Forbidden triples generating a finite set of 3-connected graphs (Q2517659)

Summary: For a graph \(G\) and a set \(\mathcal{F}\) of connected graphs, \(G\) is said be \(\mathcal{F}\)-free if \(G\) does not contain any member of \(\mathcal{F}\) as an induced subgraph. We let \(\mathcal{G}_{3}(\mathcal{F})\) denote the set of all \(3\)-connected \(\mathcal{F}\)-free graphs. This paper is concerned with sets \(\mathcal{F}\) of connected graphs such that \(|\mathcal{F}|=3\) and \(\mathcal{G}_{3}(\mathcal{F})\) is finite. Among other results, we show that for an integer \(m\geq 3\) and a connected graph \(T\) of order greater than or equal to \(4\), \(\mathcal{G}_{3}(\{K_{4},K_{2,m},T\})\) is finite if and only if \(T\) is a path of order \(4\) or \(5\).