Graph Skeletons and Diminishing Minors

TL;DR

This paper introduces coarse bottlenecking and skeletons in graphs to simplify their structure while preserving quasi-isometry, advancing understanding of graphs with excluded minors and addressing a conjecture in the field.

Contribution

It develops the concept of coarse skeletons and demonstrates their use in simplifying graphs with excluded minors, progressing towards a conjecture by Georgakopoulos and Papasoglu.

Findings

Coarse bottlenecking guarantees skeletons resemble original graphs up to quasi-isometry.

Graphs with excluded asymptotic minors can be simplified to skeletons without 3-fat minors.

The result does not extend to 2-fat minors, highlighting limitations of the approach.

Abstract

We introduce the notion of coarse bottlenecking in graphs and coarse skeletons of graphs and show how bottlenecking guarantees that a skeleton resembles (up to quasi-isometry) the original graph. We show how these tools can be used to simplify the structure of graphs upto quasi-isometry that have an excluded asymptotic minor, reducing it to a skeleton of the original containing no $3$-fat minor. We give an example to show that a similar result does not hold for $2$-fat minors. This makes progress towards a Conjecture posed by Georgakopoulos and Papasoglu.