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 $$2k+ m- 1$$2k+m-1 for positive integers k, m has a subdigraph H of minimum outdegree or minimum indegree at least $$m-1$$m-1 such that $$D - x$$D-x is k-connected for all $$x \in V(H)$$x∈V(H). For $$m = 1$$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).

中文翻译：

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k-连通有向图中的临界顶点

证明了连接数为 k 的每个非完全有限有向图都有一个最多包含 k 个临界顶点的片段 F。以下结果是一个直接结果：对于正整数 k，m 具有最小出度和入度至少 $$2k+ m- 1$$2k+m-1 的每个 k 连通的有限有向图 D 具有最小出度的子图 H或最小入度至少 $$m-1$$m-1 使得 $$D - x$$Dx 对于所有 $$x \in V(H)$$x∈V(H) 都是 k-连通的。对于 $$m = 1$$m=1，这立即意味着在 k 临界有限有向图中存在一个入度或出度小于 2k 的顶点，这在 Mader 中得到了证明（J Comb Theory (B) 53： 260–272, 1991)。