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In this work we study how some elementary graph operations (like the disjointunion) and the collapse of two vertices modify the cut ideal of a graph. Theypave the way for reducing the cut ideal of every graph to the cut ideal ofsmaller ones. To deal with the collapse operation we generalize the definitionof cut ideal given in literature, introducing the concepts of edge labeling andedge multiplicity: in fact we state the \emph{non-classical behavior} of thecut ideal. Moreover we show the transformation of the toric map hidden behindthese operations.

Vertex Collapsing and Cut Ideals