Dominating Sets With at Most \(k\) Components

Abstract

A connected dominating set \(D\) of a graph \(G\) has the property that not only does \(D\) dominate the graph but the subgraph induced by the vertices of \(D\) is also connected. We generalize this concept by allowing the subgraph induced by \(D\) to contain at most \(k\) components and examine the minimum possible order of such a set. In the case of trees, we provide lower and upper bounds and a characterization for those trees which achieve the former.