On the Minimum Hub Set Problem

Let G = (V, E) be a finite, simple, and undirected graph. A vertex subset H of V is a hub set of G if for any pair of nonadjacent vertices u, v ∈ V H, there is a u ~ v path P such that all internal vertices of P are in H. The minimum hub set problem on G is the problem of finding a hub set H of G such that |H| is the minimum among all possible hub sets of G. The minimum connected hub set problem concentrates on finding a minimum hub set H such that the subgraph induced by H is connected. In this paper, we show that the minimum (connected) hub set problem is NP-complete on split graphs and we show that it is polynomially solvable on bounded treewidth graphs. Further, we show that the minimum connected hub set problem is equivalent to the minimum connected dominating set problem on AT-free graphs.