On the vertex cover P_3 problem parameterized by treewidth

Consider a graph G . A subset of vertices, F , is called a vertex cover P_t ( VCP_t ) set if every path of order t contains at least one vertex in F . Finding a minimum VCP_t set in a graph is is NP-hard for any integer t≥ 2 and is called the MVCP_3 problem. In this paper, we study the parameterized algorithms for the MVCP_3 problem when the underlying graph G is parameterized by the treewidth. Given an n -vertex graph together with its tree decomposition of width at most p , we present an algorithm running in time 4^p· n^O(1) for the MVCP_3 problem. Moreover, we show that for the MVCP_3 problem on planar graphs, there is a subexponential parameterized algorithm running in time 2^O(√(k))· n^O(1) where k is the size of the optimal solution.