Consider a directed, weakly connected bipartite graph $G = (U, V, E)$ where $U$ and $V$ are sets comprising the nodes of $G$, and $E \subseteq U \times V$ is the set of edges. The task is to find a minimum directed cut, say $C \subseteq E$, such that the graph $G = (U, V, E - C)$ consists of two disconnected bipartite graphs each of which are (1) not edgeless, and (2) weakly connected. The cut $C$ is directed in the sense that all edges in $C$ originate from the same bipartite subgraph, and minimum in the sense that it has the lowest cardinality possible.
Is this problem studied? If so, what is the computational complexity of the best known algorithm that solves it? If not, original algorithm ideas are more than welcome!