Tree in forbidden triples generating a finite set of graphs with high connectivity

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For a graph G and a set F of connected graphs, G is said be F-free if G does not contain any member of F as an induced subgraph. For k≥3, we let Gk(F) denote the set of all k-connected F-free graphs. This paper is concerned with integers k,n, m1 and m2 and a tree T such that Gk({Kn,Km1,m2,T}) is finite. Among other results, we show that for integers k, n, m1 and m2 with k≥3, n≥3 and m2≥m1≥2, the diameter of a tree T such that Gk({Kn,Km1,m2,T}) is finite is bounded above by a constant which depends only on k but not on n, m1 or m2.