Distance domination in graphs with given minimum and maximum degree

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 ${\mathit{\gamma }}_k(G)$ 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 $\mathit{\delta }\ge 2$ and maximum degree $\mathrm{\Delta }$ and of order $n\ge \mathrm{\Delta }+k‐1$ , then ${\mathit{\gamma }}_k(G)\le \frac{n+\mathit{\delta }‐\mathrm{\Delta }}{\mathit{\delta }+k‐1}$ . This result improves existing known results.