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Polynomial Bounds for the Graph Minor Structure Theorem

Maximilian Gorsky, Michał T. Seweryn, Sebastian Wiederrecht

TL;DR

The paper resolves long-standing questions about the Graph Minor Structure Theorem by proving polynomial bounds on the structure: there exists a function clique(t) = O(t^{2300}) such that any H-minor-free graph G is a clique-sum of graphs that are clique(|H|)-near embeddable in a surface where H does not embed. The authors provide fully constructive proofs yielding polynomial-time algorithms to either find H as a minor in G or produce the near-embedding clique-sum decomposition, and they also confirm Wollan’s conjectures on Euler-genus bounds and polynomial bounds for related minor-embedding theorems. A central technical innovation is the transaction-mesh framework, which orthogonalizes crooked transactions and replaces inductive leaps with global meta-structures, avoiding exponential blowups that plagued prior approaches (KTW). This advancement has broad algorithmic consequences, improving running times for GMST-based algorithms, reducing dependencies on H in topological-minor and related results, and enabling faster isomorphism and packing-type results on minor-closed classes. Overall, the work provides a decisive, constructive pathway to polynomially-bounded structure theorems for graph minors and lays the groundwork for more efficient algorithms in minor-closed graph classes.

Abstract

The Graph Minor Structure Theorem, originally proven by Robertson and Seymour [JCTB, 2003], asserts that there exist functions $f_1, f_2 \colon \mathbb{N} \to \mathbb{N}$ such that for every non-planar graph $H$ with $t := |V(H)|$, every $H$-minor-free graph can be obtained via the clique-sum operation from graphs which embed into surfaces where $H$ does not embed after deleting at most $f_1(t)$ many vertices with up to at most $t^2-1$ many ``vortices'' which are of ``depth'' at most $f_2(t)$. In the proof presented by Robertson and Seymour the functions $f_1$ and $f_2$ are non-constructive. Kawarabayashi, Thomas, and Wollan [arXiv, 2020] found a new proof showing that $f_1(t), f_2(t) \in 2^{\mathbf{poly}(t)}$. While believing that this bound was the best their methods could achieve, Kawarabayashi, Thomas, and Wollan conjectured that $f_1$ and $f_2$ can be improved to be polynomials. In this paper we confirm their conjecture and prove that $f_1(t), f_2(t) \in \mathbf{O}(t^{2300})$. Our proofs are fully constructive and yield a polynomial-time algorithm that either finds $H$ as a minor in a graph $G$ or produces a clique-sum decomposition for $G$ as above.

Polynomial Bounds for the Graph Minor Structure Theorem

TL;DR

The paper resolves long-standing questions about the Graph Minor Structure Theorem by proving polynomial bounds on the structure: there exists a function clique(t) = O(t^{2300}) such that any H-minor-free graph G is a clique-sum of graphs that are clique(|H|)-near embeddable in a surface where H does not embed. The authors provide fully constructive proofs yielding polynomial-time algorithms to either find H as a minor in G or produce the near-embedding clique-sum decomposition, and they also confirm Wollan’s conjectures on Euler-genus bounds and polynomial bounds for related minor-embedding theorems. A central technical innovation is the transaction-mesh framework, which orthogonalizes crooked transactions and replaces inductive leaps with global meta-structures, avoiding exponential blowups that plagued prior approaches (KTW). This advancement has broad algorithmic consequences, improving running times for GMST-based algorithms, reducing dependencies on H in topological-minor and related results, and enabling faster isomorphism and packing-type results on minor-closed classes. Overall, the work provides a decisive, constructive pathway to polynomially-bounded structure theorems for graph minors and lays the groundwork for more efficient algorithms in minor-closed graph classes.

Abstract

The Graph Minor Structure Theorem, originally proven by Robertson and Seymour [JCTB, 2003], asserts that there exist functions such that for every non-planar graph with , every -minor-free graph can be obtained via the clique-sum operation from graphs which embed into surfaces where does not embed after deleting at most many vertices with up to at most many ``vortices'' which are of ``depth'' at most . In the proof presented by Robertson and Seymour the functions and are non-constructive. Kawarabayashi, Thomas, and Wollan [arXiv, 2020] found a new proof showing that . While believing that this bound was the best their methods could achieve, Kawarabayashi, Thomas, and Wollan conjectured that and can be improved to be polynomials. In this paper we confirm their conjecture and prove that . Our proofs are fully constructive and yield a polynomial-time algorithm that either finds as a minor in a graph or produces a clique-sum decomposition for as above.

Paper Structure

This paper contains 123 sections, 113 theorems, 55 equations, 53 figures.

Table of Contents

  1. Introduction
  2. A short history of explicit bounds for R&S' theory of graph minors.
  3. Applications and consequences of the GMST.
  4. Computing the GMST.
  5. Sufficiency and necessity
  6. Tree structure and the three ingredients of near embeddings.
  7. The Local Structure Theorem
  8. A short overview of our proof
  9. Crooked transactions, leap patterns and Gallai's $A$-paths theorem.
  10. Orthogonalisation of crooked transactions: The key to polynomial bounds
  11. Orthogonal crooked transactions à la KTW.
  12. Orthogonal crooked transaction: The hard way.
  13. The structure of this paper
  14. Preliminaries.
  15. Finding a $K_t$-minor.
  16. ...and 108 more sections

Key Result

Theorem 1.1

There exists a function $\mathsf{clique}\colon\mathbb{N}\to\mathbb{N}$, with $\mathsf{clique}(t)\in\mathbf{O}(t^{2300})$, such that for every non-planar graph $H$, every $H$-minor-free graph $G$ can be obtained by means of clique-sums from graphs that are $\mathsf{clique}(|H|)$-near embeddable into

Figures (53)

  • Figure 1: A timeline on explicit bounds for R&S' theory of graph minors.
  • Figure 2: Four examples illustrating that it is not possible to omit any of the three main ingredients of the GMST, namely (i) and (ii) Euler-genus, (iii) vortices, and (iv) apices.
  • Figure 3: A $\Sigma$-decomposition $\delta$ of a graph $G-A$ with two vortices, namely $v_1$ and $v_2$, and an apex set $A$. Here $\Sigma$ is the double torus, $\delta$ is centred at the flat mesh $M$, (i) illustrates the cells of $\delta$, (ii) shows the two intertwining linkages that form the mesh $M$, and (iii) illustrates the inner, path-like, structure of a vortex.
  • Figure 4: The proof of \ref{['thm:intro_local_structure']} (simplified).
  • Figure 5: (i) A mesh with some crosses spread out over its middle row together with branchsets of a $K_6$-minor, (ii) a diagram of the branchsets of the $K_6$-minor, and (iii) the final $K_6$ obtained by contracting each of the branchsets into a single vertex.
  • ...and 48 more figures

Theorems & Definitions (224)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4: Robertson and Seymour RobertsonS1990Graph (see also KTW, Lemma 3.6 KawarabayashiTW2021Quickly)
  • Proposition 2.1: Menger's Theorem Menger1927Zur
  • Lemma 2.3: folklore
  • Proposition 2.4: Robertson and Seymour RobertsonS1986Grapha
  • Proposition 2.5: Thilikos and Wiederrecht ThilikosW2024Excluding (see Theorem 4.2.)
  • Proposition 3.1: Two Paths Theorem, Jung1970VerallgemeinerungSeymour1980DisjointShiloach1980PolynomialThomassen19802LinkedRobertsonS1990Graph
  • Proposition 3.2: Erdős and Szekeres ErdosS1935Combinatorial
  • ...and 214 more