Abstract
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 $$G=(V,E)$$G=(V,E) is a subset of edges $$E' \subseteq E$$Eź⊆E which dominates all edges of G. In particular, if every edge of G is dominated by exactly one edge of $$E'$$Eź then $$E'$$Eź 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 $$O^*(1.1939^n)$$Oź(1.1939n) time and polynomial (linear) space, which improves over the existing algorithms for exactly solving this problem in general graphs.
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