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 a (c) 3/4 5, if G is a k-connected graph and G contains no subgraph D = K (1) + (K (2) a(a) K (1,2)), then G has a k-contractible edge. In this paper, by generalizing this result, we prove that for any integer t a (c) 3/4 3 and any odd integer k a (c) 3/4 2t + 1, if a k-connected graph G contains neither K (1) + (K (2) a(a) K (1,t) ), nor K (1) + (2K (2) a(a) K (1,2)), then G has a k-contractible edge.