An ${\cal O}(n^2 \log(n))$ algorithm for the weighted stable set problem in claw-free graphs

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}.