Contractible edges in some k-connected graphs

An edge e of a k-connected graph G is said to be k-contractible (or simply contractible) if the graph obtained from G by contracting e (i.e., deleting e and identifying its ends, finally, replacing each of the resulting pairs of double edges by a single edge) is still k-connected. In 2002, Kawarabayashi proved that for any odd integer k ? 5, if G is a k-connected graph and G contains no subgraph D = K ₁ + (K ₂ ∪ K _1,2), then G has a k-contractible edge. In this paper, by generalizing this result, we prove that for any integer t ? 3 and any odd integer k ? 2t + 1, if a k-connected graph G contains neither K ₁ + (K ₂ ∪ K _1,t ), nor K ₁ + (2K ₂ ∪ K _1,2), then G has a k-contractible edge.