Problems
Juergen Bierbrauer, Introduction to Coding Theory
William Kocay and Donald L.Kreher, Graphs, Algorithms, and Optimization
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DISCRETE MATHEMATICS AND ITS APPLICATIONS
Series Editor KENNETH H.ROSEN
GRAPHS, ALGORITHMS, AND OPTIMIZATION
WILLIAM KOCAY
DEPARTMENT OF COMPUTER SCIENCE
UNIVERSITY OF MANITOBA
DONALD L.KREHER
DEPARTMENT OF MATHEMATICAL SCIENCES
MICHIGAN TECHNOLOGICAL UNIVERSITY
CHAPMAN & HALL/CRC
A CRC Press Company
Boca Raton London NewYork Washington, D.C.
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Page vi
This edition published in the Taylor & Francis e-Library, 2005.
To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks
please go to www.eBookstore.tandf.co.uk.
Library of Congress Cataloging-in-Publication Data
Kocay, William.
Graphs, algorithms, and optimization/William Kocay, Donald L.Kreher.
p. cm.—(Discrete mathematics and its applications)
Includes bibliographical references and index.
ISBN 1-58488-396-0 (alk. paper)
1. Graph algorithms. I. Kreher, Donald L. II. Title. III. Series.
QA166.245.K63 2004
511′.5–dc22 2004056153
This book contains information obtained from authentic and highly regarded sources. Reprinted material
is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable
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efforts have been made to publish reliable data and information, but the author and the publisher cannot
assume responsibility for the validity of all materials or for the consequences of their use.
Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic
or mechanical, including photocopying, microfilming, and recording, or by any information storage or
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The consent of CRC Press does not extend to copying for general distribution, for promotion, for creating
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Direct all inquiries to CRC Press, 2000 N.W.Corporate Blvd., Boca Raton, Florida 33431.
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Visit the CRC Press Web site at www.crcpress.com
© 2005 by Chapman & Hall/CRC Press
No claim to original U.S. Government works
ISBN 0-203-48905-5 Master e-book ISBN
ISBN 0-203-62089-5 (OEB Format)
International Standard Book Number 1-58488-396-0 (Print Edition)
Library of Congress Card Number 2004056153
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The authors would like to take this opportunity to express their appreciation and gratitude to the following
people who have had a very significant effect on their mathematical development:
Adrian Bondy, Earl Kramer, Spyros Magliveras, Ron Read, and Ralph Stanton.
This book is dedicated to the memory of
William T.Tutte, (1917–2002)
“the greatest of the graphmen”
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Page ix
Preface
Our objective in writing this book is to present the theory of graphs from an algorithmic viewpoint. We
present the graph theory in a rigorous, but informal style and cover most of the main areas of graph theory.
The ideas of surface topology are presented from an intuitive point of view. We have also included a
discussion on linear programming that emphasizes problems in graph theory. The text is suitable for students
in computer science or mathematics programs.
Graph theory is a rich source of problems and techniques for programming and data structure development,
as well as for the theory of computing, including NP-completeness and polynomial reduction.
This book could be used a textbook for a third or fourth year course on graph algorithms which contains a
programming content, or for a more advanced course at the fourth year or graduate level. It could be used in
a course in which the programming language is any major programming language (e.g., C, C++, Java). The
algorithms are presented in a generic style and are not dependent on any particular programming language.
The text could also be used for a sequence of courses like “Graph Algorithms I” and “Graph Algorithms II”.
The courses offered would depend on the selection of chapters included. A typical course will begin with
Chapters 1, 2, 3, and 4. At this point, a number of options are available.
A possible first course would consist of Chapters 1, 2, 3, 4, 6, 8, 9, 10, 11, and 12, and a first course
stressing optimization would consist of Chapters 1, 2, 3, 4, 8, 9, 10, 14, 15, and 16. Experience indicates that
the students consider these substantial courses. One or two chapters could be omitted for a lighter course.
