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

A graph $G(V, E)$ is 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 a claw-free graph can be decomposed into claw, net-free strips and strips with stability number at most three and that, through this decomposition, the MWSS Problem can be solved in ${\cal O}(|V|(|V| \log |V| + |E|))$ time. In this paper, we describe a slightly different decomposition of a claw-free graph into 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 as in a line graph and hence that it is not harder than a matching problem.