Forbidden trees with diameter six for 3-connected graphs

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 \(Pₙ\) and \(Cₙ\) 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.