Open Neighborhood Ideals in Chordal Graphs

• Open neighborhood ideals are square-free monomial ideals that capture the adjacency structure of chordal graphs, defined by the absence of induced cycles of length ≥4.
• They are constructed using perfect elimination orderings and universal representation techniques to realize any square-free monomial ideal through chordal graph models.
• These ideals reveal key algebraic invariants such as Cohen–Macaulayness and geometric vertex decomposability, connecting combinatorial topology and commutative algebra.

An open neighborhood ideal is a square-free monomial ideal associated with a finite simple graph $G=(V,E)$, encoding the combinatorial structure of $G$ into commutative algebra. In the study of chordal graphs, which are defined by the absence of induced cycles of length at least four, open neighborhood ideals provide a powerful connection between graph theory, combinatorial commutative algebra, and simplicial complexes. These ideals play a central role in classifying and realizing square-free monomial ideals, as nearly all such ideals can be constructed from the open-neighborhood ideals of specifically designed chordal graphs. Their algebraic invariants, structural decompositions, and relationships to Cohen–Macaulayness and geometric vertex decomposability are central in current research directions (Lim, 14 Dec 2025).

1. Chordal Graphs and Perfect Elimination Orderings

A finite simple graph $G=(V,E)$ is chordal if every induced cycle in $G$ has length three. An equivalent formulation, attributed to Dirac and Rose, is that $G$ admits a perfect elimination ordering. Specifically, there exists an ordering $v_1,\dots,v_n$ of the vertices such that for each $i$, the set

$\{\,v_j : j<i\text{ and }\{v_j,v_i\}\in E\}$

forms a clique in the induced subgraph $G[\{v_1,\dots,v_{i-1}\}]$.

Two key properties of chordal graphs are:

• The absence of induced cycles of length at least four.
• The ability to construct the graph inductively: starting with a single vertex, at each step add a new vertex whose neighbors among the existing vertices form a clique.

This structure underpins the tractability and rich combinatorial properties of chordal graphs.

2. Definition and Construction of Open Neighborhood Ideals

Given a graph $G=(V,E)$ and the polynomial ring $S = k[x_v : v\in V]$ over a field $k$, the open neighborhood ideal of $G$ is

$$J(G)\;=\;\bigl\langle\;\prod_{u\in N_G(v)}x_u: v\in V\;\bigr\rangle \;\subseteq\;S,$$

where the open neighborhood $N_G(v)=\{u\in V: \{u,v\}\in E\}$ consists of the vertices adjacent to $v$.

This ideal is always square-free, and its generators reflect the adjacency structure of $G$. The open neighborhood ideal encodes dominating and total-dominating set combinatorics, forming a bridge to algebraic invariants.

3. Universality: Realization of Square-Free Monomial Ideals

A fundamental result is the universality theorem: every square-free monomial ideal can be realized, up to adjoining dummy variables, as the open neighborhood ideal of a chordal graph.

Let $I\subseteq k[x_1,\dots,x_n]$ be a square-free monomial ideal with irredundant primary decomposition $I = \cap_{i=1}^t (A_i)$, where each $A_i\subseteq \{x_1,\dots,x_n\}$ and $|A_i|>1$. Denote $\mathcal{A} = \{A_1,\dots,A_t\}$ and compute its block-transversals $\tau(\mathcal{A}) = \{T_1,\dots,T_m\}$. Introduce new variables $y_1,\dots,y_m$ and construct graph $G$ by:

• Beginning with the complete graph on $\{x_1,\dots,x_n\}$,
• For each $j=1,\dots,m$, adding vertex $y_j$ joined precisely to the clique $T_j$.

$G$ is chordal, and for the extended ring $S' = k[x_1,\dots,x_n,y_1,\dots,y_m]$, the open neighborhood ideal satisfies: $J(G) = \langle X_{T_1},\dots,X_{T_m} \rangle,$

and $I \cong J(G) + (y_1,\dots,y_m) \subseteq S'$. The minimal total dominating sets of $G$ correspond to the $A_i$. Thus, the class of open neighborhood ideals of chordal graphs is universal for square-free monomial ideals (Lim, 14 Dec 2025).

4. Primary Decomposition, Cohen–Macaulayness, and Stanley–Reisner Complexes

For any graph $G$, let $\TD(G)$ denote the set of minimal total dominating sets. The open neighborhood ideal admits the decomposition: $J(G) = \bigcap_{D \in \TD(G)} (\,x_v : v \in D)\,.$ The Stanley–Reisner complex of $J(G)$, denoted $\Delta_G$, consists of subsets $F\subseteq V$ such that $F$ is disjoint from some $D \in \TD(G)$. Purity of $\Delta_G$ (i.e., all facets have the same dimension) is characterized by $G$ being well-totally-dominated (all minimal total dominating sets have the same size). $J(G)$ is Cohen–Macaulay if and only if $\Delta_G$ is pure and shellable, with shellability often accessible via vertex decomposability.

5. Geometric Vertex Decomposability (GVD) and Shedding Vertices

A square-free monomial ideal $I$ is geometrically vertex decomposable (GVD) if it can be obtained through a recursive sequence of decompositions

$I = C_{x,I} \cap (N_{x,I} + (x)),$

with $C_{x,I}, N_{x,I}$ again GVD. GVD for $J(G)$ is equivalent to vertex decomposability of the Stanley–Reisner complex $\Delta_G$. In practice, GVD can be tested by identifying “shedding” vertices in $G$ that correspond to decomposable structure in $\Delta_G$ (Lim, 14 Dec 2025).

6. Illustrative Example: The Path $P_3$

Consider $P_3$ with vertices $\{1,2,3\}$ and edges $\{1,2\},\{2,3\}$. The open neighborhood ideal is

$$J(P_3) = \langle x_2, x_1 x_3 \rangle \subseteq k[x_1, x_2, x_3].$$

Its minimal primes are $(x_2)$ and $(x_1, x_3)$, so

$J(P_3) = (x_2) \cap (x_1, x_3).$

The Stanley–Reisner complex has facets $\{1\}$ and $\{3\}$, i.e., two disjoint vertices—clearly vertex decomposable. Thus, $J(P_3)$ is GVD and Cohen–Macaulay (Lim, 14 Dec 2025).

7. Context, Significance, and Research Directions

Open neighborhood ideals of chordal graphs serve as a unifying structure connecting the study of combinatorial topology (via simplicial complexes), commutative algebra (through monomial ideals and their decompositions), and graph theory (especially domination and total domination problems). The universality result indicates that the landscape of square-free monomial ideals can be explored through graph-theoretic constructions, deepening understanding of algebraic invariants in a combinatorial framework. Current research investigates finer invariants, explicit constructions, and algorithmic applications leveraging the well-structured nature of chordal graphs and their associated ideals (Lim, 14 Dec 2025).