文摘
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 \(E' \subseteq E\) 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 counting the number of dominating induced matchings and finding a minimum weighted dominating induced matching, if any, of a graph with weighted edges. We describe three exact algorithms for general graphs. The first runs in linear time for a given vertex dominating set of fixed size of the graph. The second runs in polynomial time if the graph admits a polynomial number of maximal independent sets. The third one is an \(O^*(1.1939^n)\) time and polynomial (linear) space, which improves over the existing algorithms for exactly solving this problem in general graphs.