  Let $G=(V,E)$ be a directed graph. A $2$-edge-twinless block in $G$ is a
maximal vertex set $C^{t}\subseteq V$ with $|C^{t}|>1$ such that for any
distinct vertices $v,w \in C^{t}$, and for every edge $e\in E$, the vertices
$v,w$ are in the same twinless strongly connected component of $G\setminus\left
\lbrace e \right\rbrace $.
  In this paper we study this concept and describe algorithms for computing
$2$-edge-twinless blocks.
