Graph minors and metric spaces

TL;DR

This paper explores the intersection of graph minors and coarse geometry, addressing whether certain metric spaces are quasi-isometric to graphs without specific minors, and introduces metric analogues of classical theorems.

Contribution

It provides affirmative results for small minors and proposes new metric analogues of fundamental graph theorems, advancing understanding in geometric graph theory.

Findings

Confirmed quasi-isometry for small graph minors

Developed a metric version of Menger's theorem

Proposed conjectures on metric analogues of classical theorems

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.