Algorithmica

, Volume 77, Issue 3, pp 642–660 | Cite as

Exact Algorithms for Minimum Weighted Dominating Induced Matching

  • Min Chih Lin
  • Michel J. Mizrahi
  • Jayme L. Szwarcfiter
Article

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.

Keywords

Exact algorithms Graph theory Dominating induced matchings 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Min Chih Lin
    • 1
    • 2
  • Michel J. Mizrahi
    • 1
    • 2
  • Jayme L. Szwarcfiter
    • 3
    • 4
  1. 1.CONICET, Instituto de CálculoBuenos AiresArgentina
  2. 2.Departamento de ComputaciónUniversidad de Buenos AiresBuenos AiresArgentina
  3. 3.I. Mat., COPPE and NCE, Universidade Federal do Rio de JaneiroRio de JaneiroBrazil
  4. 4.Instituto Nacional de MetrologiaQualidade e TecnologiaRio de JaneiroBrazil

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