Barile-Macchia Resolutions and the closed neighborhood ideal

TL;DR

Problem: understand minimal free resolutions of closed neighborhood ideals NI(G) via Barile-Macchia resolutions, with focus on trees. The approach develops a linear extension and critical-set analysis to establish minimal BM resolutions for NI(T) and to connect projective dimension to the independence number α(T). It derives explicit Betti numbers for NI(P_n) using Betti-splitting and BM techniques, and it situates NI(T) outside several previously known minimal BM classes. The work also investigates bridge-friendly properties for chordal and bipartite graphs and discusses regularity relations, opening questions about broader applicability and recursive Betti formulas.

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 $α(T)$, leading to the equality $\mathrm{pdim}(R/NI(T)) = α(T)$, where $α(T)$ denotes the independence number of $T$ and $\mathrm{pdim}$ 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.

Figures (7)

• Figure 1: A rooted tree $T$
• Figure 2: The three cases depicting the vertices $v_2,v_3,w_2$ and $w_3$.
• Figure 3: Bridge Friendly Chordal Graph that does not satisfy condition in \ref{['bridge friendly sufficient condition']}
• Figure 4: Tree in \ref{['proposition:counter_ex_rooted_hypertree']}
• Figure 5: The vertices in red color forms $V_{\sigma}$ as an output of Algorithm \ref{['Algorithm:MaxCritTree']}

Theorems & Definitions (49)

• Definition 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• Proposition 2.4
• Lemma 3.1
• proof
• Remark 3.2
• Lemma 3.3
• proof