Graphs and complete intersection toric ideals
Isabel Bermejo, Ignacio Garc\'ia-Marco, Enrique Reyes

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 , checks whether its toric ideal is a complete intersection or not. Whenever is a complete intersection, the algorithm also returns a minimal set of generators of . Moreover, we prove that if is a connected graph and is a complete intersection, then there exist two induced subgraphs and of such that the vertex set of is the disjoint union of and , where is a bipartite ring graph and is either the empty graph, an odd primitive cycle, or consists of two odd primitive cycles properly connected. Finally, if is -connected and is connected, we list the families of graphs whose toric…
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