Open Neighborhood Ideals of Well Totally Dominated Trees are Cohen-Macaulay
Authors
Jounglag Lim
James Gossell
Keri Ann Sather-Wagstaff
Devin Adams
Vi Anh Nguyen
Suzanna Castro-Tarabulsi
Aayahna Herbert
Yifan Qian
Matthew Schaller
Zoe Zhou
Yuyang Zhuo
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 (TD-sets) of $G$. Then we prove that the open neighborhood ideal of a tree is Cohen-Macaulay if and only if the tree is unmixed (well totally dominated) and calculate the Cohen-Macaulay type. We also give a descriptive characterization of all unmixed trees which takes polynomial time to verify.
