Sandra Albrechtsen, Raphael W. Jacobs, Paul Knappe, Paul Wollan
{"title":"A characterisation of graphs quasi-isometric to $K_4$-minor-free graphs","authors":"Sandra Albrechtsen, Raphael W. Jacobs, Paul Knappe, Paul Wollan","doi":"arxiv-2408.15335","DOIUrl":null,"url":null,"abstract":"We prove that there is a function $f$ such that every graph with no $K$-fat\n$K_4$ minor is $f(K)$-quasi-isometric to a graph with no $K_4$ minor. This\nsolves the $K_4$-case of a general conjecture of Georgakopoulos and Papasoglu.\nOur proof technique also yields a new short proof of the respective\n$K_4^-$-case, which was first established by Fujiwara and Papasoglu.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.15335","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.