Dominating ideals and closed neighborhood ideals of graphs

TL;DR

This paper investigates the algebraic properties of closed neighborhood and dominating ideals in graphs, focusing on trees and cycles, revealing torsion-freeness and persistence properties.

Contribution

It establishes torsion-free and persistence properties for these ideals in trees and cycles, highlighting differences between the two graph classes.

Findings

Closed neighborhood ideals of trees are normally torsion-free.

Dominating ideals of trees are normally torsion-free.

Cycle ideals generally lack torsion-freeness but exhibit near-normal torsion-freeness.

Abstract

We study the closed neighborhood ideals and the dominating ideals of graphs, in particular, of trees and cycles. We prove that the closed neighborhood ideals and the dominating ideals of trees are normally torsion-free. The closed neighborhood ideals and the dominating ideals of cycles fail to be normally torsion-free. However, we prove that the closed neighborhood ideals of cycles admit the (strong) persistence property and the dominating ideals of cycles are nearly normally torsion-free.