Total domination number of products of two directed cycles

Abstract

Let D = (V, A) be a digraph of order n. A subset S of the vertex set V(D) is a total dominating set of D if for each vertex vεD there exists a vertex uεS such that (u, v) is an arc of D. The total domination number of D, γ_t(D), is the cardinality of the smallest total dominating set of D. In this paper we calculate the total domination number of the cardinal product and cartesian product of two directed cycles C_m and C_n for some values of m and arbitrary n.