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Exact Algorithms for Minimum Weighted Dominating Induced Matching

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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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Authors and Affiliations

  1. CONICET, Instituto de Cálculo, Buenos Aires, Argentina

    Min Chih Lin & Michel J. Mizrahi

  2. Departamento de Computación, Universidad de Buenos Aires, Buenos Aires, Argentina

    Min Chih Lin & Michel J. Mizrahi

  3. I. Mat., COPPE and NCE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

    Jayme L. Szwarcfiter

  4. Instituto Nacional de Metrologia, Qualidade e Tecnologia, Rio de Janeiro, Brazil

    Jayme L. Szwarcfiter

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  1. Min Chih Lin
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  2. Michel J. Mizrahi
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  3. Jayme L. Szwarcfiter
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Correspondence to Michel J. Mizrahi.

Additional information

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

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