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

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