Finding the densest subgraph in a bi-partite graph is a polynomial time problem. Also, each bi-partite graph has a densest connected subgraph. In this paper, we first prove that each bi-partite graph has a densest connected subgraph. This proof is different than that of an undirected graph, since our definition of the density is different. We then provide a max-flow min-cut algorithm for finding a densest subgraph of a bi-partite graph and prove te correctness of this binary search algorithm.
Shah, Dhara; Prasad, Sushil; and Aghajarian, Danial, "Finding densest subgraph in a bi-partite graph" (2019). Computer Science Technical Reports. 1.