Graph minors and metric spaces
Agelos Georgakopoulos, Panos Papasoglu

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 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.
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Taxonomy
TopicsTopological and Geometric Data Analysis · Computational Geometry and Mesh Generation · Geometric Analysis and Curvature Flows
