Algebraic implications of neighborhood hypergraphs and their transversal hypergraphs

TL;DR

This paper explores the algebraic properties of neighborhood hypergraphs and their transversals, revealing new classes of monomial ideals and characterizing specific graph classes with normally torsion-free ideals.

Contribution

It introduces new classes of normally torsion-free monomial ideals derived from neighborhood hypergraphs and characterizes cycles with such properties.

Findings

Closed neighborhood and dominating ideals of strongly chordal graphs are normally torsion-free.

Characterization of cycles with normally torsion-free dominating ideals.

Analysis of stable sets of associated primes in cycle graphs.

Abstract

In this paper, we unfold balanced and totally balanced neighborhood hypergraphs to discover new classes of normally torsion-free monomial ideals. As a consequence, we establish that the closed neighborhood ideals and the dominating ideals of strongly chordal graphs are normally torsion-free. We discuss the stable sets of associated primes of the dominating ideals of cycles and characterize all the cycles with normally torsion-free dominating ideals.