{"title":"关于具有最大卡片冗余数的树","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":"{\"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}","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}
On the trees with maximum Cardinality-Redundance number
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.