Planar graphs in blowups of fans

TL;DR

This paper demonstrates that all n-vertex planar graphs can be embedded into blowups of fans with controlled size and bandwidth, extending the result to minor-closed classes using novel sparsification and embedding techniques.

Contribution

It introduces a new local sparsification lemma and generalizes bandwidth embedding methods to minor-closed graph classes, providing a unified framework for graph embedding.

Findings

Every n-vertex planar graph is contained in a blowup of a fan with vertex blowup size O(√n log^2 n)

Existence of a small vertex set whose removal yields a graph with bounded local density and bandwidth

Extension of results to any proper minor-closed class using graph products

Abstract

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.