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The Theory of Graphs


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Table of Contents

Introduction 1 General Definitions Sets and Multivalued Functions Paths and Circuits of a Graph Chains and Cycles of a Graph 2 Descendance Relations Weak Ordering Associated with a Graph Inductive Graphs and Bases 3 The Ordinal Function and the Grundy Function on an Infinite Graph General Remarks Concerning Infinite Graphs Ordinal Function Grundy Functions Operations over Graphs 4 The Fundamental Numbers of the Theory of Graphs Cyclomatic Number Chromatic Number Methods for Finding the Chromatic Number Coefficient of Internal Stability Coefficient of External Stability 5 Kernels of a Graph Existence and Uniqueness Theorems Application to Grundy Functions 6 Games on a Graph Nim Type Games The General Definition of a Game (with Perfect Information) Strategies 7 The Problem of the Shortest Route Stepwise Procedures Various Generalizations 8 Transport Networks The Problem of Maximum Flow The Problem of Minimum Flow Problems of Flow when a Set of Values is Associated with each Arc Infinite Transport Networks 9 The Theorem of the Demi-Degrees Inward or Outward Demi-Degree 10 Matching of a Simple Graph The Maximum Matching Problem Deficiency of a Simple Graph The Hungarian Method Extensions to the Infinite Case Application to the Theory of Matrices 11 Factors Hamiltonian Paths and Circuits Factors and Methods for Determining Them The Problem of Finding a Partial Graph with Given Demi-Degrees 12 Centres of a Graph Centrees Radius 13 The Diameter of a Strongly Connected Graph General Remarks on Strongly Connected Graphs without Loops Diameter 14 The Matrix Associated with a Graph The Use of Standard Matrix Operations Problems of Enumeration The Problem of the Leader The Use of Boolean Operations 15 Incidence Matrices Matrices with the Unimodular Property Systems with the Unimodular Property Cyclomatic Matrices 16 Trees and Arborescences Trees Analytic Treatment Arborescences 17 Euler's Problem Eulerian Cycles Eulerian Circuits 18 Matching in the General Case The Theory of Alternating Chains The Determination of a Partial Graph with Given Degrees Perfect Matching Application to the Coefficient of Internal Stability 19 Semi-Factors Hamiltonian Cycles and Semi-Factors A Necessary and Sufficient Condition for the Existence of a Semi-Factor 20 The Connectivity of a Graph Articulation Points Biconnected Graphs h-Connected Graphs 21 Planar Graphs General Properties Generalization Appendix 1 Note on the General Theory of Games Appendix 2 Note on Transport Problems Bibliography List of Symbols Index of Terms Used

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