几何图中环形分离器的不存在性
Non-existence of annular separators in geometric graphs
摘要：
Benjamini和Papasoglou（2011）证明了具有均匀多项式体积增长的平面图允许$1$ -维环形分离器：通过移除最多$O(R)$顶点，可以将距离任何顶点的图距离$R$处的顶点与距离$2R$处的顶点分离。他们问几何$d$ -维图形与均匀多项式体积增长同样承认$(d-1)$ -维环形分离器时，$d > 2$。我们表明，这是失败的，在很强的意义上：对于任何$d \geq 3$和每一个$s \geq 1$，有一个集合的相邻不相交的领域在$\mathbb{R}^d$的相切图$G$具有均匀的多项式增长，但这样，所有的环形分离器在$G$的基数至少$R^s$。
英文摘要：
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 $2R$ 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 $d \geq 3$ and every $s \geq 1$, there is a collection of interior-disjoint spheres in $\mathbb{R}^d$ whose tangency graph $G$ has uniform polynomial growth, but such that all annular separators in $G$ have cardinality at least $R^s$.
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