A characterisation of graphs quasi-isometric to $K_4$-minor-free graphs

Sandra Albrechtsen; Raphael W. Jacobs; Paul Knappe; Paul Wollan

A characterisation of graphs quasi-isometric to $K_4$-minor-free graphs

Combinatorics 2025-11-19 v3 Metric Geometry

Authors: Sandra Albrechtsen , Raphael W. Jacobs , Paul Knappe , Paul Wollan

Abstract

We prove that there is a function $f$ such that every graph with no $K$ -fat $K_4$ minor is $f(K)$ -quasi-isometric to a graph with no $K_4$ minor. This solves the $K_4$ -case of a general conjecture of Georgakopoulos and Papasoglu. Our proof technique also yields a new short proof of the respective $K_4^-$ -case, which was first established by Fujiwara and Papasoglu.

Keywords

quasi-isometric $K_4$-minor-free graphs graph theory geodesic growth planar graphs

Cite

@article{arxiv.2408.15335,
  title  = {A characterisation of graphs quasi-isometric to $K_4$-minor-free graphs},
  author = {Sandra Albrechtsen and Raphael W. Jacobs and Paul Knappe and Paul Wollan},
  journal= {arXiv preprint arXiv:2408.15335},
  year   = {2025}
}

Comments

30 pages, 12 figures, SVG files available in TeX Source; revision based on referee comment; fixed two references

A characterisation of graphs quasi-isometric to $K_4$-minor-free graphs

Abstract

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