{"title":"On the maximum number of minimum total dominating sets in forests","authors":"Michael A. Henning, Elena Mohr, D. Rautenbach","doi":"10.23638/DMTCS-21-3-3","DOIUrl":null,"url":null,"abstract":"Fricke, Hedetniemi, Hedetniemi, and Hutson asked whether every tree with domination number $\\gamma$ has at most $2^\\gamma$ minimum dominating sets. Bien gave a counterexample, which allows to construct forests with domination number $\\gamma$ and $2.0598^\\gamma$ minimum dominating sets. We show that every forest with domination number $\\gamma$ has at most $2.4606^\\gamma$ minimum dominating sets, and that every tree with independence number $\\alpha$ has at most $2^{\\alpha-1}+1$ maximum independent sets.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"2 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2018-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discret. Math. Theor. Comput. Sci.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.23638/DMTCS-21-3-3","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 8
Abstract
Fricke, Hedetniemi, Hedetniemi, and Hutson asked whether every tree with domination number $\gamma$ has at most $2^\gamma$ minimum dominating sets. Bien gave a counterexample, which allows to construct forests with domination number $\gamma$ and $2.0598^\gamma$ minimum dominating sets. We show that every forest with domination number $\gamma$ has at most $2.4606^\gamma$ minimum dominating sets, and that every tree with independence number $\alpha$ has at most $2^{\alpha-1}+1$ maximum independent sets.