Authors–Shibata quadratic stable-set-ring conjecture for perfectly contractile graphs

Let $G$ be a perfect graph. The stable set ring $K[G]$ is the toric ring associated with the stable sets of $G$, and a ring is quadratic when its defining ideal is generated by quadratic relations. An induced subgraph is obtained by restricting $G$ to a subset of its vertices.

Authors–Shibata conjecture. The following conditions are equivalent:

1. $G$ is perfectly contractile.
2. $G$ contains no odd holes, no antiholes and no odd prisms as induced subgraphs.
3. The stable set ring $K[G]$ is quadratic.

This conjecture proposes a ring-theoretic characterization of perfectly contractile graphs and extends the Everett–Reed forbidden-subgraph conjecture. The source does not provide evidence that the conjecture has been resolved.