The k k -tuple twin domination in generalized de Bruijn and Kautz networks ?
Given a digraph (network) G=(V,A) G = ( V , A ) , a vertex u u in G G is said to out-dominate itself and all vertices v v such that the arc (u,v)?A ( u , v ) ? A ; similarly, u u in-dominates both itself and all vertices w w such that the arc (w,u)?A ( w , u ) ? A . A set D D of vertices of G G is a k k -tuple twin dominating set  if every vertex of G G is out-dominated and in-dominated by at least k k vertices in D D , respectively. The k k -tuple twin domination problem is to determine a minimum k k -tuple twin dominating set for a digraph. In this paper we investigate the k k -tuple twin domination problem in generalized de Bruijn networks GB(n,d) G B ( n , d ) and generalized Kautz GK(n,d) G K ( n , d ) networks when d d divides n n . We provide construction methods for constructing minimum k k -tuple twin dominating sets in these networks. These results generalize previous results given by Araki [T. Araki, The k k -tuple twin domination in de Bruijn and Kautz digraphs, Discrete Mathematics 308 (2008) 64066413] for de Bruijn and Kautz networks.
generalized de bruijn network k k -tuple twin domination k k k k k generalized kautz network interconnection network