Exact algorithms for dominating induced matching based on graph partition

A dominating induced matching, also called an efficient edge domination, of a graph G=(V,E) with n=|V| vertices and m=|E| edges is a subset F⊆E of edges in the graph such that no two edges in F share a common endpoint and each edge in E∖F is incident with exactly one edge in F. It is NP-hard to decide whether a graph admits a dominating induced matching or not. In this paper, we design a 1.1467nnO(1)-time exact algorithm for this problem, improving all previous results. This problem can be redefined as a partition problem that is to partition the vertex set of a graph into two parts I and F, where I induces an independent set (a 0-regular graph) and F induces a perfect matching (a 1-regular graph). After giving several structural properties of the problem, we show that the problem always contains some “good vertices,” branching on which by including them to either I or F we can effectively reduce the graph. This leads to a fast exact algorithm to this problem.