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Erschienen in:
18.11.2016
Independent sets in some classes of \(S_{i, j, k}\)-free graphs
Erschienen in: Journal of Combinatorial Optimization | Ausgabe 2/2017
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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.