There's a standard combinatorial equivalence between undirected bipartite graphs and general directed graphs: just use the biadjacency matrix of the bipartite graph as an adjacency matrix of the directed graph and vice versa. See: R. A. Brualdi, F. Harary, , and Z. Miller (1980), "Bigraphs versus digraphs via matrices", Journal of Graph Theory 4 (1): 51–73, doi:10.1002/jgt.3190040107, MR558453. But then e.g. a cycle in the bipartite graph turns into a cycle with alternating edge orientations in the directed graph, not exactly what you probably want.
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