  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 ${\mathcal A}$
of graphs that have no odd holes, no antiholes and no odd stretchers as induced
subgraphs. In particular, every graph belonging to ${\mathcal A}$ is perfect.
Everett and Reed conjectured that a graph belongs to ${\mathcal A}$ if and only
if it is perfectly contractile. In the present paper, we discuss graphs
belonging to ${\mathcal A}$ from a viewpoint of commutative algebra. In fact,
we conjecture that a perfect graph $G$ belongs to ${\mathcal A}$ if and only if
the toric ideal of the stable set polytope of $G$ 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.
