  We propose bipartite analogues of comparability and cocomparability graphs.
Surprizingly, the two classes coincide. We call these bipartite graphs
cocomparability bigraphs. We characterize cocomparability bigraphs in terms of
vertex orderings, forbidden substructures, and orientations of their
complements. In particular, we prove that cocomparability bigraphs are
precisely those bipartite graphs that do not have edge-asteroids; this is
analogous to Gallai's structural characterization of cocomparability graphs by
the absence of (vertex-) asteroids. Our characterizations imply a robust
polynomial-time recognition algorithm for the class of cocomparability
bigraphs. Finally, we also discuss a natural relation of cocomparability
bigraphs to interval containment bigraphs, resembling a well-known relation of
cocomparability graphs to interval graphs.
