Perfectly contractile graphs and quadratic toric rings

Perfect graphs form one of the distinguished classes of finite simple graphs. In 2006, Chudnovsky, Robertson, Seymour, and Thomas proved that a graph is perfect if and only if it has no odd holes and no odd antiholes as induced subgraphs, which was conjectured by Berge. We consider the class A${\mathcal {A}}$ of graphs that have no odd holes, no antiholes, and no odd stretchers as induced subgraphs. In particular, every graph belonging to A${\mathcal {A}}$ is perfect. Everett and Reed conjectured that a graph belongs to A${\mathcal {A}}$ if and only if it is perfectly contractile. In the present paper, we discuss graphs belonging to A${\mathcal {A}}$ from a viewpoint of commutative algebra. In fact, we conjecture that a perfect graph G$G$ belongs to A${\mathcal {A}}$ if and only if the toric ideal of the stable set polytope of G$G$ is generated by quadratic binomials. Especially, we show that this conjecture is true for Meyniel graphs, perfectly orderable graphs, and clique separable graphs, which are perfectly contractile graphs.