On the [formula omitted]-domination numbers of iterated line digraphs

Abstract

An (h,k)-dominating set in a digraph G is a subset D of V(G) such that the subdigraph induced by D is h-connected and for every vertex v of G, v is in-dominated and out-dominated by at least k vertices in D. The (h,k)-domination number γh,k(G) of G is the minimum cardinality of an (h,k)-dominating set in G. An (h,k)-dominating set finds applications to fault-tolerant location problems of resources in communication networks and fault-tolerant virtual backbone in wireless networks.Let G be a connected d-regular digraph and 1≤k<d. Let Lm(G) denote the m-iterated line digraph of G. In this note, we show that γh,k(Lm(G))=kdm−1|V(G)| for all m≥2 and 0≤h≤min{k,⌊d2⌋}. From our results, the (h,k)-domination numbers of d-ary (generalized) de Bruijn and Kautz digraphs are determined for 0≤h≤min{k,⌊d2⌋}, which strengthen the previously known results on (generalized) de Bruijn and Kautz digraphs.