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

TL;DR

This paper proves that graphs excluding a certain minor are quasi-isometric to minor-free graphs, confirming a special case of a broader conjecture and providing new proofs for related cases.

Contribution

It establishes a function relating graphs with no K-fat K4 minor to K4 minor-free graphs, solving a specific case of a conjecture and offering a new proof for a related case.

Findings

Graphs with no K-fat K4 minor are quasi-isometric to K4 minor-free graphs.

The proof technique simplifies understanding of minor-exclusion and quasi-isometry.

Provides a new short proof for the K4^- case originally established by Fujiwara and Papasoglu.

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.