Non-Existence of Annular Separators in Geometric Graphs

Benjamini and Papasoglou (2011) showed that planar graphs with uniform polynomial volume growth admit 1-dimensional annular separators: The vertices at graph distance R from any vertex can be separated from those at distance 2 R by removing at most O ( R ) vertices. They asked whether geometric d -dimensional graphs with uniform polynomial volume growth similarly admit (d-1) -dimensional annular separators when d>2 . We show that this fails in a strong sense: For any d⩾ 3 and every s⩾ 1 , there is a collection of interior-disjoint spheres in ℝ^d whose tangency graph G has uniform polynomial growth, but such that all annular separators in G have cardinality at least R^s .