{"title":"A Note on Oct1+-Minor-Free Graphs and Oct2+-Minor-Free Graphs","authors":"Wenyan Jia, Shuai Kou, Weihua Yang, Chengfu Qin","doi":"10.1142/s0219265921500304","DOIUrl":null,"url":null,"abstract":"Let [Formula: see text] and [Formula: see text] be the planar and non-planar graphs, respectively, obtained from the Octahedron by 3-splitting a vertex. For [Formula: see text], we prove that if a 4-connected graph is [Formula: see text]-minor-free, then it is [Formula: see text], [Formula: see text] [Formula: see text] or it is obtained from [Formula: see text] by repeatedly 4-splitting vertices. We also show that a planar graph is [Formula: see text]-minor-free if and only if it is constructed by repeatedly taking 0-, 1-, 2-sums starting from [Formula: see text], where [Formula: see text] is the set of graphs obtained by repeatedly taking the special 3-sums of [Formula: see text] and [Formula: see text] is the graph obtained from two 5-cycles [Formula: see text], [Formula: see text] by adding the five edges [Formula: see text] for all [Formula: see text]. For [Formula: see text], we prove that if a 4-connected graph is [Formula: see text]-minor-free, then it is planar, [Formula: see text] [Formula: see text], the line graph of [Formula: see text] or it is obtained from [Formula: see text] by repeatedly 4-splitting vertices.","PeriodicalId":153590,"journal":{"name":"J. Interconnect. Networks","volume":"18 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2022-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"J. Interconnect. Networks","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1142/s0219265921500304","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
Let [Formula: see text] and [Formula: see text] be the planar and non-planar graphs, respectively, obtained from the Octahedron by 3-splitting a vertex. For [Formula: see text], we prove that if a 4-connected graph is [Formula: see text]-minor-free, then it is [Formula: see text], [Formula: see text] [Formula: see text] or it is obtained from [Formula: see text] by repeatedly 4-splitting vertices. We also show that a planar graph is [Formula: see text]-minor-free if and only if it is constructed by repeatedly taking 0-, 1-, 2-sums starting from [Formula: see text], where [Formula: see text] is the set of graphs obtained by repeatedly taking the special 3-sums of [Formula: see text] and [Formula: see text] is the graph obtained from two 5-cycles [Formula: see text], [Formula: see text] by adding the five edges [Formula: see text] for all [Formula: see text]. For [Formula: see text], we prove that if a 4-connected graph is [Formula: see text]-minor-free, then it is planar, [Formula: see text] [Formula: see text], the line graph of [Formula: see text] or it is obtained from [Formula: see text] by repeatedly 4-splitting vertices.