Independent sets in some classes of $$S_{i, j, k}$$ S i , j , k -free graphs

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 $$i, j, k \ge 1$$i,j,kź1, let $$S_{i, j, k}$$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 $$S_{i,j, k}$$Si,j,k is a subdivision of a claw. The computational complexity of the MWIS problem for the class of $$S_{1, 2, 2}$$S1,2,2-free graphs, and for the class of $$S_{1, 1, 3}$$S1,1,3-free graphs are open. In this paper, we show that the MWIS problem can be solved in polynomial time for ($$S_{1, 2, 2}, S_{1, 1, 3}$$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.