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Introduction to Graph and Hypergraph Theory
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Authors: Vitaly I. Voloshin ( Dept. of Mathematics, Physics and Computer Science, Troy Univ., Troy, AL) 
Book Description:
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This book is for math and computer science majors, for students and representatives of many other disciplines (like bioinformatics, for example) taking courses in graph theory, discrete mathematics, data structures, algorithms. It is also for anyone who wants to understand the basics of graph theory, or just is curious. No previous knowledge in graph theory or any other significant mathematics is required. The very basic facts from set theory, proof techniques and algorithms are sufficient to understand it; but even those are explained in the text. Structurally, the text is divided into two parts where Part II is the generalization of Part I. The first part discusses the key concepts of graph theory
with emphasis on trees, bipartite graphs, cycles, chordal graphs, planar graphs and graph coloring. The second part considers generalizations of Part I and discusses hypertrees, bipartite hypergraphs, hypercycles, chordal hypergraphs, planar hypergraphs and hypergraph coloring. There is an interaction between the parts and within the parts to show how ideas of generalizations work. The main point is to exhibit the ways of generalizations and interactions of mathematical concepts from the very simple to the most advanced. One of the features of this text is the duality of hypergraphs. This fundamental concept is missing in graph theory (and in its introductory teaching) because dual graphs are not properly graphs, they are hypergraphs. However, as Part II shows, the duality is a very powerful tool in understanding, simplifying and unifying many combinatorial relations; it is basically a look at the same structure from the opposite (vertices versus edges) point of view.

Table of Contents:
Preface pp. i-xiv

I. Graphs pp.1-4

1. Basic Definitions and Concepts pp.5-5

1.1 Fundamentals pp.5-7
1.2 Graph modeling applications pp.8-11
1.3 Graph representations pp..12-14
1.4 Generalizations pp.15-17
1.5 Basic graph classes pp.18-24
1.6 Basic graph operations pp.25-28
1.7 Basic subgraphs pp.29-33
1.8 Separation and connectivity pp.34-38

2. Trees and Bipartite Graphs pp.39-39
2.1 Trees and cyclomatic number pp.39-40
2.2 Trees and distance pp.41-42
2.3 Minimum spanning tree pp.43-44
2.4 Bipartite graphs pp.45-50

3. Chordal Graphs pp.51-51
3.1 Preliminary pp.51-51
3.2 Separators and simplicial vertices pp.52-56
3.3 Degrees pp.57-58
3.4 Distances in chordal graphs pp.59-61
3.5 Quasi-triangulated graphs pp.62-66

4. Planar Graphs pp.67-67
4.1 Plane and planar graphs pp.67-68
4.2 Euler’s formula pp.69-70
4.3 K5 and K3 3 are not planar graphs pp.71-72
4.4 Kuratowski’s theorem and planarity testing pp.73-75
4.5 Plane triangulations and dual graphs pp.76-78
5. Graph Coloring pp.79-79
5.1 Preliminary pp.79-79
5.2 Definitions and examples pp.80-82
5.3 Structure of colorings pp.83-88
5.4 Chromatic polynomial pp.89-94
5.5 Coloring chordal graphs pp.95-101
5.6 Coloring planar graphs pp.102-107
5.7 Perfect graphs pp.108-111
5.8 Edge coloring and Vizing’s theorem pp.112-115
5.9 Upper chromatic index pp.116-122

6. Traversals and Flows pp.123-123
6.1 Eulerian graphs pp.123-124
6.2 Hamiltonian graphs pp.125-126
6.3 Network flows pp.127-130

II. Hypergraphs pp.131-134

7. Basic Hypergraph Concepts pp.135-135
7.1 Preliminary definitions pp.135-138
7.2 Incidence and duality pp.139-143
7.3 Basic hypergraph classes pp.144-145
7.4 Basic hypergraph operations pp.146-150
7.5 Subhypergraphs pp.151-153
7.6 Conformality and Helly property pp.154-160

8. Hypertrees and Chordal Hypergraphs pp.161-161
8.1 Hypertrees and chordal conformal hypergraphs pp.161-167
8.2 Algorithms on hypertrees pp.168-173
8.3 Cyclomatic number of a hypergraph pp.174-180

9. Some Other Remarkable Hypergraph Classes pp.181-181
9.1 Balanced hypergraphs pp.181-182
9.2 Interval hypergraphs pp.183-184
9.3 Normal hypergraphs pp.185-186
9.4 Planar hypergraphs pp.187-192

10. Hypergraph Coloring pp.193-193
10.1 Basic kinds of classic hypergraph coloring pp.193-196
10.2 Greedy algorithm for the lower chromatic number pp.197-200
10.3 Basic definitions of mixed hypergraph coloring pp.201-206
10.4 Greedy algorithm for the upper chromatic number pp.207-212
10.5 Splitting-contraction algorithm pp.213-218
10.6 Uncolorability pp.219-226
10.7 Unique colorability pp.227-235
10.8 Perfection pp.236-243
10.9 Chromatic spectrum pp.244-253
10.10 Coloring planar hypergraphs pp.254-262

11. Modeling with Hypergraphs pp.263-263
11.1 List colorings without lists pp.263-263
11.2 Resource allocation pp.264-266

12 Appendix pp.267-267
12.1 What is mathematical induction pp.267-268
12.2 Graph Theory algorithms and their complexity pp.269-269
12.3 Answers and hints to selected exercises pp.270-274
12.4 Glossary of additional concepts pp.275-278

References. pp.279-280

Index. pp.281-287

   Binding: Hardcover
   Pub. Date: 2012
   Pages: 301
   ISBN: 978-1-60692-372-6
   Status: AV
Status Code Description
AN Announcing
FM Formatting
PP Page Proofs
FP Final Production
EP Editorial Production
PR At Prepress
AP At Press
AV Available
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Introduction to Graph and Hypergraph Theory