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 G̃₃(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 G̃₃(₃,C₄,T\) is a finite family, where Cₙ is a cycle of order n.