Complexity of the Maximum k-Path Vertex Cover Problem

Abstract

This paper introduces the maximization version of the k-path vertex cover problem, called the Maximum K-Path Vertex Cover problem (MaxP_kVC for short): A path consisting of k vertices, i.e., a path of length k-1 is called a k-path. If a k-path P_k includes a vertex v in a vertex set S, then we say that v or S covers P_k. Given a graph G=(V, E) and an integer s, the goal of MaxP_kVC is to find a vertex subset S⊆V of at most s vertices such that the number of k-paths covered by S is maximized. The problem MaxP_kVC is generally NP-hard. In this paper we consider the tractability/intractability of MaxP_kVC on subclasses of graphs. We prove that MaxP₃VC remains NP-hard even for split graphs. Furthermore, if the input graph is restricted to graphs with constant bounded treewidth, then MaxP₃VC can be solved in polynomial time.