A structural duality for path-decompositions into parts of small radius
Abstract
It is an easy observation that if a graph~ admits a path-decomposition whose parts have small radius, then contains no large subdivision of or as a (quasi-)geodesic subgraph. We show that these are in fact the only obstructions to such path-decompositions of small radial width, and we prove analogous results for decompositions modelled on cycles and subdivided stars instead of paths. With our results we confirm in a strong form a conjecture of Georgakopoulos and Papasoglu on fat-minor-characterisations of graphs quasi-isometric to paths, cycles and paths, and subdivided stars, respectively. For this, we present a novel view on quasi-isometries between graphs by graph-decompositions of bounded radial width and spread. This new perspective enables us to prove further results in coarse graph theory, and may thus be of independent interest.
Keywords
Cite
@article{arxiv.2307.08497,
title = {A structural duality for path-decompositions into parts of small radius},
author = {Sandra Albrechtsen and Reinhard Diestel and Ann-Kathrin Elm and Eva Fluck and Raphael W. Jacobs and Paul Knappe and Paul Wollan},
journal= {arXiv preprint arXiv:2307.08497},
year = {2025}
}
Comments
This new version has a completely rewritten Introduction, which takes into account several developments occurring close or parallel to the first publication; after the Introduction, we made only restructurings, and added a few propositions and some small corrections and amendments