An \(O^*(1.1939^n)\) time algorithm for minimum weighted dominating induced matching

Abstract

Say that an edge of a graph \(G\) dominates itself and every other edge adjacent to it. An edge dominating set of a graph \(G=(V,E)\) is a subset of edges \(E' \subseteq E\) which dominates all edges of \(G\). In particular, if every edge of \(G\) is dominated by exactly one edge of \(E'\) then \(E'\) is a dominating induced matching. It is known that not every graph admits a dominating induced matching, while the problem to decide if it does admit it is NP-complete. In this paper we consider the problems of finding a minimum weighted dominating induced matching, if any, and counting the number of dominating induced matchings of a graph with weighted edges. We describe an exact algorithm for general graphs that runs in \(O^*(1.1939^n)\) time and polynomial (linear) space. This improves over any existing exact algorithm for the problems in consideration.