Approximation Algorithm and FPT Algorithm for Connected-k-Subgraph Cover on Minor-Free Graphs

Given a graph G, the minimum Connected-k-Subgraph Cover problem (MinCkSC) is to find a minimum vertex subset C of G such that every connected subgraph of G on k vertices has at least one vertex in C. If furthermore the subgraph of G induced by C is connected, then the problem is denoted as MinCkSC(con). In this paper, we first present a PTAS for MinCkSC on an H-minor-free graph, where H is a graph with a constant number of vertices. Then, we design an O((omega + 1)(2(k - 1)(omega + 2))(3 omega +3))|V|-time FPT algorithm for MinCkSC(con) on a graph with treewidth omega, based on which we further design an O(2(O(root tlog t)|V|O(1))) time subexponential FPT algorithm for MinCkSC(con) on an H-minor-free graph, where t is an upper bound of solution size.