Contractible edges in minimally k-connected graphs

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. In this paper, we prove that a (K1 + C4)-free minimally k-connected graph has a k-contractible edge, if around each vertex of degree k, there is an edge which is not contained in a triangle. This implies previous two results, one due to Thomassen and the other due to Kawarabayashi.