Forbidden Triples Containing a Complete Graph and a Complete Bipartite Graph of Small Order

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}}) contains no star, ({|mathcal{F}|=3}) and ({mathcal{G}_{3}(mathcal{F})}) is finite. Among other results, we show that for a connected graph T( ≠ K ₁) which is not a star, ({mathcal{G}_{3}({K_{4},K_{2,2},T})}) is finite if and only if T is a path of order at most 6.