Comparability of the total Betti numbers of toric ideals of graphs

TL;DR

This paper investigates how combinatorial operations affect the total Betti numbers of toric ideals of graphs, identifying specific procedures that preserve these algebraic invariants.

Contribution

It introduces a combinatorial operation that preserves the total Betti numbers of toric ideals of graphs, enhancing understanding of their stability under graph modifications.

Findings

Identified operations that change Betti numbers predictably

Discovered a procedure that preserves total Betti numbers

Provides insights into the stability of algebraic invariants under graph modifications

Abstract

The total Betti numbers of the toric ideal of a simple graph are, in general, highly sensitive to any small change of the graph. In this paper we look at some combinatorial operations that cause total Betti numbers to change in predictable ways. In particular, we focus on a procedure that preserves these invariants.