### A bound on the $k$-domination number of a graph

Let $G$ be a graph with vertex set $V\left(G\right)$, and let $k\ge 1$ be an integer. A subset $D\subseteq V\left(G\right)$ is called a $k$-dominating set if every vertex $v\in V\left(G\right)-D$ has at least $k$ neighbors in $D$. The $k$-domination number ${\gamma}_{k}\left(G\right)$ of $G$ is the minimum cardinality of a $k$-dominating set in $G$. If $G$ is a graph with minimum degree $\delta \left(G\right)\ge k+1$, then we prove that $${\gamma}_{k+1}\left(G\right)\le \frac{\left|V\left(G\right)\right|+{\gamma}_{k}\left(G\right)}{2}.$$ In addition, we present a characterization of a special class of graphs attaining equality in this inequality.