A Characterization of Block graphs that are Well-$k$-Dominated

Abstract

Let $k\ge 1$ be an integer and let $G$ be a graph. A set $D$ of vertices of $G$ is a $k$-dominating set if every vertex of $V(G)–D$ is within distance $k$ of some vertex of $D$. The graph $G$ is called well-$k$-dominated if every minimal $k$-dominating set of $G$ is of the same cardinality. A characterization of block graphs that are well-$k$-dominated is presented, where a block graph is a graph in which each of its blocks is complete.