Bounds on the domination number in oriented graphs

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 γ(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 ∈ S, |O[x]− O[S − x]| ≥ 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 β1(D) and s(D), the number of edges in a maximum matching and support vertices of the underlying graph of an oriented graphD, respectively. In this paper, we show that for every oriented graph D, s(D) ≤ ir(D) ≤ γ(D) n(D)−β1(D). We also give characterizations of oriented trees satisfying γ(T ) = n(T ) − β1(T ) and oriented graphs satisfying γ(D) = s(D) and s(D) = n(D) − β1(D), respectively.