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Graph minors and metric spaces

Agelos Georgakopoulos, Panos Papasoglu

TL;DR

This work develops a coarse-geometry framework for graph-minor theory by introducing fat minors and asymptotic minors, and studies when a geodesic space without a fat $H$-minor is quasi-isometric to a graph with no $H$-minor. It constructs coarse analogues of classical theorems—Menger, König, and Halin—and proves a range of results, including Manning's quasi-tree characterization, the star-case of the fat-minor conjecture, and several dichotomies for large-scale structure (coarse König and Halin theorems). A central theme is that absence of asymptotic minors imposes strong coarse-structural regularity, often captured by quasi-isometries to minor-free graphs and by finite-asymptotic-dimension behavior. The paper also outlines several conjectures and highlights recent progress and counterexamples that guide ongoing development of a coarse graph-theoretic analogue of minor theory and Hadwiger-style results.

Abstract

We present problems and results that combine graph-minors and coarse geometry. For example, we ask whether every geodesic metric space (or graph) without a fat $H$ minor is quasi-isometric to a graph with no $H$ minor, for an arbitrary finite graph $H$. We answer this affirmatively for a few small $H$. We also present a metric analogue of Menger's theorem and Konig's ray theorem. We conjecture metric analogues of the Erdos--Posa Theorem and Halin's grid theorem.

Graph minors and metric spaces

TL;DR

This work develops a coarse-geometry framework for graph-minor theory by introducing fat minors and asymptotic minors, and studies when a geodesic space without a fat -minor is quasi-isometric to a graph with no -minor. It constructs coarse analogues of classical theorems—Menger, König, and Halin—and proves a range of results, including Manning's quasi-tree characterization, the star-case of the fat-minor conjecture, and several dichotomies for large-scale structure (coarse König and Halin theorems). A central theme is that absence of asymptotic minors imposes strong coarse-structural regularity, often captured by quasi-isometries to minor-free graphs and by finite-asymptotic-dimension behavior. The paper also outlines several conjectures and highlights recent progress and counterexamples that guide ongoing development of a coarse graph-theoretic analogue of minor theory and Hadwiger-style results.

Abstract

We present problems and results that combine graph-minors and coarse geometry. For example, we ask whether every geodesic metric space (or graph) without a fat minor is quasi-isometric to a graph with no minor, for an arbitrary finite graph . We answer this affirmatively for a few small . We also present a metric analogue of Menger's theorem and Konig's ray theorem. We conjecture metric analogues of the Erdos--Posa Theorem and Halin's grid theorem.
Paper Structure (16 sections, 23 theorems, 5 equations, 3 figures)

This paper contains 16 sections, 23 theorems, 5 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Metric spaces
  4. Asymptotic dimension
  5. Fat minors
  6. Manning's theorem --- quasi-trees
  7. Con-jecture \ref{['conj fat min']} for stars
  8. More on Con-jecture \ref{['conj fat min']}
  9. Coarse König Theorem
  10. Coarse Halin Theorems
  11. Coarse Menger Theorem
  12. Alternative construction of the paths --- reducing to the classical Menger theorem
  13. Choosing endpoints
  14. An application --- diverging rays
  15. Further problems
  16. ...and 1 more sections

Key Result

Theorem 3.1

The following are equivalent for every graph $G$

Figures (3)

  • Figure 1: Turning a diverging ray into a fat ray minor. Points meeting the same dashed line have the same distance from $o$, indicated at the top. Branch sets are depicted in blue, if colour is shown, and branch paths in red.
  • Figure 2: One of the possibilities in the proof of Pro-position \ref{['U ii']}.
  • Figure 3: A non-locally-finite graph failing the conclusion of Corollary \ref{['cor inf Menger']}.

Theorems & Definitions (63)

  • Conjecture 1.1
  • Conjecture 1.3: Coarse Hadwiger conjecture
  • Conjecture 1.4: Coarse Menger Theorem
  • proof : Proof (sketch).
  • proof
  • Definition 1: Fat minors
  • Definition 2.3: Asymptotic minors
  • Definition 2
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['triangle-free']}
  • ...and 53 more