Relation between hardness of recognition of a graph class and forbidden subgraph characterization 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?
Although it seems intuitive that the list of forbidden (induced) subgraphs for a class $\mathscr{C}$ of graphs which has NP-hard recognition should possess some "intrinsic" complexity, I have recently found some striking negative evidence to this intuition in the literature.

Perhaps the simplest to describe is the following, taken from an article by B. Lévêque, D. Lin, F. Maffray and N. Trotignon.

Let $F$ be the family of graphs which are composed of a cycle of length at least four, plus three vertices: two adjacent to the same vertex $u$ of the cycle, and one adjacent to a vertex $v$ of the cycle, where $u$ and $v$ are not consecutive in the cycle (and no other edges).

Now let $F&#39;$ be the family of graphs which are composed exactly the same way, except that you add four vertices: two adjacent to the same vertex $u$ of the cycle (as before), but now two adjacent to the same vertex $v$ of the cycle, where again $u$ and $v$ are not consecutive.

Then the class of graphs which has $F$ as the forbidden induced subgraphs has polynomial-time recognition, whereas the recognition of the class which has $F&#39;$ as the forbidden induced subgraphs is NP-hard.

Therefore, I find it hard to conceive of any general condition that a list of forbidden induced subgraphs has to satisfy when it results in a class with (NP-) hard recognition, considering that such a condition will have to separate the "very similar" $F$ and $F&#39;$ above.