  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 at
most two clique-strips. A clique is strongly bisimplicial if its neighborhood
is partitioned into two cliques which are mutually non-adjacent and a
clique-strip is a sequence of cliques $\{H_0, \dots, H_p\}$ with the property
that $H_i$ is adjacent only to $H_{i-1}$ and $H_{i+1}$. By exploiting such a
structure we show how to solve the Maximum Weight Stable Set Problem in such a
graph in time ${\cal O}(|V|\sqrt{|E|})$.
