Independent sets in some classes of Si,j,k -free graphs

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Research Article

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Journal of Combinatorial Optimization


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≥ 1 , let Si,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 Si,j,k is a subdivision of a claw. The computational complexity of the MWIS problem for the class of S1 , 2 , 2-free graphs, and for the class of S1 , 1 , 3-free graphs are open. In this paper, we show that the MWIS problem can be solved in polynomial time for (S1 , 2 , 2, S1 , 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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