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Geometrically vertex decomposable open neighborhood ideals

Jounglag Lim

TL;DR

This work advances the study of open neighborhood ideals (ONIs) by proving that ONIs of TD-unmixed trees are geometrically vertex decomposable, which implies vertex-decomposability of their Stanley-Reisner complexes. It connects ONIs to facet ideals of simplicial trees, showing that Cohen–Macaulay ONIs of trees arise as special cases of Cohen–Macaulay facet ideals, and identifies a broad realizability result for square-free monomial ideals as ONIs of chordal graphs. The results blend combinatorial graph structure with geometric decomposability concepts, providing a pathway to classify and realize broad classes of square-free monomial ideals via graph-theoretic constructions. The findings have implications for understanding the interplay between graph neighborhoods, stable complexes, and algebraic properties like Cohen–Macaulayness and vertex decomposability in a unified framework.

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.

Geometrically vertex decomposable open neighborhood ideals

TL;DR

This work advances the study of open neighborhood ideals (ONIs) by proving that ONIs of TD-unmixed trees are geometrically vertex decomposable, which implies vertex-decomposability of their Stanley-Reisner complexes. It connects ONIs to facet ideals of simplicial trees, showing that Cohen–Macaulay ONIs of trees arise as special cases of Cohen–Macaulay facet ideals, and identifies a broad realizability result for square-free monomial ideals as ONIs of chordal graphs. The results blend combinatorial graph structure with geometric decomposability concepts, providing a pathway to classify and realize broad classes of square-free monomial ideals via graph-theoretic constructions. The findings have implications for understanding the interplay between graph neighborhoods, stable complexes, and algebraic properties like Cohen–Macaulayness and vertex decomposability in a unified framework.

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.

Paper Structure

This paper contains 9 sections, 26 theorems, 55 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Vertex decomposable complexes and the Stanley-Reisner correspondence
  4. Open neighborhood ideals and Stable complexes
  5. Characterization of TD-unmixed trees
  6. Geometrically vertex decomposability and even-stable complexes
  7. $\mathcal{N}(T)$ is Geometrically vertex decomposable
  8. Open neighborhood ideals and facet ideals of simplicial trees
  9. Open neighborhood ideals of chordal graphs

Key Result

Theorem 2.2

Let $\Delta'$ and $\Delta"$ be pure simplicial complexes with disjoint vertex sets. Then $\Delta' \ast \Delta"$ is vertex decomposable if and only if both $\Delta'$ and $\Delta"$ are vertex decomposable.

Figures (11)

  • Figure 1: Graph $T$
  • Figure 2: Graph $T$
  • Figure 3: Subgraphs $T'$ (left) and $T"$ (right) of $T$
  • Figure 4: Star graph with at least 2 leaves
  • Figure 5: Left to right: $\mathcal{O}(P_6,v_1)$, $\mathcal{O}(P_6,v_4)$, and $\mathcal{O}(P_6,v_3)$
  • ...and 6 more figures

Theorems & Definitions (66)

  • Definition 2.1: joinIsShellable
  • Theorem 2.2: joinIsShellable
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Definition 2.7
  • Example 2.8
  • Theorem 2.9: COURAGE
  • Example 2.10
  • Definition 2.11
  • ...and 56 more