The complexity of 2-vertex-connected orientation in mixed graphs

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 c V  if 

 is 2-arc-connected and D - v is strongly connected for all v E T. We deduce a characterization of the graphs admitting a 2T-connected orientation from the theorem of Thomassen.