, Volume 68, Issue 4, pp 998-1018

Dominating Induced Matchings for P 7-Free Graphs in Linear Time

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Let G be a finite undirected graph with edge set E. An edge set E′⊆E is an induced matching in G if the pairwise distance of the edges of E′ in G is at least two; E′ is dominating in G if every edge eEE′ intersects some edge in E′. The Dominating Induced Matching Problem (DIM, for short) asks for the existence of an induced matching E′ which is also dominating 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. However, its complexity was open for P k -free graphs for any k≥5; P k denotes a chordless path with k vertices and k−1 edges. We show in this paper that the weighted DIM problem is solvable in linear time for P 7-free graphs in a robust way.