GRAPHS, ALGORITHMS, AND OPTIMIZATION

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1 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 New York Washington, D.C. j ""

2 Contents 1 Graphs and Their Complements Introduction 1 Exercises Degree sequences 8 Exercises Analysis 18 Exercises Notes 22 2 Paths and Walks Introduction Complexity 27 Exercises Walks 28 Exercises The shortest-path problem Weighted graphs and Dijkstra's algorithm 33 Exercises Data structures Floyd's algorithm 42 Exercises Notes 45 3 Some Special Classes of Graphs Bipartite graphs 47 Exercises Line graphs 50 Exercises Moore graphs 52 Exercises 57

3 3.4 Euler tours An Euler tour algorithm 59 Exercises Notes 62 Trees and Cycles Introduction 63 Exercises Fundamental cycles 65 Exercises Co-trees and bonds 67 Exercises Spanning tree algorithms Prim's algorithm 72 Data structures 74 Exercises Kruskal's algorithm 76 Data structures and complexity The Cheriton-Tarjan algorithm 78 Exercises Leftist binary trees 79 Exercises Notes 87 The Structure of Trees Introduction Non-rooted trees 90 Exercises Read's tree encoding algorithm The decoding algorithm 95 Exercises Generating rooted trees 97 Exercises Generating non-rooted trees 105 Exercises Priifer sequences Spanning trees The matrix-tree theorem Ill Exercises Notes 117

4 Connectivity Introduction 119 Exercises Blocks Finding the blocks of a graph 125 Exercises The depth-first search Complexity 135 Exercises Notes 137 Alternating Paths and Matchings Introduction 139 Exercises The Hungarian algorithm Complexity 147 Exercises Perfect matchings and 1-factorizations 148 Exercises The subgraph problem Coverings in bipartite graphs Tutte's theorem 155 Exercises Notes 158 Network Flows Introduction The Ford-Fulkerson algorithm 165 Exercises Matchings and flows 176 Exercises Menger's theorems 178 Exercises Disjoint paths and separating sets 180 Exercises Notes 185 Hamilton Cycles Introduction 187 Exercises The crossover algorithm Complexity 193 Exercises 196

5 9.3 The Hamilton closure 197 Exercises The extended multi-path algorithm Data structures for the segments 204 Exercises Decision problems, NP-completeness 205 Exercises The traveling salesman problem 214 Exercises TheATSP Christofides' algorithm 218 Exercises Notes Digraphs Introduction Activity graphs, critical paths Topological order 225 Exercises Strong components 229 Exercises An application to fabrics 236 Exercises Tournaments 238 Exercises Satisfiability 240 Exercises Notes Graph Colorings Introduction Intersecting lines in the plane 247 Exercises Cliques Mycielski's construction Critical graphs 254 Exercises Chromatic polynomials 256 Exercises Edge colorings Complexity 268 Exercises 269

6 11.7 NP-completeness Notes Planar Graphs Introduction Jordan curves Graph minors, subdivisions 277 Exercises Euler's formula Rotation systems Dual graphs Platonic solids, polyhedra 290 Exercises Triangulations The sphere Whitney's theorem Medial digraphs 300 Exercises The 4-color problem 301 Exercises Straight-line drawings Kuratowski's theorem 309 Exercises The Hopcroft-Tarjan algorithm Bundles Switching bundles The general Hopcroft-Tarjan algorithm Notes Graphs and Surfaces Introduction Surfaces Handles and crosscaps The Euler characteristic and genus of a surface 337 Exercises Graph embeddings, obstructions Graphs on the torus 342 Exercises Platonic maps on the torus Drawing torus maps, triangulations 352 Exercises Graphs on the projective plane 357

7 Thefacewidth Double covers 368 Exercises Embedding algorithms 372 Exercises Heawood's map coloring theorem 382 Exercises Notes Linear Programming Introduction A simple example Simple graphical example Slack and surplus variables 391 Exercises The simplex algorithm Overview Some notation Phase 0: finding a basis solution Obtaining a basis feasible solution The tableau Phase 2: improving a basis feasible solution Unbounded solutions Conditions for optimality Phase 1: initial basis feasible solution An example Cycling 413 Exercises Notes The Primal-Dual Algorithm Introduction Alternate form of the primal and its dual Geometric interpretation Example Complementary slackness The dual of the shortest-path problem 430 Exercises The primal-dual algorithm Initial feasible solution The shortest-path problem Maximum flow 444

8 Exercises Notes Discrete Linear Programming Introduction Backtracking Branch and bound 453 Exercises Unimodular matrices 465 Exercises Notes 468 Bibliography 469 Index 477

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