Forbidden subgraphs generating a finite set

For a set F of connected graphs, a graph G is said to be F -free if G does not contain any member of F as an induced subgraph. The members of F are referred to as forbidden subgraphs. When we study the relationship between forbidden subgraphs and a certain graph property, we often allow the possibility of the existence of exceptional graphs as long as their number is finite. However, in this type of research, if the set of k -connected F -free graphs itself, denoted by G k ( F ) , is finite, then every graph in G k ( F ) logically satisfies all the graph properties, except for possibly a finite number of exceptions. If this occurs, F does not give any information about a particular property. We think that such sets F obscure the view in the study of forbidden subgraphs, and we want to remove them. With this motivation, we study the sets F with finite G k ( F ) . We prove that if | F | ≤ 2 and G k ( F ) is finite, then either K 1 , 2 ∈ F or F consists of a complete graph and a star. For each of the values of k , 1 ≤ k ≤ 6 , we then characterize all the pairs { K l , K 1 , m } such that G k ( { K l , K 1 , m } ) is finite. We also give a complete characterization of F with | F | ≤ 3 and finite G 2 ( F ) .