Bounds on the domination number in oriented graphs

Abstract

A dominating set of an oriented graph D is a set S of vertices of D such that every vertex not in S is a successor of some vertex of S. The minimum cardinality of a dominating set of D; denoted (D), is the domination number of D. An irredundant set of an oriented graph D is a set S of vertices of D such that every vertex of S has a private successor, that is, for all x 2 S; jO[x] O[S x]j 1. The irredundance number of an oriented graph, denoted ir(D), is the least number of vertices in a maximal irredundant set. We denote by 1 (D) and s(D); the number of edges in a maximum matching and support vertices of the underlyng graph of an oriented graph D; respectively. In this paper, we show that for every oriented graph D, s(D) ir(D) (D) 6 n(D) 1 (D). We also give characterizations of oriented trees satisfying (T ) = n(T ) 1 (T ) and oriented graphs satisfying (D) = s(D) and s(D) = n(D) 1 (D); respectively.