As part of their graph minor project, Robertson and Seymour showed in 1990 that the class of graphs that can be embedded in a given surface can be characterized by a finite set of minimal excluded minors. However, their proof, because existential, does not provide any information on these excluded minors. Seymour proved in 1993 the first and, until now, only known upper bound on the order of the minimal excluded minors for a given surface. This bound is double exponential in the Euler genus $g$ of the surface and, therefore, very far from the $\Omega(g)$ lower bound on the maximal order of minimal excluded minors for a surface and most likely far from the best possible bound. More than thirty years later, this paper finally makes progress in lowering this bound to a quasi-polynomial in the Euler genus of the surface. The main catalyzer to reach a quasi-polynomial bound is a breakthrough on the characteristic size of a forbidden structure for a minimal excluded minor $G$ for a surface of Euler genus $g$: although it is not hard to show that $G$ does not contain $O(g)$ disjoint cycles that are contractible and nested in some embedding of $G$ as demonstrated by Seymour, this bound can be lowered to $O(\log g)$ which is essential to obtain the quasi-polynomial bound in this paper. As subsidiary results, we also improve the current bound on the treewidth of a minimal excluded minor $G$ for a surface by improving the first and, until now, only known bound provided by Seymour.

翻译：作为图子式研究项目的一部分，Robertson和Seymour在1990年证明了可嵌入给定曲面的图类可由有限个最小排除子式刻画。然而，由于其证明是存在性的，并未提供关于这些排除子式的任何信息。Seymour在1993年证明了关于给定曲面最小排除子式阶数的首个（也是迄今为止唯一已知的）上界。该上界是关于曲面欧拉亏格$g$的双指数函数，因此与曲面最小排除子式最大阶数的$\Omega(g)$下界相距甚远，且很可能远非最优上界。三十多年后，本文最终将这一上界改进为关于曲面欧拉亏格的拟多项式。实现拟多项式上界的关键突破在于：对于欧拉亏格$g$曲面的最小排除子式$G$，其禁用结构的特征尺寸得到了显著改进。尽管如Seymour所证明，容易证明$G$不包含$O(g)$个在$G$的某个嵌入中可收缩且相互嵌套的互不相交环，但本文将该界限降低至$O(\log g)$，这对获得拟多项式上界至关重要。作为附带成果，我们还通过改进Seymour给出的首个（且迄今为止唯一已知的）界限，提升了关于曲面最小排除子式$G$树宽的现有上界。