  Edge and vertex connectivity are fundamental concepts in graph theory. While
they have been thoroughly studied in the case of undirected graphs,
surprisingly not much has been investigated for directed graphs. In this paper
we study $2$-edge connectivity problems in directed graphs and, in particular,
we consider the computation of the following natural relation: We say that two
vertices $v$ and $w$ are $2$-edge-connected if there are two edge-disjoint
paths from $v$ to $w$ and two edge-disjoint paths from $w$ to $v$. This
relation partitions the vertices into blocks such that all vertices in the same
block are $2$-edge-connected. Differently from the undirected case, those
blocks do not correspond to the $2$-edge-connected components of the graph. We
show how to compute this relation in linear time so that we can report in
constant time if two vertices are $2$-edge-connected. We also show how to
compute in linear time a sparse certificate for this relation, i.e., a subgraph
of the input graph that has $O(n)$ edges and maintains the same
$2$-edge-connected blocks as the input graph.
