{"title":"The primality graph of critical 3-hypergraphs","authors":"","doi":"10.1007/s00373-024-02772-x","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>Given a 3-hypergraph <em>H</em>, a subset <em>M</em> of <em>V</em>(<em>H</em>) is a module of <em>H</em> if for each <span> <span>\\(e\\in E(H)\\)</span> </span> such that <span> <span>\\(e\\cap M\\ne \\emptyset \\)</span> </span> and <span> <span>\\(e{\\setminus } M\\ne \\emptyset \\)</span> </span>, there exists <span> <span>\\(m\\in M\\)</span> </span> such that <span> <span>\\(e\\cap M=\\{m\\}\\)</span> </span> and for every <span> <span>\\(n\\in M\\)</span> </span>, we have <span> <span>\\((e{\\setminus }\\{m\\})\\cup \\{n\\}\\in E(H)\\)</span> </span>. For example, <span> <span>\\(\\emptyset \\)</span> </span>, <em>V</em>(<em>H</em>) and <span> <span>\\(\\{v\\}\\)</span> </span>, where <span> <span>\\(v\\in V(H)\\)</span> </span>, are modules of <em>H</em>, called trivial. A 3-hypergraph is prime if all its modules are trivial. Furthermore, a prime 3-hypergraph is critical if all its induced subhypergraphs, obtained by removing one vertex, are not prime. Lastly, we associate with a prime 3-hypergraph its primality graph the edges of which are the unordered pairs of vertices whose removal provides a prime induced subhypergraph. We characterize the critical 3-hypergraphs together with their primality graph.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-06","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/s00373-024-02772-x","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
Given a 3-hypergraph H, a subset M of V(H) is a module of H if for each \(e\in E(H)\) such that \(e\cap M\ne \emptyset \) and \(e{\setminus } M\ne \emptyset \), there exists \(m\in M\) such that \(e\cap M=\{m\}\) and for every \(n\in M\), we have \((e{\setminus }\{m\})\cup \{n\}\in E(H)\). For example, \(\emptyset \), V(H) and \(\{v\}\), where \(v\in V(H)\), are modules of H, called trivial. A 3-hypergraph is prime if all its modules are trivial. Furthermore, a prime 3-hypergraph is critical if all its induced subhypergraphs, obtained by removing one vertex, are not prime. Lastly, we associate with a prime 3-hypergraph its primality graph the edges of which are the unordered pairs of vertices whose removal provides a prime induced subhypergraph. We characterize the critical 3-hypergraphs together with their primality graph.
Abstract Given a 3-hypergraph H, a subset M of V(H) is a module of H if for each \(e\in E(H)\) such that \(e\cap Mne \emptyset \) and \(e{setminus } Mne \emptyset \) 、There exists \(m\in M\) such that \(e\cap M=\{m\}\) and for every \(n\in M\) , we have \((e{setminus }\{m\})\cup \{n\}\in E(H)\) .例如， \(emptyset \), V(H) and \({v\}\), where \(v\in V(H)\).都是 H 的模块，称为微模块。如果一个 3-hypergraph 的所有模块都是琐碎的，那么它就是素数。此外，如果删除一个顶点后得到的所有诱导子超图都不是质数，那么质数 3-hypergraph 就是临界图。最后，我们将素数 3-hypergraph 与它的素数图联系起来，素数图的边是无序的顶点对，移除这些顶点可以得到素数诱导子超图。我们将临界 3-hypergraph 连同它们的基元图一起描述。
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