图与小或大罗马{3}-支配数

Nafiseh Ebrahimi, H. A. Ahangar, M. Chellali, S. M. Sheikholeslami
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引用次数: 0

摘要

对于整数$k\geq1,$,在图$G=(V,E)$上,罗马$\{k\}$主导函数(R $\{k\}$ DF)是一个函数$f:V\rightarrow\{0,1,\dots,k\}$,使得对于每个顶点$v\in V$与$f(v)=0$, $\sum_{u\in N(v)}f(u)\geq k$,其中$N(v)$是与$v.$相邻的顶点集,R $\{k\}$ DF的权值是其在整个顶点集上的函数值之和,罗马$\{k\}$ -支配数$\gamma_{\{kR\}}(G)$是anR $\{k\}$ DF在$G$上的最小权重。在本文中,我们将专注于$k=3$的情况,其中对于每个阶为$n\geq3,$$3\leq$$\gamma_{\{kR\}}(G)\leq n.$的连通图,我们对所有阶为$n\geq3$的连通图$G$进行了表征,使得$\gamma_{\{3R\}}(G)\in\{3,n-1,n\},$,并改进了之前的下界和上界。此外,我们证明了对于每棵树$T$ oforder $n\geq3$, $\gamma_{\{3R\}}(T)\geq\gamma(T)+2$,其中$\gamma(T)$是$T,$的支配数,我们描述了达到这个界限的树。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Graphs with small or large Roman {3}-domination number
For an integer $k\geq1,$ a Roman $\{k\}$-dominating function (R$\{k\}$DF) on a graph $G=(V,E)$ is a function $f:V\rightarrow\{0,1,\dots,k\}$ such that for every vertex $v\in V$ with $f(v)=0$, $\sum_{u\in N(v)}f(u)\geq k$, where $N(v)$ is the set of vertices adjacent to $v.$ The weight of an R $\{k\}$DF is the sum of its function values over the whole set of vertices, and the Roman $\{k\}$-domination number $\gamma_{\{kR\}}(G)$ is the minimum weight of an R$\{k\}$DF on $G$. In this paper, we will be focusing on the case $k=3$, where trivially for every connected graphs of order $n\geq3,$ $3\leq$ $\gamma _{\{kR\}}(G)\leq n.$ We characterize all connected graphs $G$ of order $n\geq3$ such that $\gamma_{\{3R\}}(G)\in\{3,n-1,n\},$ and we improve the previous lower and upper bounds. Moreover, we show that for every tree $T$ of order $n\geq3$, $\gamma_{\{3R\}}(T)\geq\gamma(T)+2$, where $\gamma(T)$ is the domination number of $T,$ and we characterize the trees attaining this bound.
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