Barile-Macchia Resolutions and the closed neighborhood ideal

TL;DR

This paper studies the minimal free resolutions of closed neighborhood ideals of graphs, showing they are minimal for trees and providing explicit Betti number formulas, with insights into chordal and bipartite graphs.

Contribution

It introduces the concept of bridge-friendly ideals for trees, constructs critical cells related to independence numbers, and derives explicit Betti number formulas for path graphs.

Findings

Closed neighborhood ideals of trees are bridge-friendly and have minimal BM resolutions.

The projective dimension of $NI(T)$ equals the independence number of $T$.

Explicit formulas for Betti numbers of $NI(P_n)$ are obtained.

Abstract

We investigate the minimal free resolutions of closed neighborhood ideals of graphs within the framework of Barile-Macchia (BM) resolutions. We show that for any tree $T$, the closed neighborhood ideal $NI(T)$ is bridge-friendly, and hence its BM resolution is minimal. The combinatorial structure of trees further allows us to construct a maximal critical cell of size $\alpha(T)$, leading to the equality $\operatorname{pd}(R/NI(T)) = \alpha(T)$, where $\alpha(T)$ denotes the independence number of $T$ and $\operatorname{pd}$ is the projective dimension. Using Betti splitting techniques, we also obtain explicit formulas for the graded Betti numbers of $NI(P_n)$, where $P_n$ is the path graph on $n$ vertices. Finally, we make some observations on the bridge-friendly condition of the closed neighborhood ideals of chordal and bipartite graphs.