Excluding surfaces as minors in graphs

TL;DR

The paper addresses the constructive GMST problem by achieving a tight bound on Euler genus while preserving the global-structure, almost-embeddable framework, and it introduces a grid-encoding approach (Dyck-walls/Dyck-grids) to unify surface embeddings with minor-universality. The core method weaves together Dyck-walls, Σ-decompositions, and a local-to-global strategy to refine the local GMST to feature a tight genus bound and a minor-universal description for graphs excluding a fixed-genus surface minor. The main contributions include a tight Euler-genus bound compatible with a constructive GMST, a local GMST version tailored to grid-like, minor-universal patterns, and a polynomial-time algorithmic implementation that computes the relevant walls, decompositions, and minor models. Altogether, this advances constructive graph minor theory by providing explicit, scalable tools for decomposing H-minor-free graphs with genus-constrained embeddings and paves the way for practical algorithms targeting surface-embedded graph classes.

Abstract

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 $Σ$ where $H$ does not embed after removing a small number of \textsl{apex vertices} and confining some vertices into a bounded number of \textsl{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.

Figures (12)

• Figure 1: The annulus grid $\mathscr{A}_{9},$ the handle grid $\mathscr{H}_{9}$ and the crosscap grid $\mathscr{C}_{9}$ in order from left to right. Notice that both $\mathscr{H}_{9}$ and $\mathscr{C}_{9}$ contain two $(18\times 9)$ grids as vertex-disjoint subgraphs (depicted in different colors). Moreover, $\mathscr{C}_{9}$ contains a $(18\times 18)$-grid as a spanning subgraph.
• Figure 2: The Dyck-grid of order $8$ with one handle and two crosscaps, i.e., the graph $\mathscr{D}_{8}^{1,2}.$ The red dashed lines indicates that "same hight" lefmost and rightmost vertices are adjacent (as it is the case in \ref{['supplanted']}).
• Figure 3: Swapping the position of a crosscap and a handle. The colors show how the handle and the crosscap on the right are routed through the crosscap and the handle on the left.
• Figure 4: Three crosscaps in a row.
• Figure 5: The result of the construction from \ref{['functionaries']}.

Theorems & Definitions (53)

• Proposition 1: Graph Minor Structure Theorem (GMST), robertson2003graph
• Proposition 2: KawarabayashiTW20Quicklyexcluding
• Proposition 3
• Theorem 1.1
• Proposition 4
• Definition 1: Dyck-grid
• Theorem 3.1
• Lemma 1
• proof
• Lemma 2