{"title":"A synthesis for uniformly 3-edge-connected graphs","authors":"Carl Kingsford , Guillaume Marçais","doi":"10.1016/j.dam.2025.02.005","DOIUrl":null,"url":null,"abstract":"<div><div>A multigraph is <em>uniformly</em> <span><math><mi>k</mi></math></span>-<em>edge-connected</em> if there are exactly <span><math><mi>k</mi></math></span> edge-disjoint paths between any pair of vertices. For example, a uniformly <span><math><mi>k</mi></math></span>-edge-connected graph is obtained from a <span><math><mi>k</mi></math></span>-edge-connected graph by collapsing the nodes connected by more than <span><math><mi>k</mi></math></span> edge-disjoint paths into supernodes. We characterize the class of uniformly 3-edge-connected graphs, giving a synthesis involving two operations by which every uniformly 3-edge-connected multigraph can be generated. Slightly modified syntheses give the planar uniformly 3-edge-connected graphs and the uniformly 3-edge-connected graphs with the fewest possible edges, generalizing the well-known Harary graphs. In proving the correctness of the synthesis, we also show the existence of a particular type of induced, non-separating cycle in near 3-regular graphs, which is of interest in its own right.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"368 ","pages":"Pages 18-29"},"PeriodicalIF":1.0000,"publicationDate":"2025-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166218X25000630","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2025/2/20 0:00:00","PubModel":"Epub","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
A multigraph is uniformly -edge-connected if there are exactly edge-disjoint paths between any pair of vertices. For example, a uniformly -edge-connected graph is obtained from a -edge-connected graph by collapsing the nodes connected by more than edge-disjoint paths into supernodes. We characterize the class of uniformly 3-edge-connected graphs, giving a synthesis involving two operations by which every uniformly 3-edge-connected multigraph can be generated. Slightly modified syntheses give the planar uniformly 3-edge-connected graphs and the uniformly 3-edge-connected graphs with the fewest possible edges, generalizing the well-known Harary graphs. In proving the correctness of the synthesis, we also show the existence of a particular type of induced, non-separating cycle in near 3-regular graphs, which is of interest in its own right.
期刊介绍：
The aim of Discrete Applied Mathematics is to bring together research papers in different areas of algorithmic and applicable discrete mathematics as well as applications of combinatorial mathematics to informatics and various areas of science and technology. Contributions presented to the journal can be research papers, short notes, surveys, and possibly research problems. The "Communications" section will be devoted to the fastest possible publication of recent research results that are checked and recommended for publication by a member of the Editorial Board. The journal will also publish a limited number of book announcements as well as proceedings of conferences. These proceedings will be fully refereed and adhere to the normal standards of the journal.
Potential authors are advised to view the journal and the open calls-for-papers of special issues before submitting their manuscripts. Only high-quality, original work that is within the scope of the journal or the targeted special issue will be considered.
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