An O?(1.1939~n) Time Algorithm for Minimum Weighted Dominating Induced Matching

摘要

Say that an edge of a graph G dominates itself and every other edge sharing a vertex of it. An edge dominating set of a graph G = (V,E) is a subset of edges E' ? 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, for solving these problems. This improves over the existing exact algorithms for the problems in consideration.