Forbidden Subgraphs Generating Almost the Same Sets

Let H be a set of connected graphs. A graph G is said to be H-free if G does not contain any element of H as an induced subgraph. Let F-k(H) be the set of k-connected H-free graphs. When we study the relationship between forbidden subgraphs and a certain graph property, we often allow a finite exceptional set of graphs. But if the symmetric difference of F-k(H-1) and F-k(H-2) is finite and we allow a finite number of exceptions, no graph property can distinguish them. Motivated by this observation, we study when we obtain a finite symmetric difference. In this paper, our main aim is the following. If vertical bar H vertical bar <= 3 and the symmetric difference of F-1({H}) and F-1(H) is finite, then either H is an element of H or vertical bar H vertical bar = 3 and H = C-3. Furthermore, we prove that if the symmetric difference of F-k({H-1}) and F-k({H-2}) is finite, then H-1 = H-2.