It is proved that every non-complete, finite digraph of connectivity number k has a fragment F containing at most k critical vertices. The following result is a direct consequence: every k-connected, finite digraph D of minimum out- and indegree at least 2k+m−1 for positive integers k, m has a subdigraph H of minimum outdegree or minimum indegree at least m−1 such that D−x is k-connected for all x∈V(H). For m=1, this implies immediately the existence of a vertex of indegree or outdegree less than 2k in a k-critical, finite digraph, which was proved in Mader (J Comb Theory (B) 53:260–272, 1991). The final publication is available at Springer via https://doi.org/10.1007/s12188-016-0173-y.