Bounds on an Independent Distance Domination Parameter

Let \(m \geq 1\) be an integer and let \(G\) be a graph of order \(n\). A set \(\mathcal{D}\) of vertices of \(G\) is an \(m\)-dominating set of \(G\) if every vertex of \(V(G) – \mathcal{D}\) is within distance \(m\) from some vertex of \(\mathcal{D}\). An independent set of vertices of \(G\) is a set of vertices of \(G$ whose elements are pairwise nonadjacent. The minimum cardinality among all independent \(m\)-dominating sets of \(G\) is called the independent \(m\)-domination number and is denoted by \(id(m,G)\). We show that if \(G\) is a connected graph of order \(n \geq m + 1\), then \(id(m,G) \leq ({n+m+1-2\sqrt{n}/{m}}),\) and this bound is sharp.