For a set of graphs Π, the Stable-Π problem asks whether, given a graph G, we can find an independent set S in G, such that G−S∈Π. For instance, if Π is the set of all bipartite graphs, Stable-Π coincides with vertex 3-colourability, and if Π is the set of 1-regular graphs, the problem is known as efficient edge domination. Numerous other examples of the Stable-Π problem have been studied in the literature. In the present contribution, we systematically study the Stable-Π problem with respect to the speed (a term meaning size) of Π. In particular, we show that for all hereditary classes Π with a subfactorial speed of growth, Stable-Π is solvable in polynomial time. We then explore the problem for minimal hereditary factorial classes Π. Contrary to the conjecture proposed in Lozin (2005) [18], the complexity of Stable-Π turns out to be polynomial for nearly all minimal hereditary factorial classes Π. On the other hand, if we do not require Π to be hereditary, the complexity of the problem can jump to NP-completeness.