Abstract
The maximum weight independent set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. In 1982, Alekseev (Comb Algebraic Methods Appl Math 132:3–13, 1982) showed that the M(W)IS problem remains NP-complete on H-free graphs, whenever H is connected, but neither a path nor a subdivision of the claw. We will focus on graphs without a subdivision of a claw. For integers \(i, j, k \ge 1\), let \(S_{i, j, k}\) denote a tree with exactly three vertices of degree one, being at distance i, j and k from the unique vertex of degree three. Note that \(S_{i,j, k}\) is a subdivision of a claw. The computational complexity of the MWIS problem for the class of \(S_{1, 2, 2}\)-free graphs, and for the class of \(S_{1, 1, 3}\)-free graphs are open. In this paper, we show that the MWIS problem can be solved in polynomial time for (\(S_{1, 2, 2}, S_{1, 1, 3}\), co-chair)-free graphs, by analyzing the structure of the subclasses of this class of graphs. This also extends some known results in the literature.
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Acknowledgements
The author sincerely thanks Prof. Frédéric Maffray and Prof. Vadim V. Lozin for their valuable suggestions and comments. The author also thanks the anonymous referees for their suggestions and comments.
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Karthick, T. Independent sets in some classes of \(S_{i, j, k}\)-free graphs. J Comb Optim 34, 612–630 (2017). https://doi.org/10.1007/s10878-016-0096-7
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DOI: https://doi.org/10.1007/s10878-016-0096-7