A lower bound for the distance k-domination number of trees

Abstract

A subset D of vertices of a graph G = (V, E) is a distance k-dominating set for G if the distance between every vertex of V − D and D is at most k. The minimum size of a distance k-dominating set of G is called the distance k-domination number γk(G) of G. In this paper we prove that (2k + 1)γk(T) ≥ ¦V¦ + 2k − kn1 for each tree T = (V, E) with n1 leafs, and we characterize the class of trees that satisfy the equality (2k + 1)γk(T) = ¦V¦ + 2k − kn1. Our results generalize those of Lemanska [4] for k = 1 and of Cyman, Lemanska and Raczek [1] for k = 2.