  A directed graph $G=(V,E)$ is strongly biconnected if $G$ is strongly
connected and its underlying graph is biconnected. A strongly biconnected
directed graph $G=(V,E)$ is called $2$-vertex-strongly biconnected if $|V|\geq
3$ and the induced subgraph on $V\setminus\left\lbrace w\right\rbrace $ is
strongly biconnected for every vertex $w\in V$. In this paper we study the
following problem.
  Given a $2$-vertex-strongly biconnected directed graph $G=(V,E)$, compute an
edge subset $E^{2sb} \subseteq E$ of minimum size such that the subgraph
$(V,E^{2sb})$ is $2$-vertex-strongly biconnected.
