{"title":"树的完全意大利支配数的一个新上界","authors":"S. Nazari-Moghaddam, M. Chellali","doi":"10.7151/dmgt.2324","DOIUrl":null,"url":null,"abstract":"Abstract A perfect Italian dominating function (PIDF) on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that for every vertex u with f(u) = 0, the total weight of f assigned to the neighbors of u is exactly two. The weight of a PIDF is the sum of its functions values over all vertices. The perfect Italian domination number of G, denoted γIp(G) \\gamma _I^p\\left( G \\right) , is the minimum weight of a PIDF of G. In this paper, we show that for every tree T of order n ≥ 3, with ℓ(T) leaves and s(T) support vertices, γpI(T) ≤ γIp(T)≤4n-l(T)+2s(T-1)5 \\gamma _I^p\\left( T \\right) \\le {{4n - \\mathcal{l}\\left( T \\right) + 2s\\left( {T - 1} \\right)} \\over 5} , improving a previous bound given by T.W. Haynes and M.A. Henning in [Perfect Italian domination in trees, Discrete Appl. Math. 260 (2019) 164–177].","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"A New Upper Bound for the Perfect Italian Domination Number of a Tree\",\"authors\":\"S. Nazari-Moghaddam, M. Chellali\",\"doi\":\"10.7151/dmgt.2324\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract A perfect Italian dominating function (PIDF) on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that for every vertex u with f(u) = 0, the total weight of f assigned to the neighbors of u is exactly two. The weight of a PIDF is the sum of its functions values over all vertices. The perfect Italian domination number of G, denoted γIp(G) \\\\gamma _I^p\\\\left( G \\\\right) , is the minimum weight of a PIDF of G. In this paper, we show that for every tree T of order n ≥ 3, with ℓ(T) leaves and s(T) support vertices, γpI(T) ≤ γIp(T)≤4n-l(T)+2s(T-1)5 \\\\gamma _I^p\\\\left( T \\\\right) \\\\le {{4n - \\\\mathcal{l}\\\\left( T \\\\right) + 2s\\\\left( {T - 1} \\\\right)} \\\\over 5} , improving a previous bound given by T.W. Haynes and M.A. Henning in [Perfect Italian domination in trees, Discrete Appl. Math. 260 (2019) 164–177].\",\"PeriodicalId\":48875,\"journal\":{\"name\":\"Discussiones Mathematicae Graph Theory\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2022-06-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discussiones Mathematicae Graph Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.7151/dmgt.2324\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discussiones Mathematicae Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.7151/dmgt.2324","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
A New Upper Bound for the Perfect Italian Domination Number of a Tree
Abstract A perfect Italian dominating function (PIDF) on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that for every vertex u with f(u) = 0, the total weight of f assigned to the neighbors of u is exactly two. The weight of a PIDF is the sum of its functions values over all vertices. The perfect Italian domination number of G, denoted γIp(G) \gamma _I^p\left( G \right) , is the minimum weight of a PIDF of G. In this paper, we show that for every tree T of order n ≥ 3, with ℓ(T) leaves and s(T) support vertices, γpI(T) ≤ γIp(T)≤4n-l(T)+2s(T-1)5 \gamma _I^p\left( T \right) \le {{4n - \mathcal{l}\left( T \right) + 2s\left( {T - 1} \right)} \over 5} , improving a previous bound given by T.W. Haynes and M.A. Henning in [Perfect Italian domination in trees, Discrete Appl. Math. 260 (2019) 164–177].
期刊介绍:
The Discussiones Mathematicae Graph Theory publishes high-quality refereed original papers. Occasionally, very authoritative expository survey articles and notes of exceptional value can be published. The journal is mainly devoted to the following topics in Graph Theory: colourings, partitions (general colourings), hereditary properties, independence and domination, structures in graphs (sets, paths, cycles, etc.), local properties, products of graphs as well as graph algorithms related to these topics.