For a family \ (F\) of graphs, a graph \ (G\) is said to be \ (F\) -free if \ (G\) contains no member of \ (F\) as an induced subgraph. We let \ (G₃ (F) \) be the family of \ (3\) -connected \ (F\) -free graphs. Let \ (P₍\) and \ (C₍\) denote the path and the cycle of order \ (n\), respectively. Let \ (T₀\) be the tree of order nine obtained by joining a pendant edge to the central vertex of \ (P₇\). Let \ (T₁\) and \ (T₂\) be the trees of order ten obtained from \ (T₀\) by joining a new vertex to a vertex of \ (P₇\) adjacent to an endvertex, and to a vertex of \ (P₇\) adjacent to the central vertex, respectively. We show that \ (G₃ (\C₃, C₄, T₁\) \) and \ (G₃ (\C₃, C₄, T₂\) \) are finite families.
Yoshimi Egawa (Tue,) studied this question.