Forbidden subgraphs generating a finite set of graphs with minimum degree three and large girth

For a family H of graphs, a graph G is said to be H-free if G contains no member of H as an induced subgraph. We let G₃ (H) denote the family of connected H-free graphs having minimum degree at least 3. In this paper, we characterize the non-caterpillar trees T having diameter at least 7 such that G₃ (\C₃, C₄, T\) is a finite family, where C₍ is a cycle of order n.