Open Neighborhood Ideals of Well-Totally Dominated Trees are Cohen-Macaulay

TL;DR

This paper studies the algebraic properties of open neighborhood ideals of graphs, proving Cohen-Macaulayness characterizes well-totally dominated trees and providing their Cohen-Macaulay type.

Contribution

It characterizes when the open neighborhood ideal of a tree is Cohen-Macaulay based on the tree's domination properties and describes the ideal's primary decomposition.

Findings

Open neighborhood ideal of a tree is Cohen-Macaulay iff the tree is well-totally dominated.

Provides the minimal primary decomposition of the open neighborhood ideal.

Calculates the Cohen-Macaulay type for these ideals.

Abstract

We introduce and investigate the open neighborhood ideal $\mathcal{N}(G)$ of a finite simple graph $G$. We describe the minimal primary decomposition of $\mathcal{N}(G)$ in terms of the minimal total dominating sets (TDSs) of $G$. Then we prove that the open neighborhood ideal of a tree is Cohen-Macaulay if and only if the tree is well-totally dominated (WTD) and calculate the Cohen-Macaulay type.