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arXiv · 2510.19707

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

Jounglag Lim·James Gossell·Keri Ann Sather-Wagstaff
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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.

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Jounglag Lim, James Gossell, Keri Ann Sather-Wagstaff. 2025-10-22. Open Neighborhood Ideals of Well-Totally Dominated Trees are Cohen-Macaulay. https://arxiv.org/abs/2510.19707

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