Some forbidden subgraph conditions for a graph to have a k-contractible edge

An edge of a k-connected graph is said to be k-contractible if the contraction of the edge results in a k-connected graph. Let K"n^- stand for the graph obtained from K"n by removing one edge. Let G be a k-connected graph (k>=5). It is known that if either ''k is odd and G contains no K"4^-=K"2+2K"1'' or ''G contains no K"1+2K"2'', then G has a k-contractible edge. In this paper, we prove that if G contains neither K"2+sK"1 nor K"1+tK"2 with positive integers s,t such that s(t-1)=k+1 and G contains neither K"5^- nor 5K"1+P"3, then G has a k-contractible edge.