Bound of Distance Domination Number of Graph and Edge Comb Product Graph

Abstract

Let G = (V, E) be a simple, nontrivial, finite, connected and undirected graph. For an integer 1 ≤ k ≤ diam(G), 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)\S is at distance at most k from some vertex of S. The k-domination number, denoted by γk(G), of G is the minimum cardinality of a k-dominating set of G. In this paper, we improve the lower bound on the distance domination number of G regarding to the diameter and minimum degree as well as the upper bound regarding to the order and minimum k distance neighbourhood. In addition, we determine the bound of distance domination number of edge comb product graph.