K-domination number of products of two directed cycles and two directed paths

Abstract

Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A subset S of the vertex set V(D) is a k-dominating set (k ≥ 1) of D if for each vertex v not in S there exists k vertices {ui, ..., uk} ⊆ S such that (ui, v) is an arc of D for i = 1, ..., k. The k-domination number of D, k(D), is the cardinality of the smallest k-dominating set of D. The k-domination number (k ≥ 2) of the Cartesian products of two directed cycles, two directed paths and Cartesian products of a directed path and a cycle are determined. Also, we give k-domination number (k ≥ 2) of the direct product of two directed cycles and two directed paths.