Equitable and outdegree equitable domination number of graphs

Abstract

Let G = (V, E) be a simple graph. A subset D of V is said to be a dominating set of G, if each vertex in V is either in D or has a neighbour in D. A subset D of V is said to be an equitable dominating set of G, if for every v ? V ? D, there exists a vertex u ? D such that uv ? E(G) and |deg(u) ? deg(v)| ? 1. The minimum cardinality of an equitable dominating set of G, denoted by ??(G), is called the equitable domination number of G. The edges from a vertex u ? D to V ? D are called the dominating edges of u from D. A dominating set D is called an outdegree equitable dominating set if the difference between the cardinalities of the sets of dominating edges from any two points of D is atmost one. The minimum cardinality of an outdegree equitable dominating set of G, denoted by ?oe(G) is called the outdegree equitable domination number of G. In this paper we study equitable domination number and outdegree equitable domination number of some graphs.