Splittings of toric ideals

Let I⊆R=K[x1,…,xn] be a toric ideal, i.e., binomial prime ideal. We investigate when the ideal I can “split” into sum of two smaller ideals. For general I, we give sufficient condition for this splitting in terms integer matrix that defines I. When I=IG is finite simple graph G, additional splittings IG related to subgraphs G. there exists I=I1+I2 show some cases describe (multi-)graded Betti numbers I1 and I2.