Computing $2$-twinless blocks

Let $G=(V,E))$ be a directed graph. A $2$-twinless block in $G$ is a maximal vertex set $B\subseteq V$ of size at least $2$ such that for each pair of distinct vertices $x,y \in B$, and for each vertex $w\in V\setminus\left\lbrace x,y \right\rbrace$, the vertices $x,y$ are in the same twinless strongly connected component of $G\setminus\left \lbrace w \right\rbrace$. In this paper we present algorithms for computing the $2$-twinless blocks of a directed graph.