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 Fk(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 Fk(H1) and Fk(H2) 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 ∗[email protected], This work was supported by JSPS KAKENHI Grant Number 23740095. †[email protected] ‡[email protected]