Barile-Macchia Resolutions and the closed neighborhood ideal

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.