{"title":"Removable Edges in Claw-Free Bricks","authors":"","doi":"10.1007/s00373-024-02769-6","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>An edge <em>e</em> in a matching covered graph <em>G</em> is <em>removable</em> if <span> <span>\\(G-e\\)</span> </span> is matching covered. Removable edges were introduced by Lovász and Plummer in connection with ear decompositions of matching covered graphs. A <em>brick</em> is a non-bipartite matching covered graph without non-trivial tight cuts. The importance of bricks stems from the fact that they are building blocks of matching covered graphs. Lovász proved that every brick other than <span> <span>\\(K_4\\)</span> </span> and <span> <span>\\(\\overline{C_6}\\)</span> </span> has a removable edge. It is known that every 3-connected claw-free graph with even number of vertices is a brick. By characterizing the structure of adjacent non-removable edges, we show that every claw-free brick <em>G</em> with more than 6 vertices has at least 5|<em>V</em>(<em>G</em>)|/8 removable edges.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00373-024-02769-6","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
An edge e in a matching covered graph G is removable if \(G-e\) is matching covered. Removable edges were introduced by Lovász and Plummer in connection with ear decompositions of matching covered graphs. A brick is a non-bipartite matching covered graph without non-trivial tight cuts. The importance of bricks stems from the fact that they are building blocks of matching covered graphs. Lovász proved that every brick other than \(K_4\) and \(\overline{C_6}\) has a removable edge. It is known that every 3-connected claw-free graph with even number of vertices is a brick. By characterizing the structure of adjacent non-removable edges, we show that every claw-free brick G with more than 6 vertices has at least 5|V(G)|/8 removable edges.
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