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

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.

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

Abstract

We introduce and investigate the open neighborhood ideal of a finite simple graph . We describe the minimal primary decomposition of in terms of the minimal total dominating sets (TD-sets) of . 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.
Paper Structure (16 sections, 53 theorems, 141 equations, 28 figures)

This paper contains 16 sections, 53 theorems, 141 equations, 28 figures.

Table of Contents

  1. Introduction
  2. Total Dominating Sets and Open Neighborhood Ideals
  3. Foundations
  4. Decompositions
  5. Characterizations of unmixed trees
  6. Domination selector
  7. Balanced trees, RD-, and BD-sets
  8. Descriptive characterization of unmixed balanced trees
  9. Descriptive characterization of arbitrary trees via interior graphs
  10. Constructing unmixed balanced trees
  11. Shellability of Stable complexes
  12. Stable complexes of graphs
  13. Shellability of even stable complexes
  14. Cohen-Macaulay Type of unmixed trees
  15. Cohen-Macaulay type of balanced trees
  16. ...and 1 more sections

Key Result

Lemma 2.2.1

Let $S,V' \subseteq V$ and $R = A[V]$. Then $V'$ is an $S$-TD-set of $G$ if and only if $\mathcal{N}_S(G) \subseteq \left\langle V' \right\rangle$.

Figures (28)

  • Figure 1: Trees with 2-colorings
  • Figure 2: Graph $Y$ with a 2-coloring
  • Figure 3: Branch and radar example on $G$
  • Figure 4: Tree $T$ with vertices labeled and colored
  • Figure 5: Tree $T$ with branches (there are branches growing from all $s_i$'s like $s_m$)
  • ...and 23 more figures

Theorems & Definitions (141)

  • Definition 2.1.1
  • Definition 2.1.2
  • Example 2.1.3
  • Lemma 2.2.1
  • proof
  • Theorem 2.2.2
  • proof
  • Definition 2.2.3
  • Definition 2.2.4
  • Example 2.2.5
  • ...and 131 more