Fixed-Radius Neighbourhoods in Three-Dimensional Unit-Cube Graphs

Abstract

We prove Conjecture 4 of Chan, Chang, Gao, Kisfaludi-Bak, Le, and Zheng (SoCG 2026): for every fixed integer $r\ge0$, the radius-$r$ neighbourhoods of a unit-cube graph in $\mathbb{R}^3$ have bounded VC-dimension. Our bound is $2^{2^{O(r)}}$, independent of the number of vertices. The main ingredient is a VC-dimension bound for reachability through arbitrary fixed-length sequences of three-dimensional generalized dominance relations. Its proof uses groups ordered in two coordinates, with neighbouring groups acting as barriers that preserve this invariant when one layer is removed. Combining these bounds with the existing interval-representation framework, we obtain a Las Vegas algorithm that, for every fixed integer $Δ\ge1$, decides whether a graph given by $n$ cube centres has diameter at most $Δ$ in expected $O_Δ(n^{2-\varepsilon_Δ})$ time, for an explicit constant $\varepsilon_Δ>0$. Finally, for every integer $r\ge7$, we construct unit-cube graphs with rational centres whose radius-$r$ neighbourhoods have VC-dimension $Ω(r^3)$.