Complete Graph Minors and the Graph Minor Structure Theorem

TL;DR

This paper investigates the maximum size of complete graph minors in graphs built from surface embeddings, vortices, apex vertices, and clique-sums, providing bounds up to a constant factor.

Contribution

It establishes bounds on the size of complete graph minors in graphs constructed via the Graph Minor Structure Theorem's ingredients, addressing a key converse question.

Findings

Bounds on the order of complete graph minors in such graphs

Answer to the converse question of the structure theorem

Results are tight up to a constant factor

Abstract

The graph minor structure theorem by Robertson and Seymour shows that every graph that excludes a fixed minor can be constructed by a combination of four ingredients: graphs embedded in a surface of bounded genus, a bounded number of vortices of bounded width, a bounded number of apex vertices, and the clique-sum operation. This paper studies the converse question: What is the maximum order of a complete graph minor in a graph constructed using these four ingredients? Our main result answers this question up to a constant factor.