标题： 排除曲面作为图的子式 显示英文标题

标题： Excluding Surfaces as Minors in Graphs

摘要： 图素数结构定理（GMST）由罗伯逊和塞缪尔森提出，该定理指出，对于每个图$H,$，任何$H$-无小图的图$G$都具有有界粘合度的树分解，使得每个袋子的边沿可以嵌入到一个表面$\Sigma$中，其中在移除少量顶点后$H$无法嵌入，并将一些顶点限制到有限数量的有限深度涡旋中。 然而，该陈述原始形式中涉及的函数并不是显式的。 在巨大的努力下，河原桥、托马斯和沃尔兰证明了一个类似的陈述，具有显式的（并且在$|V(H)|$上是单指数）界限。 然而，他们的证明将“一个无法嵌入$H$的曲面”替换为“欧拉-亏格为$\mathcal{O}(|H|^2)$的曲面”。 在本文中，我们填补了这一空白，并证明 Kawarabayashi、Thomas 和 Wollan 的界限可以通过欧拉-亏格的紧致界限来实现。 此外，我们提供了一个更精细的 GMST 版本，专门针对排除网格状图，而不是单个图，这些网格状图对于给定的一组曲面是小图通用的。 这使我们能够以 Robertson 和 Seymour 的风格描述排除固定欧拉-亏格图作为小图的图，而不是专注于图的大小。 显示英文摘要

摘要： 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 $\Sigma$ where $H$ does not embed after removing a small number of apex vertices and confining some vertices into a bounded number of 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.