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

A claw-free graph G(V,E) is said to be basic if there exists a matching M ⊆ E whose edges are strongly bisimplicial and such that each connected component C of G − M is either a clique or a {claw, net}-free graph or satisfies α(G[C \ V (M)]) ≤ 3. The Maximum Weight Stable Set (MWSS) Problem in a basic claw-free graph can be easily solved by first solving at most four MWSS problems in each connected component of G−M in O(|V | log(|V |)) time ([7,8]) and then solving the MWSS Problem on a suitable line graph constructed from G in O(|V | log(|V |)) time. In this paper we show that, by means of lifting operations, every claw-free graph G(V,E) can be transformed, in O(|V |) time, into a basic claw-free graph Ḡ(V̄ , Ē) such that |V̄ | = O(|V |) and a MWSS of G can be obtained from a MWSS of Ḡ. This shows that the complexity of solving the MWSS Problem in a claw-free graph G(V,E) is O(|V | log(|V |)), the same as in line graphs.