We would like to thank the many people who provided encouragement while we wrote this book, pointed out
typos and errors, and gave useful suggestions. In particular, we would like to convey our thanks to Ben Li and
John van Rees of the University of Manitoba for proofreading some chapters.
William Kocay
Donald L.Kreher
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William Kocay obtained his Ph.D. in Combinatorics and Optimization from the University of Waterloo in
1979. He is currently a member of the Computer Science Department, and an adjunct member of the
Mathematics Department, at the University of Manitoba, and a member of St. Paul’s College, a college
affiliated with the University of Manitoba. He has published numerous research papers, mostly in graph theory
and algorithms for graphs. He was managing editor of the mathematics journal Ars Combinatoria from 1988 to
1997. He is currently on the editorial board of that journal. He has had extensive experience developing
software for graph theory and related mathematical structures.
Donald L.Kreher obtained his Ph.D. from the University of Nebraska in 1984. He has held academic
positions at Rochester Institute of Technology and the University of Wyoming. He is currently a University
Professor of Mathematical Sciences at Michigan Technological University, where he teaches and conducts
research in combinatorics and combinatorial algorithms. He has published numerous research papers and is a
co-author of the internationally acclaimed text “Combinatorial Algorithms: Generation Enumeration and
Search”, CRC Press, 1999. He serves on the editorial boards of two journals.
Professor Kreher is the sole recipient of the 1995 Marshall Hall Medal, awarded by the Institute of
Combinatorics and its Applications.
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Contents

1 Graphs and Their Complements 1
1.1 Introduction 1
Exercises 6
1.2 Degree sequences 8
Exercises 17
1.3 Analysis 18
Exercises 21
1.4 Notes 22
2 Paths and Walks 23
2.1 Introduction 23
2.2 Complexity 27
Exercises 27
2.3 Walks 28
Exercises 29
2.4 The shortest-path problem 30
2.5 Weighted graphs and Dijkstra’s algorithm 33
Exercises 36
2.6 Data structures 37
2.7 Floyd’s algorithm 42
Exercises 44
2.8 Notes 45
3 Some Special Classes of Graphs 47
3.1 Bipartite graphs 47
Exercises 49
3.2 Line graphs 50
Exercises 51
3.3 Moore graphs 52
Exercises 57
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3.4 Euler tours 58
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3.4.1 An Euler tour algorithm 59
Exercises 62
3.5 Notes 62
4 Trees and Cycles 63
4.1 Introduction 63
Exercises 64
4.2 Fundamental cycles 65
Exercises 65
4.3 Co-trees and bonds 67
Exercises 69
4.4 Spanning tree algorithms 70
4.4.1 Prim’s algorithm 72
Data structures 74
Exercises 75
4.4.2 Kruskal’s algorithm 76
Data structures and complexity 77
4.4.3 The Cheriton-Tarjan algorithm 78
Exercises 79
4.4.4 Leftist binary trees 79
Exercises 86
4.5 Notes 87
5 The Structure of Trees 89
5.1 Introduction 89
5.2 Non-rooted trees 90
Exercises 92
5.3 Read’s tree encoding algorithm 92
5.3.1 The decoding algorithm 95
Exercises 96
5.4 Generating rooted trees 97
Exercises 105
5.5 Generating non-rooted trees 105
Exercises 106
5.6 Prüfer sequences 106
5.7 Spanning trees 109
5.8 The matrix-tree theorem 111
Exercises 116
5.9 Notes 117
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6 Connectivity 119
6.1 Introduction 119
Exercises 121
6.2 Blocks 122
6.3 Finding the blocks of a graph 125
Exercises 128
6.4 The depth-first search 128
6.4.1 Complexity 135
Exercises 136
6.5 Notes 137
7 Alternating Paths and Matchings 139
7.1 Introduction 139
Exercises 143
7.2 The Hungarian algorithm 143
7.2.1 Complexity 147
Exercises 148
7.3 Perfect matchings and 1-factorizations 148
Exercises 151
7.4 The subgraph problem 152
7.5 Coverings in bipartite graphs 154
