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The Local Structure Theorem for Graph Minors with Finite Index

Christophe Paul, Evangelos Protopapas, Dimitrios M. Thilikos, Sebastian Wiederrecht

TL;DR

The paper advances the graph minor theory by strengthening the Local Structure Theorem to incorporate finite-index information on boundaried subgraphs, enabling a bidimensional representation of colorings across a Σ-decomposition. It introduces boundaried indices, boundaried representations via grids, and a three-step proof strategy—homogenization, representation, and coarsening—to obtain a final Σ-schema with controlled vortex depth. The results yield polynomial-time algorithms for constructing the indexed LST and provide a framework for grid-model representations of colors, facilitating applications in colorful minors and related algorithmic problems. Overall, the work bridges local structures witnessed by walls with global grid-like representations, enhancing both structural understanding and algorithmic tractability in minor-closed graph families.

Abstract

The Local Structure Theorem (LST) for graph minors roughly states that every $H$-minor free graph $G$ that contains a sufficiently large wall $W$, there is a set of few vertices $A$ such that, upon removing $A$, the resulting graph $G':=G - A$ admits an "almost embedding" $δ$ into a surface $Σ$ in which $H$ does not embed. By almost embedding, we mean that there exists a hypergraph $\mathcal{H}$ whose vertex set is a subset of the vertex set of $G$ and an embedding of $\mathcal{H}$ in $Σ$ such that 1) the drawing of each hyperedge of $\mathcal{H}$ corresponds to a cell of $δ$, 2) the boundary of each cell intersects only the vertices of the corresponding hyperedge, and 3) all remaining vertices and edges of $G'$ are drawn in the interior of cells. The cells corresponding to hyperedges of arity at least $4$, called vortices, are few in number and have small "depth", while a "large" part of the wall $W$ is drawn outside the vortices and is "grounded" in the embedding $δ$. Now suppose that the subgraphs drawn inside each of the non-vortex cells are equipped with some finite index, i.e., each such cell is assigned a color from a finite set. We prove a version of the LST in which the set $C$ of colors assigned to the non-vortex cells exhibits "large" bidimensionality: The graph $G'$ contains a minor model of a large grid $Γ$ where each bag corresponding to a vertex $v$ of $Γ$, contains the subgraph drawn within a cell carrying color $α$, for every color $α\in C$. Moreover, the grid $Γ$ can be chosen in a way that is "well-connected" to the original wall $W$.

The Local Structure Theorem for Graph Minors with Finite Index

TL;DR

The paper advances the graph minor theory by strengthening the Local Structure Theorem to incorporate finite-index information on boundaried subgraphs, enabling a bidimensional representation of colorings across a Σ-decomposition. It introduces boundaried indices, boundaried representations via grids, and a three-step proof strategy—homogenization, representation, and coarsening—to obtain a final Σ-schema with controlled vortex depth. The results yield polynomial-time algorithms for constructing the indexed LST and provide a framework for grid-model representations of colors, facilitating applications in colorful minors and related algorithmic problems. Overall, the work bridges local structures witnessed by walls with global grid-like representations, enhancing both structural understanding and algorithmic tractability in minor-closed graph families.

Abstract

The Local Structure Theorem (LST) for graph minors roughly states that every -minor free graph that contains a sufficiently large wall , there is a set of few vertices such that, upon removing , the resulting graph admits an "almost embedding" into a surface in which does not embed. By almost embedding, we mean that there exists a hypergraph whose vertex set is a subset of the vertex set of and an embedding of in such that 1) the drawing of each hyperedge of corresponds to a cell of , 2) the boundary of each cell intersects only the vertices of the corresponding hyperedge, and 3) all remaining vertices and edges of are drawn in the interior of cells. The cells corresponding to hyperedges of arity at least , called vortices, are few in number and have small "depth", while a "large" part of the wall is drawn outside the vortices and is "grounded" in the embedding . Now suppose that the subgraphs drawn inside each of the non-vortex cells are equipped with some finite index, i.e., each such cell is assigned a color from a finite set. We prove a version of the LST in which the set of colors assigned to the non-vortex cells exhibits "large" bidimensionality: The graph contains a minor model of a large grid where each bag corresponding to a vertex of , contains the subgraph drawn within a cell carrying color , for every color . Moreover, the grid can be chosen in a way that is "well-connected" to the original wall .

Paper Structure

This paper contains 82 sections, 25 theorems, 42 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Tangles.
  3. The local structure theorem.
  4. Boundaried subgraphs with respect to a fixed tangle.
  5. An indexed extension of the LST.
  6. Brief outline.
  7. Future work.
  8. Preliminaries
  9. Minors.
  10. Separations.
  11. Paths and linkages.
  12. Extended graphs.
  13. Grids and walls
  14. Tangles and the minors they control
  15. Tangles.
  16. ...and 67 more sections

Key Result

Proposition 1

There exist functions $\mathsf{lst}^{1} \colon \mathbb{N} \to \mathbb{N}$ and $\mathsf{wall}^{1} \colon \mathbb{N}^{2} \to \mathbb{N}$ with $\mathsf{lst}^{1}(t) \in \mathsf{poly}(k)$ and $\mathsf{wall}^{1}(k, r) \in \mathsf{poly}(k) \cdot r$, such that for every positive integer $r$, every $k$-verte

Figures (15)

  • Figure 1: A $(5 \times 10)$-grid on the left and an elementary $(5 \times 5)$-wall on the right.
  • Figure 2: An elementary $(5 \times 5)$-wall segment $W_{1}$. The top/bottom boundary vertices of $W_{1}$ are depicted in either green or orange while the left/right boundary vertices of $W_{2}$ are depicted in either green or blue. Note that vertices in green are both top/bottom and left/right boundary vertices.
  • Figure 3: An elementary $(5 \times 5)$-annulus wall.
  • Figure 4: An elementary $(5, 5)$-handle segment $W$. The left enclosure of $W$ bounds the left rainbow of $W$ and is depicted in blue while the right enclosure of $W$ bounds the right rainbow of $W$ and is depicted in magenta.
  • Figure 5: An elementary $(5 \times 5)$-crosscap segment $W$. The enclosure of $W$ bounds the rainbow of $W$ and is depicted in magenta.
  • ...and 10 more figures

Theorems & Definitions (43)

  • Proposition 1: GorskySW2025Polynomial
  • Theorem 1.1
  • Proposition 2
  • Proposition 3: Local Structure Theorem GorskySW2025Polynomial, Theorem 15.1
  • Lemma 1
  • proof
  • Corollary 1
  • Lemma 2: Homegenization lemma
  • proof
  • Proposition 4: Lemma 22, BasteST19HittingMinors
  • ...and 33 more