A short derivation of the structure theorem for graphs with excluded topological minors

TL;DR

This paper provides a concise proof of a structure theorem for graphs excluding a fixed topological minor, refining bounds on parameters and enhancing understanding of their decomposition properties.

Contribution

It offers a shorter proof of Grohe and Marx's theorem, improving bounds on parameters for graphs excluding a fixed topological minor.

Findings

Shorter proof of the structure theorem

Improved bounds on parameters

Enhanced understanding of graph decompositions

Abstract

As a major step in their proof of Wagner's conjecture, Robertson and Seymour showed that every graph not containing a fixed graph $H$ as a minor has a tree-decomposition in which each torso is almost embeddable in a surface of bounded genus. Recently, Grohe and Marx proved a similar result for graphs not containing $H$ as a topological minor. They showed that every graph which does not contain $H$ as a topological minor has a tree-decomposition in which every torso is either almost embeddable in a surface of bounded genus, or has a bounded number of vertices of high degree. We give a short proof of the theorem of Grohe and Marx, improving their bounds on a number of the parameters involved.