We consider two possible extensions of a theorem of Thomassen characterizing the graphs admitting a 2-vertex-connected orientation. First, we show that the problem of deciding whether a mixed graph has a 2-vertex-connected orientation is NP-hard. This answers a question of Bang-Jensen, Huang and Zhu. For the second part, we call a directed graph $D=(V,A)$ $2T$-connected for some $T\subseteq V$ if $D$ is 2-arc-connected and $D-v$ is strongly connected for all $v\in T$. We deduce a characterization of the graphs admitting a $2T$-connected orientation from the theorem of Thomassen.

中文翻译：

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混合图中 2-vertex-connected 方向的复杂性

我们考虑了 Thomassen 定理的两个可能扩展，该定理表征了允许 2 顶点连接方向的图。首先，我们表明决定混合图是否具有 2 顶点连接方向的问题是 NP 难的。这回答了 Bang-Jensen、Huang 和 Zhu 的问题。对于第二部分，我们称有向图$丁=(V,A)$ $\mathrm{2个}吨$- 连接一些$吨\subseteq V$如果$丁$是 2-arc-connected 和$丁-v$与所有人紧密相连$v\in 吨$. 我们推导出图表的特征，其中包含$\mathrm{2个}吨$-来自 Thomassen 定理的连接方向。