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Algorithmica

, Volume 77, Issue 4, pp 1283–1302 | Cite as

Finding Dominating Induced Matchings in \(P_8\)-Free Graphs in Polynomial Time

  • Andreas BrandstädtEmail author
  • Raffaele Mosca
Article

Abstract

Let \(G=(V,E)\) be a finite undirected graph. An edge set \(E' \subseteq E\) is a dominating induced matching (d.i.m.) in G if every edge in E is intersected by exactly one edge of \(E'\). The Dominating Induced Matching (DIM) problem asks for the existence of a d.i.m. in G; this problem is also known as the Efficient Edge Domination problem. The DIM problem is related to parallel resource allocation problems, encoding theory and network routing. It is \({\mathbb {NP}}\)-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three and is solvable in linear time for \(P_7\)-free graphs. However, its complexity was open for \(P_k\)-free graphs for any \(k \ge 8\); \(P_k\) denotes the chordless path with k vertices and \(k-1\) edges. We show in this paper that the weighted DIM problem is solvable in polynomial time for \(P_8\)-free graphs.

Keywords

Dominating induced matching Efficient edge domination  \(P_8\)-free graphs Polynomial time algorithm 

Notes

Acknowledgments

The authors gratefully thank three anonymous reviewers for their helpful comments. The second author would like to witness that he just tries to pray a lot and is not able to do anything without that.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institut für InformatikUniversität RostockRostockGermany
  2. 2.Dipartimento di EconomiaUniversitá degli Studi “G. D’Annunzio”PescaraItaly

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