{"title":"On the trees with maximum Cardinality-Redundance number","authors":"Elham Mohammadi, N. J. Rad","doi":"10.56415/csjm.v32.03","DOIUrl":null,"url":null,"abstract":"A vertex $v$ is said to be over-dominated by a set $S$ if $|N[u]\\cap S|\\geq 2$. The cardinality--redundance of $S$, $CR(S)$, is the number of vertices of $G$ that are over-dominated by $S$. The cardinality--redundance of $G$, $CR(G)$, is the minimum of $CR(S)$ taken over all dominating sets $S$. A dominating set $S$ with $CR(S) = CR(G)$ is called a $CR(G)$-set. In this paper, we prove an upper bound for the cardinality--redundance in trees in terms of the order and the number of leaves, and characterize all trees achieving equality for the proposed bound.","PeriodicalId":42293,"journal":{"name":"Computer Science Journal of Moldova","volume":null,"pages":null},"PeriodicalIF":0.2000,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computer Science Journal of Moldova","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.56415/csjm.v32.03","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
A vertex $v$ is said to be over-dominated by a set $S$ if $|N[u]\cap S|\geq 2$. The cardinality--redundance of $S$, $CR(S)$, is the number of vertices of $G$ that are over-dominated by $S$. The cardinality--redundance of $G$, $CR(G)$, is the minimum of $CR(S)$ taken over all dominating sets $S$. A dominating set $S$ with $CR(S) = CR(G)$ is called a $CR(G)$-set. In this paper, we prove an upper bound for the cardinality--redundance in trees in terms of the order and the number of leaves, and characterize all trees achieving equality for the proposed bound.