Faster computation of the maximum dissociation set and minimum 3-path vertex cover in graphs

Abstract

A dissociation set in a graph G = (V,E) is a vertex subset D such that the subgraph G[D] induced on D has vertex degree at most 1. A 3-path vertex cover in a graph is a vertex subset C such that every path of three vertices contains at least one vertex from C. Clearly, a vertex set D is a dissociation set if and only if V\D is a 3-path vertex cover. There are many applications for dissociation sets and 3- path vertex covers. However, it is NP-hard to compute a dissociation set of maximum size or a 3-path vertex cover of minimum size in graphs. Several exact algorithms have been proposed for these two problems and they can be solved in O∗ (1.4658n) time in n-vertex graphs. In this paper, we reveal some interesting structural properties of the two problems, which allow us to solve them in O∗ (1.4656n) time and polynomial space or O∗ (1.3659n) time and exponential space.