Vertex Collapsing and Cut Ideals

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