  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.
