On the sum of the total domination numbers of a digraph and its converse

Abstract

A vertex subset S of a digraph D is called a dominating set of D if every vertex not in S has an in-neighbor in S. A dominating set S of D is called a total dominating set of D if the subdigraph induced by S has no isolated vertices. The total domination number of D, denoted by γt(D), is the minimum cardinality of a total dominating set of D. We show that for any connected digraph D of order n≥3, γt(D)+γt(D⁻ )≤5n/3, where D− is the converse of D. Furthermore, we characterize the oriented trees for which the equality holds.