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7.6 Tutte’s theorem 155
Exercises 158
7.7 Notes 158
8 Network Flows 161
8.1 Introduction 161
8.2 The Ford-Fulkerson algorithm 165
Exercises 175
8.3 Matchings and flows 176
Exercises 177
8.4 Menger’s theorems 178
Exercises 180
8.5 Disjoint paths and separating sets 180
Exercises 183
8.6 Notes 185
9 Hamilton Cycles 187
9.1 Introduction 187
Exercises 190
9.2 The crossover algorithm 191
9.2.1 Complexity 193
Exercises 196
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9.3 The Hamilton closure 197
Exercises 199
9.4 The extended multi-path algorithm 200
9.4.1 Data structures for the segments 204
Exercises 204
9.5 Decision problems, NP-completeness 205
Exercises 213
9.6 The traveling salesman problem 214
Exercises 216
9.7 The ΔTSP 216
9.8 Christofides’ algorithm 218
Exercises 220
9.9 Notes 221
10 Digraphs 223
10.1 Introduction 223
10.2 Activity graphs, critical paths 223
10.3 Topological order 225
Exercises 228
10.4 Strong components 229
Exercises 230
10.4.1 An application to fabrics 236
Exercises 236
10.5 Tournaments 238
Exercises 240
10.6 2-Satisfiability 240
Exercises 243
10.7 Notes 243
11 Graph Colorings 245
11.1 Introduction 245
11.1.1 Intersecting lines in the plane 247
Exercises 248
11.2 Cliques 249
11.3 Mycielski’s construction 253
11.4 Critical graphs 254
Exercises 255
11.5 Chromatic polynomials 256
Exercises 258
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11.6 Edge colorings 258
11.6.1 Complexity 268
Exercises 269
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11.7 NP-completeness 269
11.8 Notes 274
12 Planar Graphs 275
12.1 Introduction 275
12.2 Jordan curves 276
12.3 Graph minors, subdivisions 277
Exercises 282
12.4 Euler’s formula 282
12.5 Rotation systems 284
12.6 Dual graphs 286
12.7 Platonic solids, polyhedra 290
Exercises 291
12.8 Triangulations 292
12.9 The sphere 295
12.10 Whitney’s theorem 297
12.11 Medial digraphs 300
Exercises 301
12.12 The 4-color problem 301
Exercises 305
12.13 Straight-line drawings 305
12.14 Kuratowski’s theorem 309
Exercises 312
12.15 The Hopcroft-Tarjan algorithm 312
12.15.1 Bundles 316
12.15.2 Switching bundles 318
12.15.3 The general Hopcroft-Tarjan algorithm 321
12.16 Notes 325
13 Graphs and Surfaces 327
13.1 Introduction 327
13.2 Surfaces 329
13.2.1 Handles and crosscaps 336
13.2.2 The Euler characteristic and genus of a surface 337
Exercises 340
13.3 Graph embeddings, obstructions 341
13.4 Graphs on the torus 342
Exercises 349
13.4.1 Platonic maps on the torus 349
13.4.2 Drawing torus maps, triangulations 352
Exercises 357
13.5 Graphs on the projective plane 357
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13.5.1 The facewidth 364
13.5.2 Double covers 368
Exercises 370
13.6 Embedding algorithms 372
Exercises 381
13.7 Heawood’s map coloring theorem 382
Exercises 384
13.8 Notes 385
14 Linear Programming 387
14.1 Introduction 387
14.1.1 A simple example 387
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14.1.2 Simple graphical example 389
14.1.3 Slack and surplus variables 391
Exercises 394
14.2 The simplex algorithm 395
14.2.1 Overview 395
14.2.2 Some notation 395
14.2.3 Phase 0: finding a basis solution 396
14.2.4 Obtaining a basis feasible solution 397
14.2.5 The tableau 398
14.2.6 Phase 2: improving a basis feasible solution 399
14.2.7 Unbounded solutions 403
14.2.8 Conditions for optimality 405
14.2.9 Phase 1: initial basis feasible solution 407
14.2.10 An example 411
14.3 Cycling 413
Exercises 415
14.4 Notes 416
15 The Primal-Dual Algorithm 417
15.1 Introduction 417
15.2 Alternate form of the primal and its dual 423
15.3 Geometric interpretation 424
15.3.1 Example 424
15.4 Complementary slackness 429
15.5 The dual of the shortest-path problem 430
Exercises 433
15.6 The primal-dual algorithm 434
15.6.1 Initial feasible solution 437
15.6.2 The shortest-path problem 440
15.6.3 Maximum flow 444
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Exercises 446
