Dominating sets with at most k components

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.