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We introduce a class of ideals generated by a set of 2-minors of $m\timesn$-matrix of indeterminates indexed by a pair of graphs. This class of idealsis a natural common generalization of binomial edge ideals and ideals generatedby adjacent minors. We determine the minimal prime ideals of such ideals andgive a lower bound for their degree of nilpotency. In some special cases wecompute their Gr\"obner basis and characterize unmixedness andCohen--Macaulayness.

The binomial edge ideal of a pair of graphs