A note on 2-vertex-connected orientations

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.