A polynomial bound for the minimal excluded minors for a surface

2026-04-03

AI Summary

This work addresses the problem of bounding the order of minimal excluded minors (obstructions) for graphs embeddable on a surface of genus \(g\). Bridging the substantial gap between the previously best-known quasi-polynomial upper bound and the linear lower bound \(\Omega(g)\), we introduce a novel forbidden structure based on separators, shifting the analytical focus from surface genus to separator size in subgraphs. By developing a separator-dependent framework of nested contractible cycles, we establish the first polynomial upper bound of \(O(g^{8+\varepsilon})\) on the order of minimal excluded minors. This result significantly narrows the gap with the known lower bound and confirms the long-standing conjecture that a polynomial upper bound exists.

Abstract

As part of the 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, provides no explicit information about these excluded minors. In 1993, Seymour established the first upper bound on the order of such minimal excluded minors. Very recently, Houdaigoui and Kawarabayashi improved this result by deriving a quasi-polynomial upper bound. Despite this progress, the gap between this bound and the known linear lower bound $Ω(g)$ (where $g$ denotes the genus) remains substantial. In particular, they conjectured that a polynomial upper bound should hold. In this paper, we confirm this conjecture by showing that the order of the minimal excluded minors for a surface of genus $g$ is $O(g^{8+\varepsilon})$ for every $\varepsilon >0$. This result significantly narrows the gap between the known lower and upper bounds, bringing the asymptotic behavior much closer to the conjectured optimum. Our approach introduces a new forbidden structure of minimal excluded minors. Let $G$ be a minimal excluded minor for a surface of Euler genus $g$. Houdaigoui and Kawarabayashi showed that $G$ contains $O(\log g)$ pairwise disjoint cycles that are contractible and nested in some embedding of $G$. We strengthen this result by proving a separator-based variant: for any contractible subgraph $H \subseteq G$ with a separator of size $s$ (with $H$ completely contained in one side), the subgraph $H$ contains $O(\log s)$ disjoint cycles that are contractible and nested in some embedding of $G$. This allows us to replace a genus-dependent bound with a separator-dependent one, which is the main new ingredient in deriving our polynomial bound.

Problem

• graph minors
• surface embedding
• excluded minors
• genus
• polynomial bound

Innovation

• minimal excluded minors
• polynomial bound
• graph embedding
• separator theorem
• Euler genus