Forbidden-subgraph characterization of perfectly contractile perfect graphs

Characterize perfectly contractile perfect graphs by proving that a perfect graph is perfectly contractile if and only if it contains no odd holes, no antiholes, and no odd prisms as induced subgraphs, equivalently if and only if its stable-set toric ideal is generated by quadratic binomials.

Background

Perfectly contractile graphs are defined through successive contractions of even pairs, and every perfectly contractile graph is perfect. The paper states that a forbidden-graph characterization of this class is not known in general.

The cited conjecture proposes equivalence among perfect contractibility, the absence of odd holes, antiholes, and odd prisms as induced subgraphs, and quadratic generation of the stable-set toric ideal. The present paper proves the relevant equivalence for line graphs of line-perfect graphs, but does not resolve the conjecture for all perfect graphs.

References

In contrast to the strong perfect graph theorem, a forbidden graph characterization of perfectly contractile graphs is still open. However, there is a conjecture of this problem.

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

In contrast to the strong perfect graph theorem, a forbidden graph characterization of perfectly contractile graphs is still open. However, there is a conjecture of this problem. An odd prism is a graph consisting of two disjoint triangles with three disjoint induced paths of odd length between them. Everett and Reed conjectured that a graph $G$ is perfectly contractile if and only if $G$ contains no odd holes, no antiholes and no odd prisms as induced subgraphs.

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