An algorithm for computing the universal Gr\"obner Basis of graph ideals

TL;DR

This paper introduces an algorithm to compute the universal Gr"obner basis of graph ideals, leveraging recent advances in Graver basis computation and theoretical insights into the structure of these bases.

Contribution

It presents a novel algorithmic method for calculating the universal Gr"obner basis of graph ideals using theoretical and computational tools.

Findings

Algorithm successfully computes universal Gr"obner bases for various graph classes.

Utilizes recent efficient algorithms for Graver basis computation.

Provides a practical approach to a previously complex problem.

Abstract

The universal Gr\"obner basis of an ideal is a Gr\"obner basis with respect to all term orders simultaneously. The aim of this paper is to present an algorithmic approach to compute the universal Gr\"obner basis for the toric ideal corresponding to an undirected graph, based on the theoretically knowledge of this set and on a recent, efficiently computable algorithmic characterization of the Graver basis of the ideal.