{"title":"Double total domination number of Cartesian product of paths","authors":"Linyu Li, Jun Yue, Xia Zhang","doi":"10.3934/math.2023479","DOIUrl":null,"url":null,"abstract":"A vertex set $ S $ of a graph $ G $ is called a double total dominating set if every vertex in $ G $ has at least two adjacent vertices in $ S $. The double total domination number $ \\gamma_{\\times 2, t}(G) $ of $ G $ is the minimum cardinality over all the double total dominating sets in $ G $. Let $ G \\square H $ denote the Cartesian product of graphs $ G $ and $ H $. In this paper, the double total domination number of Cartesian product of paths is discussed. We determine the values of $ \\gamma_{\\times 2, t}(P_i\\square P_n) $ for $ i = 2, 3 $, and give lower and upper bounds of $ \\gamma_{\\times 2, t}(P_i\\square P_n) $ for $ i \\geq 4 $.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.8000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"AIMS Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/math.2023479","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A vertex set $ S $ of a graph $ G $ is called a double total dominating set if every vertex in $ G $ has at least two adjacent vertices in $ S $. The double total domination number $ \gamma_{\times 2, t}(G) $ of $ G $ is the minimum cardinality over all the double total dominating sets in $ G $. Let $ G \square H $ denote the Cartesian product of graphs $ G $ and $ H $. In this paper, the double total domination number of Cartesian product of paths is discussed. We determine the values of $ \gamma_{\times 2, t}(P_i\square P_n) $ for $ i = 2, 3 $, and give lower and upper bounds of $ \gamma_{\times 2, t}(P_i\square P_n) $ for $ i \geq 4 $.
如果$ G $中的每个顶点在$ S $中至少有两个相邻顶点,则图$ G $的顶点集$ S $称为双共支配集。$ G $的双总支配数$ \gamma_{\times 2, t}(G) $是$ G $中所有双总支配集的最小基数。设$ G \square H $表示图$ G $和$ H $的笛卡尔积。本文讨论了路径笛卡尔积的双总支配数。我们确定了$ i = 2, 3 $的$ \gamma_{\times 2, t}(P_i\square P_n) $值,并给出了$ i \geq 4 $的$ \gamma_{\times 2, t}(P_i\square P_n) $的下界和上界。
期刊介绍:
AIMS Mathematics is an international Open Access journal devoted to publishing peer-reviewed, high quality, original papers in all fields of mathematics. We publish the following article types: original research articles, reviews, editorials, letters, and conference reports.