Excluding surfaces as minors in graphs
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917225028386816 |
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| author | Thilikos, Dimitrios M. Wiederrecht, Sebastian |
| author_facet | Thilikos, Dimitrios M. Wiederrecht, Sebastian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19230 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Excluding surfaces as minors in graphs Thilikos, Dimitrios M. Wiederrecht, Sebastian Combinatorics 05C10, 05C83, 05C75, 68R10 G.2.2 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. |
| title | Excluding surfaces as minors in graphs |
| topic | Combinatorics 05C10, 05C83, 05C75, 68R10 G.2.2 |
| url | https://arxiv.org/abs/2601.19230 |