A Characterization of Block graphs that are Well-\(k\)-Dominated

Abstract

Let \(k \geq 1\) 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.