A Characterisation of Graphs Quasi-isometric to $$K_4$$-minor-free Graphs

Abstract We prove that there is a function f such that every graph with no K -fat $$K_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> minor is f ( K )-quasi-isometric to a graph with no $$K_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> minor. This solves the $$K_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> -case of a general conjecture of Georgakopoulos and Papasoglu. Our proof technique also yields a new short proof of the respective $$K_4^-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>K</mml:mi> <mml:mn>4</mml:mn> <mml:mo>-</mml:mo> </mml:msubsup> </mml:math> -case, which was first established by Fujiwara and Papasoglu.