{"title":"关于图形中的完全隔离","authors":"Geoffrey Boyer, Wayne Goddard, Michael A. Henning","doi":"10.1007/s00010-024-01057-1","DOIUrl":null,"url":null,"abstract":"<div><p>An isolating set in a graph is a set <i>S</i> of vertices such that removing <i>S</i> and its neighborhood leaves no edge; it is total isolating if <i>S</i> induces a subgraph with no vertex of degree 0. We show that most graphs have a partition into two disjoint total isolating sets and characterize the exceptions. Using this we show that apart from the 7-cycle, every connected graph of order <span>\\(n\\ge 4\\)</span> has a total isolating set of size at most <i>n</i>/2, which is best possible.\n</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 2","pages":"623 - 633"},"PeriodicalIF":0.7000,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On total isolation in graphs\",\"authors\":\"Geoffrey Boyer, Wayne Goddard, Michael A. Henning\",\"doi\":\"10.1007/s00010-024-01057-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>An isolating set in a graph is a set <i>S</i> of vertices such that removing <i>S</i> and its neighborhood leaves no edge; it is total isolating if <i>S</i> induces a subgraph with no vertex of degree 0. We show that most graphs have a partition into two disjoint total isolating sets and characterize the exceptions. Using this we show that apart from the 7-cycle, every connected graph of order <span>\\\\(n\\\\ge 4\\\\)</span> has a total isolating set of size at most <i>n</i>/2, which is best possible.\\n</p></div>\",\"PeriodicalId\":55611,\"journal\":{\"name\":\"Aequationes Mathematicae\",\"volume\":\"99 2\",\"pages\":\"623 - 633\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-04-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Aequationes Mathematicae\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00010-024-01057-1\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Aequationes Mathematicae","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00010-024-01057-1","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
图中的隔离集是这样一个顶点集 S：移除 S 及其邻域不会留下任何边；如果 S 引发了一个没有顶点度为 0 的子图，那么它就是全隔离集。 我们证明了大多数图都有一个分割成两个互不相交的全隔离集，并描述了例外情况的特征。利用这一点，我们证明除了 7 循环之外，每个阶为 \(n\ge 4\) 的连通图最多都有一个大小为 n/2 的全孤立集，这是最好的情况。
An isolating set in a graph is a set S of vertices such that removing S and its neighborhood leaves no edge; it is total isolating if S induces a subgraph with no vertex of degree 0. We show that most graphs have a partition into two disjoint total isolating sets and characterize the exceptions. Using this we show that apart from the 7-cycle, every connected graph of order \(n\ge 4\) has a total isolating set of size at most n/2, which is best possible.
期刊介绍：
aequationes mathematicae is an international journal of pure and applied mathematics, which emphasizes functional equations, dynamical systems, iteration theory, combinatorics, and geometry. The journal publishes research papers, reports of meetings, and bibliographies. High quality survey articles are an especially welcome feature. In addition, summaries of recent developments and research in the field are published rapidly.
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