Analyzing the 3-path vertex cover problem in selected graph classes

Abstract

In this paper, we focus on analyzing the 3-path vertex cover (3PVC) problem in a number of graph classes. Let \(G=(V,E)\) be a simple graph. A set \(C \subseteq V\) is called a k-path vertex cover of G, if each path of order k in G, contains at least one vertex from C. In the k-path vertex cover problem, we are given a graph G, and asked to find a k-path vertex cover of minimum size. For \(k=3\), the problem becomes the well-known 3PVC problem. A problem that is closely related to the 3PVC problem is the dissociation set (DS) problem. Given a graph \(G=(V,E)\), a dissociation set is any \(D \subseteq V\), such that the vertex-induced subgraph \(G'= (D,E')\) consists of vertices having degree 0 or 1. In the dissociation set problem, we are required to find a dissociation set of maximum cardinality. Both these problems have also been studied extensively as per the literature. In this paper, we focus on pipartite (planar and bipartite) graphs for the most part. We first show that the 3PVC problem is NP-hard, even in pipartite graphs having maximum degree 4. We then show that the 3PVC problem on this class of graphs admits a linear time \(\frac{8}{5}\)-approximation algorithm. Next, we show that the 3PVC problem is APX-complete in bipartite graphs having maximum degree 4 and cubic graphs. Finally, we discuss an elegant and alternative proof for the APX-completeness of the vertex cover problem in cubic graphs and establish lower bounds for the 3PVC problem in special graph classes. It is important to note that our work is the first of its kind to establish APX-completeness of the 3PVC problem in graphs.