A characterisation of graphs quasi-isometric to K₄-minor-free graphs

We prove that there is a function f such that every graph with no K-fat K₄ minor is f (K) -quasi-isometric to a graph with no K₄ minor. This solves the K₄-case of a general conjecture of Georgakopoulos and Papasoglu. Our proof technique also yields a new short proof of the respective K₄^--case, which was first established by Fujiwara and Papasoglu.