Graphs and complete intersection toric ideals

TL;DR

This paper develops a polynomial-time algorithm to determine if a graph's toric ideal is a complete intersection, characterizes such graphs through their subgraph structures, and classifies specific families of these graphs.

Contribution

It introduces an efficient algorithm for checking complete intersection property of graph toric ideals and provides a combinatorial characterization of such graphs.

Findings

Polynomial-time algorithm for complete intersection detection

Structural characterization involving bipartite ring graphs and cycles

Classification of specific graph families with complete intersection toric ideals

Abstract

Our purpose is to study the family of simple undirected graphs whose toric ideal is a complete intersection from both an algorithmic and a combinatorial point of view. We obtain a polynomial time algorithm that, given a graph $G$, checks whether its toric ideal $P_G$ is a complete intersection or not. Whenever $P_G$ is a complete intersection, the algorithm also returns a minimal set of generators of $P_G$. Moreover, we prove that if $G$ is a connected graph and $P_G$ is a complete intersection, then there exist two induced subgraphs $R$ and $C$ of $G$ such that the vertex set $V(G)$ of $G$ is the disjoint union of $V(R)$ and $V(C)$, where $R$ is a bipartite ring graph and $C$ is either the empty graph, an odd primitive cycle, or consists of two odd primitive cycles properly connected. Finally, if $R$ is $2$-connected and $C$ is connected, we list the families of graphs whose toric ideals are complete intersection.