A quadratic inequality among minors is valid for all totally nonnegative matrices iff, for every feasible planar matching, the matching occurs at least as many times on the left side as on the right.
Dodgson, Condensation of determinants, Proc
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On universal quadratic inequalities for minors of TNN matrices
A quadratic inequality among minors is valid for all totally nonnegative matrices iff, for every feasible planar matching, the matching occurs at least as many times on the left side as on the right.