  In this paper we study the primality of weakly connected collections of
cells, showing that the ideal generated by inner 2-minors attached to a weakly
connected and simple collection of cells is the toric ideal of the edge ring of
a weakly chordal bipartite graph. As an application of this result we
characterize the primality of the polyomino ideals of weakly closed paths, a
new class of non simple polyominoes.
