Planar graphs in blowups of fans

2024-07-08

ACM-SIAM Symposium on Discrete Algorithms

AI Summary

This paper studies the embedding problem of planar graphs into fan graphs whose vertices are inflated into cliques. It proves that any $n$-vertex planar graph embeds into a fan graph with each vertex replaced by a clique of size $O(sqrt{n} log^2 n)$. The core technical step is constructing a vertex deletion set $X$ of size $O(sqrt{n} log^2 n)$ such that $G - X$ has bandwidth of the same asymptotic order—a structural characterization achieving the first tight bound. The approach integrates local sparsification lemmas, Feige–Rao-type bandwidth–density analysis, volume-preserving Euclidean embeddings, and graph product techniques. The result extends to all proper minor-closed graph classes, and establishes a precise trade-off between deletion set size and residual graph bandwidth. This work provides new upper-bound tools and structural insights for graph embedding and structural graph theory.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Bridging structured and unstructured data

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 $O((nlog n)/delta)$ 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.

Problem

• Embed planar graphs in fan blowups efficiently
• Optimize vertex removal for bandwidth reduction
• Extend results to minor-closed graph classes

Innovation

• Local sparsification lemma reduces graph density
• Generalized Feige-Rao method for bandwidth optimization
• Graph products extend results to minor-closed classes