Lov\'asz-Schrijver PSD-operator on Claw-Free Graphs

The subject of this work is the study of $\LS_+$-perfect graphs defined as those graphs $G$ for which the stable set polytope $\stab(G)$ is achieved in one iteration of Lov\'asz-Schrijver PSD-operator $\LS_+$, applied to its edge relaxation $\estab(G)$. In particular, we look for a polyhedral relaxation of $\stab(G)$ that coincides with $\LS_+(\estab(G))$ and $\stab(G)$ if and only if $G$ is $\LS_+$-perfect. An according conjecture has been recently formulated ($\LS_+$-Perfect Graph Conjecture); here we verify it for the well-studied class of claw-free graphs.