Complete Intersection Toric Ideals of Oriented Graphs and Chorded-Theta Subgraphs

TL;DR

This paper characterizes graphs whose associated toric ideals are complete intersections for all orientations, using recursive constructions and forbidden subgraph characterizations involving chorded-theta subgraphs.

Contribution

It introduces the class of CI_O graphs, characterizes them via chorded-theta subgraphs with transversal triangles, and provides explicit forbidden subgraph characterizations.

Findings

CI_O graphs are constructed from clique-sums of cycles and complete graphs.

A graph is CI_O if every chorded-theta has a transversal triangle.

Forbidden subgraphs include prisms, pyramids, thetas, and theta-partial wheels.

Abstract

Let $G=(V,E)$ be a finite, simple graph. We consider for each oriented graph $G_{\cal O}$ associated to an orientation ${\cal O}$ of the edges of $G$, the toric ideal $P_{G_{\cal O}}$. In this paper we study those graphs with the property that $P_{G_{\cal O}}$ is a binomial complete intersection, for all ${\cal O}$. These graphs are called $\text{CI}{\cal O}$ graphs. We prove that these graphs can be constructed recursively as clique-sums of cycles and/or complete graphs. We introduce the chorded-theta subgraphs and their transversal triangles. Also we establish that the $\text{CI}{\cal O}$ graphs are determined by the property that each chorded-theta has a transversal triangle. As a consequence, we obtain that the tournaments hold this property. Finally we explicitly give the minimal forbidden induced subgraphs that characterize these graphs, these families of graphs are: prisms, pyramids, thetas and a particular family of wheels that we call $\theta-$partial wheels.