  Wu and Grumbach introduced the concept of strongly biconnected directed
graphs. A directed graph $G=(V,E)$ is called strongly biconnected if the
directed graph $G$ is strongly connected and the underlying undirected graph of
$G$ is biconnected. A strongly biconnected directed graph $G=(V,E)$ is said to
be $2$- edge strongly biconnected if it has at least three vertices and the
directed subgraph $(V,E\setminus\left\lbrace e\right\rbrace )$ is strongly
biconnected for all $e \in E$. Let $G=(V,E)$ be a $2$-edge-strongly biconnected
directed graph. In this paper we study the problem of computing a minimum size
subset $H \subseteq E$ such that the directed subgraph $(V,H)$ is $2$- edge
strongly biconnected.
