An 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 nonadjacent 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 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 O(|V |) time. In two companion papers we showed that the MWSS Problem can be solved in O(|E| log |V |) time in claw-free graphs with α(G) ≤ 3 and in O(|V | √ |E|) time in {claw, net}-free graphs with α(G) ≥ 4. These results prove that the MWSS Problem in a claw-free graph can be solved in O(|V | log |V |) time as in a line graph and hence that it is not harder than a matching problem.