15.7 Notes 446
16 Discrete Linear Programming 449
16.1 Introduction 449
16.2 Backtracking 450
16.3 Branch and bound 453
Exercises 463
16.4 Unimodular matrices 465
Exercises 467
16.5 Notes 468

Bibliography 469
Index 477
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Page 1
1
Graphs and Their Complements
1.1 Introduction
The diagram in Figure 1.1 illustrates a graph. It is called the graph of the cube. The edges of the geometric
cube correspond to the line segments connecting the nodes in the graph, and the nodes correspond to the
corners of the cube where the edges meet. They are the vertices of the cube.
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FIGURE 1.1
The graph of a cube
This diagram is drawn so as to resemble a cube, but if we were to rearrange it, as in Figure 1.2, it would still
be the graph of the cube, although it would no longer look like a cube. Thus, a graph is a graphical
representation of a relation in which edges connect pairs of vertices.
DEFINITION 1.1: A simple graph G consists of a vertex set V(G) and an edge set E(G), where each edge is a
pair {u,υ} of vertices u,
We denote the set of all pairs of a set V by Then In
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FIGURE 1.2
The graph of the cube
the example of the cube, V(G)={1, 2, 3, 4, 5, 6, 7, 8}, and E(G)={12, 23, 34, 14, 15, 26, 37, 48, 56, 67, 78,
58}, where we have used the shorthand notation uυ to stand for the pair {u,υ}. If u, then
means that u is joined to υ by an edge. We say that u and υ are adjacent. We use this notation to
remind us of the linked list data structure that we will use to store a graph in the computer. Similarly,
means that u is not joined to υ. We can also express these relations by writing or
respectively. Note that in a simple graph if then If u is adjacent to each of u1, u2,…, uk,
then we write
These graphs are called simple graphs because each pair u, υ of vertices is joined by at most one edge.
Sometimes we need to allow several edges to join the same pair of vertices. Such a graph is also called a
multigraph. An edge can then no longer be defined as a pair of vertices, (or the multiple edges would not be
distinct), but to each edge there still corresponds a pair {u,υ}. We can express this formally by saying that a
graph G consists of a vertex set V(G), an edge set E(G), and a correspondence Given an
edge ψ(e) is a pair {u,υ} which are the endpoints of e. Different edges can then have the same
endpoints. We shall use simple graphs most of the time, which is why we prefer the simpler definition, but
many of the theorems and techniques will apply to multigraphs as well.
This definition can be further extended to graphs with loops as well. A loop is an edge in which both
endpoints are equal. We can include this in the general definition of a graph by making the mapping
An edge for which defines a loop. Figure 1.2 shows
a graph with multiple edges and loops. However, we shall use simple graphs most of the time, so that an
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edge will be considered to be a pair of vertices.
The number of vertices of a graph G is denoted |G|. It is called the order of G.
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FIGURE 1.3
A multigraph
The number of edges is ε(G). If G is simple, then obviously since We shall
often use node or point as synonyms for vertex.
Many graphs have special names. The complete graph Kn is a simple graph with |Kn|=n and The
empty graph is a graph with and ε=0. is the complement of Kn.
FIGURE 1.4
The complete graph K5
DEFINITION 1.2: Let G be a simple graph. The complement of G is where and
consists of all those pairs uυ which are not edges of G. Thus,
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if and only if Figure 1.5 show a graph and its complement.
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