  Inspired by a concept in comparative genomics, we investigate properties of
randomly chosen members of G_1(m,n,t), the set of bipartite graphs with $m$
left vertices, n right vertices, t edges, and each vertex of degree at least
one. We give asymptotic results for the number of such graphs and the number of
$(i,j)$ trees they contain. We compute the thresholds for the emergence of a
giant component and for the graph to be connected.
