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

TL;DR

This paper investigates the properties of -perfect graphs, focusing on claw-free graphs, and verifies a conjecture relating the Love1sz-Schrijver PSD-operator to stable set polytopes within this class.

Contribution

It confirms the -perfect graph conjecture for claw-free graphs, advancing understanding of polyhedral relaxations in graph theory.

Findings

Verification of the -perfect graph conjecture for claw-free graphs

Characterization of -perfect graphs in relation to the Love1sz-Schrijver PSD-operator

Enhanced understanding of stable set polytopes in claw-free graphs

Abstract

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.