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

AbstractAn 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 Kn− stand for the graph obtained from Kn 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 K4−=K2+2K1” or “G contains no K1+2K2”, then G has a k-contractible edge. In this paper, we prove that if G contains neither K2+sK1 nor K1+tK2 with positive integers s,t such that s(t−1)