For a family H of connected graphs and an integer k≥1, let Gk(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 K1,3 as an induced subgraph. In this paper, for each integer k≥1, we characterize the families H⊆H+ such that the symmetric difference of Gk(K1,3) and Gk(H) is finite.