I'm struggling with writing a detailed proof for this.
Let D be a digraph with at least two nodes, and let G be the underlying graph of D.
An edge-cut of G is a set S of edges of G such that G−S is disconnected, where G−S denotes the graph obtained from G by deleting all edges in S. Now,
Prove that D is strongly connected if and only if for every edge-cut S of the underlying graph G of D that separates V (G − S) into two sets A and B, there is an arc in D directed from a node in A to a node in B and an arc in D directed from a node in B to a node in A.
So far, i have assumed D is strongly connected. Taken u and v as two vertices in D, and since D is strongly connected both vertices are reachable from one another (path from u to v and a path from v to u).
Then an edge cut is performed on G, such that v and u are separated into two connected components, then v $\in$ A and u $\in$ B. I know that the edges that are cut represent the path from u to v and from v to u, but don't know how to explain via a proof. Also how do i explain the only if part?