Excluding surfaces as minors in graphs

Abstract: The Graph Minors Structure Theorem (GMST) of Robertson and Seymour states that for every graph $H,$ any $H$-minor-free graph $G$ has a tree-decomposition of bounded adhesion such that the torso of every bag embeds in a surface $Σ$ where $H$ does not embed after removing a small number of \textsl{apex vertices} and confining some vertices into a bounded number of \textsl{bounded depth} vortices. However, the functions involved in the original form of this statement were not explicit. In an enormous effort Kawarabayashi, Thomas, and Wollan proved a similar statement with explicit (and single-exponential in $|V(H)|$) bounds. However, their proof replaces the statement "a surface where $H$ does not embed'' with "a surface of Euler-genus in $\mathcal{O}(|H|^2)$''. In this paper we close this gap and prove that the bounds of Kawarabayashi, Thomas, and Wollan can be achieved with a tight bound on the Euler-genus. Moreover, we provide a more refined version of the GMST focussed exclusively on excluding, instead of a single graph, grid-like graphs that are minor-universal for a given set of surfaces. This allows us to give a description, in the style of Robertson and Seymour, of graphs excluding a graph of fixed Euler-genus as a minor, rather than focussing on the size of the graph.