{"title":"d元树中支配集的数目","authors":"Opeyemi Oyewumi, Adriana Roux, Stephan Wagner","doi":"10.1007/s13370-025-01360-3","DOIUrl":null,"url":null,"abstract":"<div><p>An (unrooted) <i>d</i>-<i>ary tree</i> is a tree in which every internal vertex has degree <span>\\(d+1\\)</span>. In this paper, we show for every fixed <span>\\(d\\ge 2\\)</span> that <i>d</i>-ary caterpillars have the minimum number of dominating sets among <i>d</i>-ary trees of a given order. We also determine the maximum number of dominating sets in binary trees (the special case <span>\\(d=2\\)</span>) and classify the extremal trees, which are also unique.</p></div>","PeriodicalId":46107,"journal":{"name":"Afrika Matematika","volume":"36 3","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s13370-025-01360-3.pdf","citationCount":"0","resultStr":"{\"title\":\"The number of dominating sets in d-ary trees\",\"authors\":\"Opeyemi Oyewumi, Adriana Roux, Stephan Wagner\",\"doi\":\"10.1007/s13370-025-01360-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>An (unrooted) <i>d</i>-<i>ary tree</i> is a tree in which every internal vertex has degree <span>\\\\(d+1\\\\)</span>. In this paper, we show for every fixed <span>\\\\(d\\\\ge 2\\\\)</span> that <i>d</i>-ary caterpillars have the minimum number of dominating sets among <i>d</i>-ary trees of a given order. We also determine the maximum number of dominating sets in binary trees (the special case <span>\\\\(d=2\\\\)</span>) and classify the extremal trees, which are also unique.</p></div>\",\"PeriodicalId\":46107,\"journal\":{\"name\":\"Afrika Matematika\",\"volume\":\"36 3\",\"pages\":\"\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2025-08-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s13370-025-01360-3.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Afrika Matematika\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s13370-025-01360-3\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Afrika Matematika","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s13370-025-01360-3","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
An (unrooted) d-ary tree is a tree in which every internal vertex has degree \(d+1\). In this paper, we show for every fixed \(d\ge 2\) that d-ary caterpillars have the minimum number of dominating sets among d-ary trees of a given order. We also determine the maximum number of dominating sets in binary trees (the special case \(d=2\)) and classify the extremal trees, which are also unique.
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