A hypergraph is Helly if every family of hyperedges of it, formed by pairwise
intersecting hyperedges, has a common vertex. We consider the concepts of
bipartite-conformal and (colored) bipartite-Helly hypergraphs. In the same way as
conformal hypergraphs and Helly hypergraphs are dual concepts, bipartite-conformal and
bipartite-Helly hypergraphs are also dual. They are useful for characterizing biclique
matrices and biclique graphs, that is, the incident biclique-vertex incidence matrix and
the intersection graphs of the maximal bicliques of a graph, respectively. These concepts
play a similar role for the bicliques of a graph, as do clique matrices and clique graphs,
for the cliques of the graph. We describe polynomial time algorithms for recognizing
bipartite-conformal and bipartite-Helly hypergraphs as well as biclique matrices.