Complete Graph Minors and the Graph Minor Structure Theorem
Gwena\"el Joret, David R. Wood

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
