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

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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.

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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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Correspondence to Andreas Brandstädt.

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Brandstädt, A., Mosca, R. Finding Dominating Induced Matchings in \(P_8\)-Free Graphs in Polynomial Time. Algorithmica 77, 1283–1302 (2017). https://doi.org/10.1007/s00453-016-0150-y

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Keywords

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