Geometrically vertex decomposable open neighborhood ideals

TL;DR

This paper proves that open neighborhood ideals of certain trees are geometrically vertex decomposable, linking graph properties with algebraic and combinatorial structures, and explores their realization in chordal graphs.

Contribution

It establishes the geometric vertex decomposability of open neighborhood ideals of TD-unmixed trees and connects Cohen-Macaulay properties with simplicial trees and chordal graphs.

Findings

Open neighborhood ideals of TD-unmixed trees are geometrically vertex decomposable.

Cohen-Macaulay open neighborhood ideals of trees are special cases of Cohen-Macaulay facet ideals.

Almost all square-free monomial ideals can be realized as open neighborhood ideals of chordal graphs.

Abstract

In this paper, we prove that the open neighborhood ideal of a TD-unmixed tree is geometrically vertex decomposable. This result implies that the associated Stanley-Reisner complex is vertex decomposable. We further demonstrate that Cohen-Macaulay open neighborhood ideals of trees are special cases of Cohen-Macaulay facet ideals of simplicial trees. Finally, we investigate open neighborhood ideals of chordal graphs and establish that almost all square-free monomial ideal can be realized as the open neighborhood ideal of a chordal graph.