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 \(G=(V,E)\) 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.
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M.C. Lin was partially supported by UBACyT Grant 20020120100058, and PICT ANPCyT Grants 2010-1970 and 2013-2205. M.J. Mizrahi was partially supported by PICT ANPCyT Grants 2010-1970 and 2013-2205. J.L. Szwarcfiter was partially supported by CNPq, CAPES and FAPERJ, research agencies.
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Lin, M.C., Mizrahi, M.J. & Szwarcfiter, J.L. Exact Algorithms for Minimum Weighted Dominating Induced Matching. Algorithmica 77, 642–660 (2017). https://doi.org/10.1007/s00453-015-0095-6
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DOI: https://doi.org/10.1007/s00453-015-0095-6