Fat minors are a coarse analogue of graph minors where the subgraphs modeling vertices and edges of the embedded graph are required to be distant from each other, instead of just being disjoint. In this paper, we give a coarse analogue of the classic theorem that an $n$-vertex graph excluding a fixed minor admits a balanced separator of size $O(\sqrt{n})$. Specifically, we prove that for every integer $d$, real $\varepsilon>0$, and graph $H$, there exist constants $c$ and $r$ such that every $n$-vertex graph $G$ excluding $H$ as a $d$-fat minor admits a set $S \subseteq V(G)$ that is a balanced separator of $G$ and can be covered by $c n^{\frac{1}{2}+\varepsilon}$ balls of radius $r$ in $G$. Our proof also works in the weighted setting where the balance of the separator is measured with respect to any weight function on the vertices, and is effective: we obtain a randomized polynomial-time algorithm to compute either such a balanced separator, or a $d$-fat model of $H$ in $G$.

翻译：胖主子图是图子式的一种粗类比，其中嵌入图中模拟顶点和边的子图之间需要保持距离，而非仅仅互不相交。本文给出了一个经典定理的粗类比：每个固定子式免图的$n$顶点图存在大小为$O(\sqrt{n})$的平衡分割子。具体而言，我们证明了对任意整数$d$、实数$\varepsilon>0$及图$H$，存在常数$c$和$r$，使得每个不含$H$作为$d$-胖主子图的$n$顶点图$G$，存在集合$S \subseteq V(G)$，该集合是$G$的平衡分割子，且能被$G$中$c n^{\frac{1}{2}+\varepsilon}$个半径为$r$的球覆盖。我们的证明同样适用于赋权情形，其中分割子的平衡性通过顶点上的任意权函数度量，且具有有效性：我们给出了随机多项式时间算法，可计算此类平衡分割子或$G$中$H$的$d$-胖模型。