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<p>We prove that there is a function <i>f</i> such that every graph with no <i>K</i>-fat <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(K_4\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation> minor is <i>f</i>(<i>K</i>)-quasi-isometric to a graph with no <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(K_4\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation> minor. This solves the&#xa0;<InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(K_4\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation>-case of a general conjecture of Georgakopoulos and Papasoglu. Our proof technique also yields a new short proof of the respective&#xa0;<InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(K_4^-\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>K</mi> <mn>4</mn> <mo>-</mo> </msubsup> </math></EquationSource> </InlineEquation>-case, which was first established by Fujiwara and Papasoglu.</p>

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A Characterisation of Graphs Quasi-isometric to \(K_4\)-minor-free Graphs

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年11月7日

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  • Sandra Albrechtsen,
  • Raphael W. Jacobs,
  • Paul Knappe,
  • Paul Wollan

摘要

We prove that there is a function f such that every graph with no K-fat \(K_4\) K 4 minor is f(K)-quasi-isometric to a graph with no \(K_4\) K 4 minor. This solves the  \(K_4\) K 4 -case of a general conjecture of Georgakopoulos and Papasoglu. Our proof technique also yields a new short proof of the respective  \(K_4^-\) K 4 - -case, which was first established by Fujiwara and Papasoglu.