Forbidden Subgraphs Generating Almost All Claw-Free Graphs With High Connectivity

For a family H of connected graphs and an integer k >= 1, let G(k)(H) denote the family of k-connected graphs which contain no element of H as an induced subgraph. Let H+ be the family of those connected graphs of order 5 which contain K-1,K-3 as an induced subgraph. In this paper, for each integer k >= 1, we characterize the families H subset of H+ such that the symmetric difference of G(k)(K-1,K-3) and Gk (H) is finite.