On computing the minimum 3-path vertex cover and dissociation number of graphs

The dissociation number of a graph G is the number of vertices in a maximum size induced subgraph of G with vertex degree at most 1. A k-path vertex cover of a graph G is a subset S of vertices of G such that every path of order k in G contains at least one vertex from S. The minimum 3-path vertex cover is a dual problem to the dissociation number. For this problem, we present an exact algorithm with a running time of O∗(1.5171n) on a graph with n vertices. We also provide a polynomial time randomized approximation algorithm with an expected approximation ratio of for the minimum 3-path vertex cover.