Relation between hardness of recognition of a graph class and forbidden subgraph characterization
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.