A Short Derivation of the Structure Theorem for Graphs with Excluded Topological Minors

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.