Hypergraph Theory: An Introduction

This book provides an introduction to hypergraphs, its aim being to overcome the lack of recent manuscripts on this theory. In the literature hypergraphs have many other names such as set systems and families of sets. This work presents the theory of hypergraphs in its most original aspects, while also introducing and assessing the latest concepts on hypergraphs. The variety of topics, their originality and novelty are intended to help readers better understand the hypergraphs in all their diversity in order to perceive their value and power as mathematical tools. This book will be a great asset to upper-level undergraduate and graduate students in computer science and mathematics. It has been the subject of an annual Master's course for many years, making it also ideally suited to Master's students in computer science, mathematics, bioinformatics, engineering, chemistry, and many other fields. It will also benefit scientists, engineers and anyone else who wants to understand hypergraphs theory. Table of Contents Cover Hypergraph Theory - An Introduction ISBN 9783319000794 ISBN 9783319000800 Preface Acknowledgments Contents 1 Hypergraphs: Basic Concepts 1.1 First Definitions 1.2 Example of Hypergraph 1.2.1 Simple Reduction Hypergraph Algorithm 1.3 Algebraic Definitions for Hypergraphs 1.3.1 Matrices, Hypergraphs and Entropy 1.3.2 Similarity and Metric on Hypergraphs 1.3.3 Hypergraph Morphism; Groups and Symmetries 1.4 Generalization of Hypergraphs References 2 Hypergraphs: First Properties 2.1 Graphs versus Hypergraphs 2.1.1 Graphs 2.1.2 Graphs and Hypergraphs 2.2 Intersecting Families, Helly Property 2.2.1 Intersecting Families 2.2.2 Helly Property 2.3 Subtree Hypergraphs 2.4 Conformal Hypergraphs 2.5 Stable (or Independent), Transversal and Matching 2.5.1 Examples: 2.6 K�nig Property and Dual K�nig Property 2.7 linear Spaces References 3 Hypergraph Colorings 3.1 Coloring 3.2 Particular Colorings 3.2.1 Strong Coloring 3.2.2 Equitable Coloring 3.2.3 Good Coloring 3.2.4 Uniform Coloring 3.2.5 Hyperedge Coloring 3.2.6 Bicolorable Hypergraphs 3.3 Graph and Hypergraph Coloring Algorithm References 4 Some Particular Hypergraphs 4.1 Interval Hypergraphs 4.2 Unimodular Hypergraphs 4.2.1 Unimodular Hypergraphs and Discrepancy of Hypergraphs 4.3 Balanced Hypergraphs 4.4 Normal Hypergraphs 4.5 Arboreal Hypergraphs, Acyclicity and Hypertree Decomposition 4.5.1 Acyclic Hypergraph 4.5.2 Arboreal and Co-Arboreal Hypergraphs 4.5.3 Tree and Hypertree Decomposition 4.6 Planar Hypergraphs References 5 Reduction-Contraction of Hypergraph 5.1 Introduction 5.2 Reduction Algorithms 5.2.1 A Generic Algorithm 5.2.2 A Minimum Spanning Tree Algorithm (HR-MST) References 6 Dirhypergraphs: Basic Concepts 6.1 Basic Definitions 6.2 Basic Properties of Directed Hypergraphs 6.3 Hypercycles in a Dirhypergraph 6.4 Algebraic Representation of Dirhypergraphs 6.4.1 Dirhypergraphs Isomorphism 6.4.2 Algebraic Representation: Definition 6.4.3 Algebraic Representation Isomorphism References 7 Applications of Hypergraph Theory: A Brief Overview 7.1 Hypergraph Theory and System Modeling for Engineering 7.1.1 Chemical Hypergraph Theory 7.1.2 Hypergraph Theory for Telecomunmications 7.1.3 Hypergraph Theory and Parallel Data Structures 7.1.4 Hypergraphs and Constraint Satisfaction Problems 7.1.5 Hypergraphs and Database Schemes 7.1.6 Hypergraphs and Image Processing 7.1.7 Other Applications References Index