A polynomial bound for the minimal excluded minors for a surface

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.

Figures (17)

• Figure 1: Almost disjoint cycles. The almost disjoint cycles are depicted in solid black lines. The black dots are vertices shared by several almost disjoint cycles.
• Figure 2: Cycles on a spanning tree rooted in $a$. The three colored subgraphs are cycles on a spanning tree rooted in $a$.
• Figure 3: The subgraph $\text{Int}(C,\Pi)$ of $G$ is shown embedded in $\Pi$. The cycles $C$ and $C_e$ are represented in blue and green, respectively, and the edge $e$ is represented in red. The bridges of $\mathcal{B} - B_C$ are highlighted in pale blue, while the bridge $B_C$ consists of the remaining part of $G- (C_e \cup e)$.
• Figure 4: Two cycles that have the same relative orientation. The cycles $C$ and $C'$ are $\Pi_H$-contractible (see Subfigure (a)) and $\Pi_H'$-noncontractible homotopic (see Subfigure (b)). They have the same relative orientation in $\Pi_H$ and $\Pi_H'$.
• Figure 5: The subgraph $B$ is 2-separated from the rest of the graph by $\{a, b\}$ and is contained in a disk in $\Pi$.

Theorems & Definitions (100)

• Theorem 1.1: seymour
• Theorem 1.2: HK2026
• Theorem 1.3
• Definition 4.1: Flipping
• Proposition 4.1: Whitney's Theorem graphs_on_surfaces
• Proposition 4.2: graphs_on_surfaces
• Definition 4.2: Almost disjoint cycles
• Proposition 4.3: Variant of \ref{['homotopic_cycles']}, HK2026
• Definition 4.3: Cycles on a spanning tree
• Proposition 4.4: Variant of \ref{['homotopic_cycles']}, HK2026