{"title":"论森林中最小支配集和总支配集的数量","authors":"Jan Petr, Julien Portier, Leo Versteegen","doi":"10.1002/jgt.23107","DOIUrl":null,"url":null,"abstract":"<p>We show that the maximum number of minimum dominating sets of a forest with domination number <span></span><math>\n \n <mrow>\n <mi>γ</mi>\n </mrow></math> is at most <span></span><math>\n \n <mrow>\n <msup>\n <msqrt>\n <mn>5</mn>\n </msqrt>\n \n <mi>γ</mi>\n </msup>\n </mrow></math> and construct for each <span></span><math>\n \n <mrow>\n <mi>γ</mi>\n </mrow></math> a tree with domination number <span></span><math>\n \n <mrow>\n <mi>γ</mi>\n </mrow></math> that has more than <span></span><math>\n \n <mrow>\n <mfrac>\n <mn>2</mn>\n \n <mn>5</mn>\n </mfrac>\n \n <msup>\n <msqrt>\n <mn>5</mn>\n </msqrt>\n \n <mi>γ</mi>\n </msup>\n </mrow></math> minimum dominating sets. Furthermore, we disprove a conjecture about the number of minimum total dominating sets in forests by Henning, Mohr and Rautenbach.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 4","pages":"976-993"},"PeriodicalIF":0.9000,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23107","citationCount":"0","resultStr":"{\"title\":\"On the number of minimum dominating sets and total dominating sets in forests\",\"authors\":\"Jan Petr, Julien Portier, Leo Versteegen\",\"doi\":\"10.1002/jgt.23107\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We show that the maximum number of minimum dominating sets of a forest with domination number <span></span><math>\\n \\n <mrow>\\n <mi>γ</mi>\\n </mrow></math> is at most <span></span><math>\\n \\n <mrow>\\n <msup>\\n <msqrt>\\n <mn>5</mn>\\n </msqrt>\\n \\n <mi>γ</mi>\\n </msup>\\n </mrow></math> and construct for each <span></span><math>\\n \\n <mrow>\\n <mi>γ</mi>\\n </mrow></math> a tree with domination number <span></span><math>\\n \\n <mrow>\\n <mi>γ</mi>\\n </mrow></math> that has more than <span></span><math>\\n \\n <mrow>\\n <mfrac>\\n <mn>2</mn>\\n \\n <mn>5</mn>\\n </mfrac>\\n \\n <msup>\\n <msqrt>\\n <mn>5</mn>\\n </msqrt>\\n \\n <mi>γ</mi>\\n </msup>\\n </mrow></math> minimum dominating sets. Furthermore, we disprove a conjecture about the number of minimum total dominating sets in forests by Henning, Mohr and Rautenbach.</p>\",\"PeriodicalId\":16014,\"journal\":{\"name\":\"Journal of Graph Theory\",\"volume\":\"106 4\",\"pages\":\"976-993\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-04-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23107\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Graph Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23107\",\"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":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23107","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
On the number of minimum dominating sets and total dominating sets in forests
We show that the maximum number of minimum dominating sets of a forest with domination number is at most and construct for each a tree with domination number that has more than minimum dominating sets. Furthermore, we disprove a conjecture about the number of minimum total dominating sets in forests by Henning, Mohr and Rautenbach.
期刊介绍:
The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences.
A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .