Graphs Theory and Applications

Introduction 17 Chapter 1. Basic Concepts 21 1.1 The origin of the graph concept 21 1.2 Definition of graphs 24 1.3 Subgraphs 28 1.4 Paths and cycles 29 1.5 Degrees 33 1.6 Connectedness 35 1.7 Bipartite graphs 36 1.8 Algorithmic aspects 37 1.9 Exercises 41 Chapter 2. Trees 45 2.1 Definitions and properties 45 2.2 Spanning trees 49 2.3 Application: minimum spanning tree problem 54 2.4 Connectivity 59 2.5 Exercises 66 Chapter 3. Colorings 71 3.1 Coloring problems 71 3.2 Edge coloring 71 3.3 Algorithmic aspects 73 3.4 The timetabling problem 75 3.5 Exercises 81 Chapter 4. Directed Graphs 83 4.1 Definitions and basic concepts 83 4.2 Acyclic digraphs 90 4.3 Arborescences 92 4.4 Exercises 95 Chapter 5. Search Algorithms 97 5.1 Depth-first search of an arborescence 97 5.2 Optimization of a sequence of decisions 103 5.3 Depth-first search of a digraph 109 5.4 Exercises 117 Chapter 6. Optimal Paths 119 6.1 Distances and shortest paths problems 119 6.2 Case of non-weighted digraphs: breadth-first search 120 6.3 Digraphs without circuits 125 6.4 Application to scheduling 128 6.5 Positive lengths 134 6.6 Other cases 142 6.7 Exercises 143 Chapter 7. Matchings 149 7.1 Matchings and alternating paths 149 7.2 Matchings in bipartite graphs 152 7.3 Assignment problem 156 7.4 Optimal assignment problem 164 7.5 Exercises 171 Chapter 8. Flows 173 8.1 Flows in transportation networks 173 8.2 The max-flow min-cut theorem 177 8.3 Maximum flow algorithm 180 8.4 Flow with stocks and demands 188 8.5 Revisiting theorems 191 8.6 Exercises 194 Chapter 9. Euler Tours 197 9.1 Euler trails and tours 197 9.2 Algorithms 201 9.3 The Chinese postman problem 207 9.4 Exercises 212 Chapter 10. Hamilton Cycles 215 10.1 Hamilton cycles 215 10.2 The traveling salesman problem 218 10.3 Approximation of a difficult problem 220 10.4 Approximation of themetric TSP 223 10.5 Exercises 234 Chapter 11. Planar Representations 237 11.1 Planar graphs 237 11.2 Other graph representations 242 11.3 Exercises 244 Chapter 12. Problems with Comments 247 12.1 Problem 1: A proof of k-connectivity 247 12.2 Problem2: An application to compiler theory 249 12.3 Problem3: Kernel of a digraph 251 12.4 Problem 4: Perfect matching in a regular bipartite graph 253 12.5 Problem5: Birkhoff-Von Neumann’s theorem 254 12.6 Problem 6: Matchings and tilings 256 12.7 Problem7: Strip mining 258 Appendix A. Expression of Algorithms 261 Appendix B. Bases of Complexity Theory 267 Bibliography 277 Index 279