In many cases, it is possible to test in polynomial time whether a given graph contains any of the members of the obstruction set, and therefore whether it belongs to the family defined by that obstruction set.
In order for a family to have a forbidden graph characterization, with a particular type of substructure, the family must be closed under substructures.
When this is true, there always exists an obstruction set (the set of graphs that are not in the family but whose smaller substructures all belong to the family).
However, for some notions of what a substructure is, this obstruction set could be infinite.
The Robertson–Seymour theorem proves that, for the particular case of graph minors, a family that is closed under minors always has a finite obstruction set.