Abstract
In a finite undirected graph \(G=(V,E)\), a vertex \(v \in V\) dominates itself and its neighbors in G. A vertex set \(D \subseteq V\) is an efficient dominating set (e.d.s. for short) of G if every \(v \in V\) is dominated in G by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d.s. in G, is known to be \(\mathbb {NP}\)-complete for \(P_7\)-free graphs and solvable in polynomial time for \(P_5\)-free graphs. The \(P_6\)-free case was the last open question for the complexity of ED on F-free graphs.
Recently, Lokshtanov, Pilipczuk and van Leeuwen showed that weighted ED is solvable in polynomial time for \(P_6\)-free graphs, based on their quasi-polynomial algorithm for the Maximum Weight Independent Set problem for \(P_6\)-free graphs. Independently, by a direct approach which is simpler and faster, we found an \(\mathcal{O}(n^5 m)\) time solution for weighted ED on \(P_6\)-free graphs. Moreover, we showed that weighted ED is solvable in linear time for \(P_5\)-free graphs which solves another open question for the complexity of (weighted) ED.
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Acknowledgments
The first author thanks Martin Milanič for discussions and comments about the WED problem for \(P_5\)-free graphs and for some subclasses of \(P_6\)-free graphs.
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Brandstädt, A., Mosca, R. (2016). Weighted Efficient Domination for \(P_6\)-Free and for \(P_5\)-Free Graphs. In: Heggernes, P. (eds) Graph-Theoretic Concepts in Computer Science. WG 2016. Lecture Notes in Computer Science(), vol 9941. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53536-3_4
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DOI: https://doi.org/10.1007/978-3-662-53536-3_4
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