Analyzing the 3-path Vertex Cover Problem in Planar Bipartite Graphs

Abstract

Let $$G=(V,E)$$ be a simple graph. A set $$C \subseteq V$$ is called a k-path vertex cover of G, if each k-path 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 cardinality. For $$k=3$$ , the problem becomes the well-known 3-path vertex cover (3PVC) problem, which has been widely studied, as per the literature. In this paper, we focus on the 3PVC problem in planar bipartite (pipartite) graphs for the most part. We first show that the 3PVC problem is NP-hard, even in pipartite graphs in which the degree of all vertices is bounded by 4. We then show that the 3PVC problem on this class of graphs admits a linear time 1.5-approximation algorithm. Finally, we show that the 3PVC problem is APX-complete in bipartite graphs. The last result is particularly interesting, since the vertex cover problem in bipartite graphs is solvable in polynomial time.