A New Upper Bound for the Perfect Italian Domination Number of a Tree

IF 0.5 4区 数学 Q3 MATHEMATICS
S. Nazari-Moghaddam, M. Chellali
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引用次数: 1

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].
树的完全意大利支配数的一个新上界
图G上的一个完全意大利支配函数(PIDF)是一个函数f: V (G)→{0,1,2},满足对于f(u) = 0的每个顶点u,分配给u的邻点f的总权值恰好为2。PIDF的权重是其所有顶点上的函数值的和。G的完美义大利支配数,记作γIp(G) \gamma _I^p \left (G \right),是G的PIDF的最小权值。在本文中,我们证明了对于n≥3阶的树T,有n (T)个叶子和s(T)个支持顶点,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,}改进了T.W. Haynes和M.A. Henning在[树的完美意大利支配,离散应用]中给出的前界。数学,260(2019)164-177]。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
2.20
自引率
0.00%
发文量
22
审稿时长
53 weeks
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.
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