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- W1761342833 abstract "A graph $G(V, E)$ is emph{claw-free} if no vertex has three pairwise non-adjacent neighbours. The Maximum Weight Stable Set (MWSS) Problem in a claw-free graph is a natural generalization of the Matching Problem and has been shown to be polynomially solvable by Minty and Sbihi in 1980. In a remarkable paper, Faenza, Oriolo and Stauffer have shown that, in a two-step procedure, a claw-free graph can be first turned into a quasi-line graph by removing strips containing all the irregular nodes and then decomposed into emph{{claw, net}-free} strips and strips with stability number at most three. Through this decomposition, the MWSS Problem can be solved in ${cal O}(|V|(|V| log |V| + |E|))$ time. In this paper, we describe a direct decomposition of a claw-free graph into emph{{claw, net}-free} strips and strips with stability number at most three which can be performed in ${cal O}(|V|^2)$ time. In two companion papers we showed that the MWSS Problem can be solved in ${cal O}(|E| log |V|)$ time in claw-free graphs with $alpha(G) le 3$ and in ${cal O}(|V| sqrt{|E|})$ time in {claw, net}-free graphs with $alpha(G) ge 4$. These results prove that the MWSS Problem in a claw-free graph can be solved in ${cal O}(|V|^2 log |V|)$ time, the same complexity of the best and long standing algorithm for the MWSS Problem in emph{line graphs}." @default.
- W1761342833 created "2016-06-24" @default.
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- W1761342833 date "2015-01-23" @default.
- W1761342833 modified "2023-09-27" @default.
- W1761342833 title "An ${cal O}(n^2 log(n))$ algorithm for the weighted stable set problem in claw-free graphs" @default.
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