{"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":12811,"journal":{"name":"Graphs and Combinatorics","volume":"87 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Graphs and Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00373-024-02772-x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
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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。
期刊介绍：
Graphs and Combinatorics is an international journal devoted to research concerning all aspects of combinatorial mathematics. In addition to original research papers, the journal also features survey articles from authors invited by the editorial board.
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