Finiteness of vertex-critical graphs for the eight remaining five-vertex forbidden graphs

Determine whether, for every integer k ≥ 1, there are finitely many k-vertex-critical (P5, H)-free graphs when H is one of the eight five-vertex graphs co-gem, chair, diamond + P1, C4 + P1, bull, P3 + 2P1, K5 − e, or K5.

Background

The paper establishes that, for every fixed integer k ≥ 1, only finitely many k-vertex-critical (P5, W4)-free graphs exist, where W4 is the four-wheel, and it completely characterizes the 5-vertex-critical graphs in this class. This resolves the corresponding finiteness question for H = W4 within the broader program of determining which five-vertex forbidden induced subgraphs H yield finite families of k-vertex-critical (P5, H)-free graphs.

The conclusion identifies eight five-vertex graphs for which the finiteness question remains unresolved: co-gem, chair, diamond + P1, C4 + P1, bull, P3 + 2P1, K5 − e, and K5. Resolving the problem requires determining finiteness for all k ≥ 1 in each of these eight hereditary graph classes. The authors note that the techniques developed for the dense graphs P3 + 2P1, K5 − e, and K5 may be especially relevant.