An $$\mathcal{O}(m\log n)$$ O ( m log n ) algorithm for the weighted stable set problem in claw-free graphs with $$\alpha ({G}) \le 3$$ α ( G ) ≤ 3

Abstract

In this paper we show how to solve the Maximum Weight Stable Set Problem in a claw-free graph G(V, E) with $$\alpha (G) \le 3$$ź(G)≤3 in time $$\mathcal{O}(|E|\log |V|)$$O(|E|log|V|). More precisely, in time $$\mathcal{O}(|E|)$$O(|E|) we check whether $$\alpha (G) \le 3$$ź(G)≤3 or produce a stable set with cardinality at least 4; moreover, if $$\alpha (G) \le 3$$ź(G)≤3 we produce in time $$\mathcal{O}(|E|\log |V|)$$O(|E|log|V|) a maximum weight stable set of G. This improves the bound of $$\mathcal{O}(|E||V|)$$O(|E||V|) due to Faenza, Oriolo and Stauffer.