{"title":"A note on connectivity in directed graphs","authors":"Stelios Stylianou","doi":"arxiv-2409.12137","DOIUrl":null,"url":null,"abstract":"We say a directed graph $G$ on $n$ vertices is irredundant if the removal of\nany edge reduces the number of ordered pairs of distinct vertices $(u,v)$ such\nthat there exists a directed path from $u$ to $v$. We determine the maximum\npossible number of edges such a graph can have, for every $n \\in \\mathbb{N}$.\nWe also characterize the cases of equality. This resolves, in a strong form, a\nquestion of Crane and Russell.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"50 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.12137","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 say a directed graph $G$ on $n$ vertices is irredundant if the removal of
any edge reduces the number of ordered pairs of distinct vertices $(u,v)$ such
that there exists a directed path from $u$ to $v$. We determine the maximum
possible number of edges such a graph can have, for every $n \in \mathbb{N}$.
We also characterize the cases of equality. This resolves, in a strong form, a
question of Crane and Russell.
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