I'm considering graph classes that can be characterized by forbidden subgraphs.
If a graph class has a finite set of forbidden subgraphs, then there is a trivial polynomial time recognition algorithm (one can just use brute force). But an infinite family of forbidden subgraphs does not imply hardness: there are some classes with infinite list of forbidden subgraphs such that the recognition can also be tested in polynomial time. Chordal and Perfect graphs are examples but, in those cases, there is a "nice" structure on the forbidden family.
Is there any know relation between the hardness of the recognition of a class and the "bad behavior" of the forbidden family? Such a relation should exist? This "bad behavior" has been formalized somewhere?