On homological invariants and Cohen-Macaulayness of closed neighborhood ideals

TL;DR

This work links the homological invariants of the closed neighborhood ideal $NI(G)$ to rich graph-theoretic structure. It establishes a sharp equality $\mathrm{reg}(S/NI(G))=\tau(G)$ for chordal graphs, extending the known tree case, and analyzes when this equality fails for other graph families, along with bounds on the projective dimension and depth. The paper provides a detailed CM characterization for very well-covered graphs, showing CM occurs precisely when $G$ is well-dominated (i.e., $G$ is $C_4$ or a whisker graph); in non-$C_4$ cases, $NI(G)$ is a complete intersection and $S/NI(G)$ is Gorenstein, with $\gamma(G)=\alpha(G)$. These results deepen the connection between dominating-set combinatorics and algebraic properties of $NI(G)$, yielding structural criteria and implications for bipartite and chordal graphs.

Abstract

Let $G$ be a finite simple graph and $NI(G)$ be the closed neighborhood ideal of $G$ in the polynomial ring $S=K[V(G)]$. In this paper, we study the Castelnuovo-Mumford regularity, projective dimension and Cohen-Macaulayness of this ideal. For any chordal graph $G$, we show that $\text{reg}(S/NI(G))=τ(G)$, where $τ(G)$ denotes the vertex cover number of $G$. This generalizes the corresponding result for trees shown in \cite{CJRS}, as in trees $τ(G)$ is the same as the matching number of $G$. When $G$ is a bipartite graph or a very well-covered graph, we notice that $\text{reg}(S/NI(G))\geq τ(G)$ and that this inequality can be strict in general. Moreover, we describe the projective dimension of $S/NI(G)$ for some families of graphs. Finally, we give a characterization of very well-covered graphs $G$ for which the ring $S/NI(G)$ is Cohen-Macaulay.

Theorems & Definitions (30)

• Remark 2.1
• Lemma 3.1
• Theorem 3.2
• proof
• Example 3.3
• Proposition 3.4
• proof
• Proposition 3.5
• proof
• Theorem 3.6