  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.
