This book provides a comprehensive and pedagogical introduction to graph theory and its applications. It contains all the standard basic material and develops significant topics and applications, such as: colorings and the timetabling problem, matchings and the optimal assignment problem, and Hamiltonian cycles and the travelling salesman problem, to name but a few. Exercises at various levels are given at the end of each chapter, and a final chapter presents a few general problems with hints for solutions, thus providing the reader with the opportunity to test and refine their knowledge on the subject. An appendix outlines the basis of computational complexity theory, in particular the definition of NP-completeness, which is essential for algorithmic 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 Neumanns 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
