The k-hop connected dominating set problem: hardness and polyhedra

Let G=(V,E) be a connected graph, and k a positive integer. A subset D⊆V is a k-hop connected dominating set (k-CDS) if the subgraph of G induced by D is connected and, for every vertex v in G, there is a vertex u in D such that the distance between v and u is at most k. We study the problem of finding a minimum k-hop connected dominating set, denoted by the acronym Mink-CDS. Firstly, we prove that Mink-CDS is NP-hard on planar bipartite graphs of maximum degree 4 and on planar biconnected graphs of maximum degree 5. We present an inapproximability threshold for Mink-CDS on bipartite and on (1, 2)-split graphs, and we also prove that Mink-CDS is APX-hard on bipartite graphs of maximum degree 4. These results are shown to hold for every positive integer k. For k=1, the classical minimum connected dominating set problem, we present an integer linear programming formulation and show some classes of inequalities that define facets of the corresponding polytope. We also present an approximation algorithm for this case.