Twin domination in digraphs

A vertex v in a digraph D out-dominates itself as well as all vertices u such that (v,u) is an arc of D; while v in-dominates both itself and all vertices w such that (w,v) is an arc of D. A set 5 of vertices of D is a twin dominating set of D if every vertex of D is out-dominated by some vertex of S and in-dominated by some vertex of S. The minimum cardinality of a twin dominating set is the twin domination number γ*(D) of D. It is shown that γ*(D) ≤ 2p/3 for every digraph D of order p having no vertex of in-degree 0 or out-degree 0. Moreover, we give a Nordhaus-Gaddum type bound for γ*, and for transitive digraphs we give a sharp upper bound for the twin domination number in terms of order and minimum degree. For a graph G, the upper orientable twin domination number DOM* (G) is the maximum twin domination number γ*(D) over all orientations D of G; while the lower orientable twin domination number dom*(G) of G is the minimum such twin domination number. It is shown that for each graph G and integer c with dom*(G) ≤ c ≤ DOM*(G), there exists an orientation D of G such that γ*(D) = c.