An improved algorithm for the vertex cover P3 problem on graphs of bounded treewidth

Given a graph G = (V,E) and a positive integer t ≥ 2, the task in the vertex cover Pt (V CPt) problem is to find a minimum subset of vertices F ⊆ V such that every path of order t in G contains at least one vertex from F . The V CPt problem is NP-complete for any integer t ≥ 2. Recently, the authors presented a dynamic programming algorithm with runtime 4 · n that can solve the V CP3 problem in any n-vertex graph given together with its tree decomposition of width at most p. In this paper, we propose an improvement of it and improved the time-complexity to 3 · n. The connected vertex cover P3 (CV CP3) problem is the connected variation of the V CP3 problem where G[F ] is required to be connected. Using the Cut&Count technique, we give a randomized algorithm with runtime 4 · n that can solve the CV CP3 problem in any n-vertex graph given together with its tree decomposition of width at most p.