A quasi-polynomial bound for the minimal excluded minors for a surface

TL;DR

This work achieves a major tightening of the long-standing bound on the size of minimal excluded minors for graphs embeddable on a surface. By proving a structural refinement that reduces the forbidden configuration from $O(g)$ disjoint contractible cycles to $O(\, ext{log} g\,)$ and by bounding key parameters such as maximum degree and face size to $ extDelta(g)=g^{O( extlog^2 g)}$, the authors derive a quasi-polynomial bound $U(g)=g^{O(\, ext{log}^3 g\,)}$ on the order of any excluded minor for a surface with Euler genus $g$; they also improve the treewidth bound to $O(g\log g)$ and bound the height of tree decompositions to a quasi-polynomial in $g$. The approach hinges on two structural breakthroughs: (i) a logarithmic bound on the number of nested cycles that are contractible or noncontractible, and (ii) a refined method to bound the height of tree decompositions, which together translate into a quasi-polynomial bound on the order of excluded minors, with significant algorithmic implications for minor-closed classes. The paper also provides a tight lower bound from an appendix, showing $L(g)= heta(\, ext{sqrt}(g)\,)$ for the order of some excluded minor on a given surface. Overall, the result narrows the gap between lower and upper bounds and informs potential future progress toward polynomial bounds, with practical impact on embedding and decomposition algorithms relying on excluded-minor characterizations.

Abstract

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 $Ω(g)$ 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.

Figures (19)

• 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 $B$ is 2-separated from the rest of the graph by $\{a, b\}$ and is contained in a disk in $\Pi$.
• Figure 4: Isolated paths. The solid lines indicate paths, whereas the dotted lines show the boundaries of the faces which the isolated paths use.
• Figure 5: Illustration for the second part of the proof of Proposition \ref{['isolated_paths']}. The disk in which $\text{int}(C, \Pi)$ can be reembedded is depicted in blue in every one of the cases.

Theorems & Definitions (117)

• Theorem 1.1: seymour
• Theorem 1.2
• Theorem 1.3: seymour
• Theorem 1.4
• Theorem 1.5: Thomassen
• Theorem 1.6
• Theorem 1.7
• Definition 4.1: Flipping
• Proposition 4.1: Whitney's Theorem graphs_on_surfaces
• Proposition 4.2: graphs_on_surfaces