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

Primality of weakly connected collections of cells and weakly closed path polyominoes