Cut ideals of K4-minor free graphs are generated by quadrics

TL;DR

This paper proves that for K4-minor free graphs, the associated cut ideals are generated solely by quadratic binomials, confirming a longstanding conjecture in algebraic statistics.

Contribution

The paper introduces a new toric fiber product theorem to fully prove that cut ideals of K4-minor free graphs are generated by quadrics, resolving the conjecture.

Findings

Cut ideals of K4-minor free graphs are generated by quadrics.

A new toric fiber product theorem is developed.

The conjecture by Sturmfels and Sullivant is fully proven.

Abstract

Cut ideals are used in algebraic statistics to study statistical models defined by graphs. Intuitively, topological restrictions on the graphs should imply structural statements about the corresponding cut ideals. Several theorems and many computer calculations support that. Sturmfels and Sullivant conjectured that the cut ideal is generated by quadrics if and only if the graph is free of K4-minors. Parts of the conjecture has been resolved by Brennan and Chen, and later by Nagel and Petrovic. We prove the full conjecture by introducing a new type of toric fiber product theorem.