Distance domination in graphs with given minimum and maximum degree

Abstract

For an integer $k \ge 1$, 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 $\gamma _k(G)$ 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 $k \ge 2$, if G is a connected graph with minimum degree $\delta \ge 2$ and maximum degree $\Delta$ and of order $n \ge \Delta + k - 1$, then $\gamma _k(G) \le \frac{n + \delta - \Delta }{\delta + k - 1}$. This result improves existing known results.