Distance domination in graphs with given minimum and maximum degree

For an integer k≥ 1 , a distance k-dominating set of a connected graph G is a set S of vertices of G such that every vertex of V(G) is at distance at most k from some vertex of S. The distance k-domination number γ_k(G) of G is the minimum cardinality of a distance k-dominating set of G. In this paper, we establish an upper bound on the distance k-domination number of a graph in terms of its order, minimum degree and maximum degree. We prove that for k≥ 2 , if G is a connected graph with minimum degree δ≥ 2 and maximum degree Δ and of order n≥ Δ + k- 1 , then γk(G)≤n+δ-Δδ+k-1. This result improves existing known results.