The kk-tuple twin domination in generalized de Bruijn and Kautz networks

Given a digraph (network) G=(V,A)G=(V,A), a vertex uu in GG is said to out-dominate itself and all vertices vv such that the arc (u,v)∈A(u,v)∈A; similarly, uu in-dominates both itself and all vertices ww such that the arc (w,u)∈A(w,u)∈A. A set DD of vertices of GG is a kk-tuple twin dominating set   if every vertex of GG is out-dominated and in-dominated by at least kk vertices in DD, respectively. The kk-tuple twin domination problem is to determine a minimum kk-tuple twin dominating set for a digraph. In this paper we investigate the kk-tuple twin domination problem in generalized de Bruijn networks GB(n,d)GB(n,d) and generalized Kautz GK(n,d)GK(n,d) networks when dd divides nn. We provide construction methods for constructing minimum kk-tuple twin dominating sets in these networks. These results generalize previous results given by Araki [T. Araki, The kk-tuple twin domination in de Bruijn and Kautz digraphs, Discrete Mathematics 308 (2008) 6406–6413] for de Bruijn and Kautz networks.