Projective dimension of closed neighborhood hypergraphs via extended double covers

Abstract: Let $G$ be a finite and simple graph without isolated vertices. We investigate the projective dimension of the closed neighborhood hypergraph $\mathcal{N}[G]$ and its relationship with the Castelnuovo-Mumford regularity of the extended bipartite double cover $\mathfrak{B}_e(G)$ of $G$. We establish the general upper bound $$\operatorname{prod-dim} (\mathcal{N}[G]) \leq \operatorname{reg}(\mathfrak{B}_e(G))$$ for all graphs. Furthermore, we prove that the exact equalities $$\operatorname{prod-dim} (\mathcal{N}[G]) = \operatorname{reg}(\mathfrak{B}_e(G)) =α(G)$$ hold when $G$ belongs to several prominent graph classes, including König-Egerváry (contains all bipartite graphs), cographs, co-chordal, chordal and comparability graphs, where $α(G)$ denotes the independence number. Our method of proofs relies on connecting algebraic invariants to the underlying combinatorial structure of graphs through covering, domination and matching parameters, together with the use of homology tools.