Determine allowable blow-ups of the computed configurations

Determine whether replacing vertices of the 449 nonisomorphic allowable graphs by complete graphs or independent sets, while preserving the original external adjacencies, always yields allowable graphs, and classify the blow-up configurations for which the resulting graphs remain allowable.

Background

The paper computationally generates 449 nonisomorphic allowable graphs for graphs of girth three with mz(G) at most 3. The authors then consider a graph operation in which a vertex is replaced by either a complete graph or an independent set, with every new vertex inheriting the original vertex's external neighbors.

The authors explain that it remains unresolved whether such blow-ups preserve allowability in general and indicate that additional forbidden graphs may arise during this process. Their computation shows that, among the configurations examined, the vertex V_t is the only one that in some cases admits both types of blow-up, but this observation does not resolve the general question.