An $O^{\ast }({1.1939}^n)$ time algorithm for minimum weighted dominating induced matching

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 $E^{\mathrm{\prime }}\subseteq E$ which dominates all edges of $G$. In particular, if every edge of $G$ is dominated by exactly one edge of $E^{\mathrm{\prime }}$ then $E^{\mathrm{\prime }}$ 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 $O^{\ast }({1.1939}^n)$ time and polynomial (linear) space. This improves over any existing exact algorithm for the problems in consideration.