Finiteness when the forbidden graph contains C5

Determine for which graphs H containing an induced C5 the set of 5-vertex-critical (P5,H)-free graphs is finite.

Background

The paper notes that the known infinite construction for (P5,C5)-free graphs applies for k≥6, whereas only finitely many 5-vertex-critical (P5,C5)-free graphs are known. This creates a distinct unresolved case at k=5.

The problem asks for a classification of all forbidden graphs containing an induced C5 for which finiteness still holds at this critical chromatic level.

References

Note that $\mathrm{crit}{5}{P_5, C_5}<\infty$, and thus it is open to determine the finiteness of $\mathrm{crit}{k}{P_5, H}$ when $H$ contains an induced $C_5$ which we now pose. For which graphs $H$ is $\mathrm{crit}_{5}{P_5, H}<\infty$ when $H$ contains an induced $C_5$?

— A dichotomy for the number of vertex-critical ($P_5$, $H$)-free graphs when $H$ is bipartite  (2608.20045 - Beaton et al., 20 Aug 2026) in Problem 3, Section 5 (Conclusion)