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<p>Let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2948_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\( K_{1,1,5} \)</EquationSource> </InlineEquation> be the graph obtained from <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2948_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\( K_{2,5} \)</EquationSource> </InlineEquation> by adding an edge connecting two vertices of degree 5. In this paper, we prove that 4-connected planar <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2948_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\( K_{1,1,5} \)</EquationSource> </InlineEquation>-minor-free graphs contain no vertex of degree more than five. We also provide a complete characterization of 4-connected, 4-regular, planar <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="373_2025_2948_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\( K_{1,1,5} \)</EquationSource> </InlineEquation>-minor-free graphs.</p>

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A Note on 4-connected Planar K1,1,5-minor-free Graphs

  • 期刊论文

  • 发表日期：

    2025年7月25日

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  • Yuqi Xu,
  • Weihua Yang,
  • Shuang Zhao

摘要

Let \( K_{1,1,5} \) be the graph obtained from \( K_{2,5} \) by adding an edge connecting two vertices of degree 5. In this paper, we prove that 4-connected planar \( K_{1,1,5} \) -minor-free graphs contain no vertex of degree more than five. We also provide a complete characterization of 4-connected, 4-regular, planar \( K_{1,1,5} \) -minor-free graphs.