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

T. Karthick ()
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T. Karthick: Indian Statistical Institute, Chennai Centre

Journal of Combinatorial Optimization, 2017, vol. 34, issue 2, No 22, 612-630

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

Keywords: Graph algorithms; Independent sets; Claw-free graphs; Fork-free graphs (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10878-016-0096-7

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