Toric-ideal characterization of perfectly contractile perfect graphs

Prove that for every perfect graph G, the following three conditions are equivalent: G is perfectly contractile; G contains no odd holes, no antiholes, and no odd prisms; and the toric ideal of the stable-set polytope of G is generated by quadratic binomials.

Background

The cited conjecture strengthens the forbidden-subgraph characterization by connecting perfect contractibility with quadratic generation of stable-set toric ideals. The paper proves this claim for line-perfect graphs by analyzing their line graphs, but leaves the stated equivalence for general perfect graphs unresolved.

References

Let $G$ be a perfect graph. Then the following conditions are equivalent\rm :

— Toric ideal of matching polytopes and edge colorings  (2501.19209 - Mori et al., 31 Jan 2025) in Conjecture labeled conj:second, Section matching