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In this paper we show that a connected {claw, net}-free graph $G(V, E)$ with$\alpha(G) \ge 4$ is the union of a strongly bisimplicial clique $Q$ and atmost two clique-strips. A clique is strongly bisimplicial if its neighborhoodis partitioned into two cliques which are mutually non-adjacent and aclique-strip is a sequence of cliques $\{H_0, \dots, H_p\}$ with the propertythat $H_i$ is adjacent only to $H_{i-1}$ and $H_{i+1}$. By exploiting such astructure we show how to solve the Maximum Weight Stable Set Problem in such agraph in time ${\cal O}(|V|\sqrt{|E|})$.

An ${\cal O}(n\sqrt{m})$ algorithm for the weighted stable set problem in {claw, net}-free graphs with $\alpha(G) \ge 4$