  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.
