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 com- mon graph, and compare the classes of graphs satisfying each of the two forbidden subgraph conditions. Our main result is the following: Let H₁;H₂;H₃be connected graphs of order at least three, and suppose that H1 is twin-less. If the symmetric difference of F(fH₁;H₂g) and F(fH₁;H₃g) is finite and the tuple (H₁;H₂;H₃) is non-trivial in a sense, then H₂and H₃are 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.