Characterize perfectly contractile graphs by forbidden induced subgraphs

Establish a forbidden-induced-subgraph characterization of perfectly contractile graphs, specifically determine whether a graph is perfectly contractile if and only if it contains no odd holes, no antiholes, and no odd prisms as induced subgraphs.

Background

A perfectly contractile graph is defined by the existence of a sequence of contractions of even pairs reducing the graph to a complete graph, with the property required for every induced subgraph. The paper notes that, unlike the strong perfect graph theorem, no forbidden-graph characterization of this class is known. It records the Everett–Reed conjecture as a proposed resolution of this open characterization problem.

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. 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 4, paragraph preceding Conjecture \ref{conj:second}