The kk-tuple twin domination in de Bruijn and Kautz digraphs

A vertex uu in a digraph GG out-dominates itself and all vertices vv such that (u,v)(u,v) is an arc of GG, similarly, uu in-dominates both itself and all vertices ww such that (w,u)(w,u) is an arc of GG. A set DD of vertices of GG is a twin dominating set of GG if every vertex of GG is out-dominated by a vertex of DD and in-dominated by a vertex in DD. In this paper, we introduce the kk-tuple twin domination in directed graphs. A set DD of vertices of GG is a kk-tuple twin dominating set if every vertex of GG is out-dominated by at least kk vertices in DD and in-dominated by at least kk vertices in DD. We consider the problem of the kk-tuple twin domination in de Bruijn and Kautz digraphs, and give construction methods for constructing minimum kk-tuple twin dominating sets in these digraphs.