A Complete Classification of 2-Linear Neighborhood Complexes

The neighborhood complex $N(G)$ and the dominance complex $D(G)$ are fundamental simplicial complexes associated with a graph $G$. We characterize precisely when the Stanley-Reisner ring $k[N(G)]$ admits a $2$-linear resolution, thereby answering an open question posed by Fröberg. We prove that this occurs if and only if $G$ is neighborhood conformal and its common neighbor graph is chordal. Equivalently, $G$ is a bipartite graph whose only induced cycles are $4$-cycles. As a consequence, we show that Katzman’s lower bound becomes an equality for this class, yielding $\mathrm{reg}(S\mathrm{/}I(G))=\mathrm{im}(G)$. Using recent results on glued clique complexes, we derive explicit combinatorial formulas for the exact graded Betti numbers of these neighborhood complexes. Finally, utilizing combinatorial Alexander duality, we obtain a corresponding classification of Cohen-Macaulay dominance complexes, and prove that the dominance complex of a graph without isolated vertices admits a $2$-linear resolution if and only if the graph is a star graph.