Vertex Collapsing and Cut Ideals

TL;DR

This paper investigates how elementary graph operations, especially vertex collapsing, affect cut ideals, introducing new concepts like edge labeling and multiplicity to generalize existing definitions and analyze their algebraic transformations.

Contribution

It introduces a generalized framework for cut ideals incorporating edge labeling and multiplicity, and studies the effects of graph operations like vertex collapse on these ideals.

Findings

Collapse operations can reduce complex cut ideals to simpler ones.

Generalized cut ideals exhibit non-classical behavior under graph operations.

Transformation of the toric map reveals algebraic changes due to graph modifications.

Abstract

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.