Chapter 1 Graph theoretic foundations

Abstract

This chapter presents basic definitions and notations of graph theory. A function that is both injective and surjective is called a bijection. A permutation is simply a bijection from a set to itself. A binary relation R on X may satisfy one or more properties. Such a relation is said to be equivalence if it is reflexive, symmetric, and transitive. A binary relation is called a strict partial order if it is irreflexive and transitive. It is a simple exercise to show that a strict partial order will also be antisymmetric. A graph is defined as a set and a certain relation on that set. It is often convenient to draw a “picture” of the graph. This may be done in many ways. Usually one draws a circle for each vertex and connects vertex x and vertex y with a directed arrow whenever xy is an edge. If both xy and yx are edges, then sometimes a single line joins x and y without arrows.