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

Abstract

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.