An O(n2logn) algorithm for the weighted stable set problem in claw-free graphs - Nobili, Paolo; Sassano, Antonio (Mathematical programming, 2021)
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, 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 { claw , net }- free strips and strips with stability number at most three. Through this decomposition, the MWSS problem can be solved in O ( | V | ( | V | log | V | + | E | ) ) time. In this paper, we describe a direct 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 | 2 ) 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 | 2 log | V | ) time, the same complexity of the best and long standing algorithm for the MWSS problem in line graphs .