  Given a $2$-vertex-twinless connected directed graph $G=(V,E)$, the minimum
$2$-vertex-twinless connected spanning subgraph problem is to find a minimum
cardinality edge subset $E^{t} \subseteq E$ such that the subgraph $(V,E^{t})$
is $2$-vertex-twinless connected. Let $G^{1}$ be a minimal $2$-vertex-connected
subgraph of $G$. In this paper we present a $(2+a_{t}/2)$-approximation
algorithm for the minimum $2$-vertex-twinless connected spanning subgraph
problem, where $a_{t}$ is the number of twinless articulation points in
$G^{1}$.
