Forbidden Pairs with a Common Graph Generating Almost the Same Sets

Let H be a family of connected graphs. A graph G is said to be H-free if G does not contain any members of H as an induced subgraph. Let F(H) be the family of connected H-free graphs. In this context, the members of H are called forbidden subgraphs. In this paper, we focus on two pairs of forbidden subgraphs containing a common graph, and compare the classes of graphs satisfying each of the two forbidden subgraph conditions. Our main result is the following: Let H1,H2,H3 be connected graphs of order at least three, and suppose that H1 is twin-less. If the symmetric difference of F({H1,H2}) and F({H1,H3}) is finite and the tuple (H1;H2,H3) is non-trivial in a sense, then H2 and H3 are obtained from the same vertex-transitive graph by successively replacing a vertex with a clique and joining the neighbors of the original vertex and the clique. Furthermore, we refine a result in [Combin. Probab. Comput. 22 (2013) 733–748] concerning forbidden pairs.