Planar graphs in blowups of fans

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 $O((n\log n)/\delta)$ vertices whose removal results in a graph with local density at most $\delta$. 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.