图中的克制{2}支配

K. Haghparast, J. Amjadi, M. Chellali, S. M. Sheikholeslami
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引用次数: 0

摘要

图$G=(V,E)$上的一个约束$\{2\}$支配函数(R $\{2\}$ -DF)是一个函数$f:V\rightarrow\{0,1,2\}$,这样\textrm{(i)}$f(N[v])\geq2$对于所有$v\in V,$,其中$N[v]$是包含$v$和所有与$v;$\textrm{相邻的顶点的集合;(ii)}由$f$下分配的顶点0诱导的子图没有孤立的顶点。R $\{2\}$ -DF的权值是其在所有顶点上的函数值之和,约束的$\{2\}$ -支配数$\gamma_{r\{2\}}(G)$是R $\{2\}$ -DF在$G.$上的最小权值,本文开始了约束的$\{2\}$ -支配数的研究。我们首先证明了计算这个参数的问题是np完全的,即使限制在二部图上也是如此。然后我们给出这个参数的各种边界。特别地,我们根据树的阶数、叶数和支持顶点的数量,建立了树的约束$\{2\}$ -支配数$T$的上界和下界。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Restrained {2}-domination in graphs
A restrained $\{2\}$-dominating function (R$\{2\}$-DF) on a graph $G=(V,E)$ is a function $f:V\rightarrow\{0,1,2\}$ such that : \textrm{(i)} $f(N[v])\geq2$ for all $v\in V,$ where $N[v]$ is the set containing $v$ and all vertices adjacent to $v;$ \textrm{(ii)} the subgraph induced by the vertices assigned 0 under $f$ has no isolated vertices. The weight of an R$\{2\}$-DF is the sum of its function values over all vertices, and the restrained $\{2\}$-domination number $\gamma_{r\{2\}}(G)$ is the minimum weight of an R$\{2\}$-DF on $G.$ In this paper, we initiate the study of the restrained $\{2\}$-domination number. We first prove that the problem of computing this parameter is NP-complete, even when restricted to bipartite graphs. Then we give various bounds on this parameter. In particular, we establish upper and lower bound on the restrained $\{2\}$-domination number of a tree $T$ in terms of the order, the numbers of leaves and support vertices.
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