We show that every $n$-vertex planar graph is contained in the graph obtained from a fan by blowing up each vertex by a complete graph of order $O(\sqrt{n}\log^2 n)$. Equivalently, every $n$-vertex planar graph $G$ has a set $X$ of $O(\sqrt{n}\log^2 n)$ vertices such that $G-X$ has bandwidth $O(\sqrt{n}\log^2 n)$. We in fact prove the same result for any proper minor-closed class, and we prove more general results that explore the trade-off between $X$ and the bandwidth of $G-X$. The proofs use three key ingredients. The first is a new local sparsification lemma, which shows that every $n$-vertex planar graph $G$ has a set of $O((n\log n)/δ)$ vertices whose removal results in a graph with local density at most $δ$. The second is a generalization of a method of Feige and Rao that relates bandwidth and local density using volume-preserving Euclidean embeddings. The third ingredient is graph products, which are a key tool in the extension to any proper minor-closed class.

翻译：我们证明每个具有$n$个顶点的平面图都包含于通过对扇图的每个顶点进行$O(\sqrt{n}\log^2 n)$阶完全图膨胀后得到的图中。等价地，每个具有$n$个顶点的平面图$G$都存在一个包含$O(\sqrt{n}\log^2 n)$个顶点的集合$X$，使得$G-X$具有$O(\sqrt{n}\log^2 n)$的带宽。实际上，我们对任意真子式封闭类证明了相同的结果，并进一步建立了$X$的规模与$G-X$的带宽之间权衡关系的更一般性结论。证明依赖于三个核心要素：首先是一个新的局部稀疏化引理，表明每个$n$顶点平面图$G$都存在一个包含$O((n\log n)/δ)$个顶点的集合，移除该集合后所得图的局部密度不超过$δ$；其次是Feige和Rao方法的推广，该方法通过保体积的欧几里得嵌入建立了带宽与局部密度的联系；第三个要素是图乘积运算，这是将结论推广至任意真子式封闭类的关键工具。