On the total \(\{ k\} \)-domination and total \(\{ k\} \)-domatic number of graphs
type: Item
instance of: scholarly article
title: On the total \(\{ k\} \)-domination and total \(\{ k\} \)-domatic number of graphs
zbMATH Open document ID: 1261.05074
author: Lutz Volkmann
author: H. Aram
author: Seyed Mahmood Sheikholeslami
published in: Bulletin of the Malaysian Mathematical Sciences Society. Second Series
publication date: 2013-03-11T00:00:00Z
full work available at URL: vol36 1 4.html
review text: A total \(\{k\}\)-dominating function of a graph \(G\) (without isolated vertices) is a function \(f : V(G) \to \{0,1,\dots,k\}\) such that \(\sum_{\{u\mid uv\in E(G)\}}f(u)\geq k\) for all \(v\in V(G)\); the weight of \(f\) is \(\omega(f) = \sum_{v\in V(G)}f(v)\). The total \(\{k\}\)-domination number, \(\gamma_t^{\{k\}}(G)\), is the minimum weight of all such functions \(f\). A set \(\{f_1,f_2,\dots,f_d\}\) of \(k\) distinct such functions such that \(\sum_{i=1}^df_i(v)\leq k\) for all \(v\in V(G)\) is a total \(\{k\}\)-dominating family on \(G\), whose maximum number of members is the total \(\{k\}\)-domatic number, \(d_t^{\{k\}}(G)\) [\textit{S. M. Sheikholeslami} and \textit{L. Volkmann}, J.\ Comb.\ Optim.\ 23, No.\ 2, 252--260 (2012; Zbl 1243.90229)]. The Cartesian product of a cycle and a path is called a cylinder. From the authors' abstract: ``In this paper we present bounds for the total \(\{k\}\)-domination number and total \(\{k\}\)-domatic number. In addition, we determine the total \(\{k\}\)-domatic number of cylinders, and we give a Nordhaus-Gaddum type result.''
Mathematics Subject Classification ID: 05C69
zbMATH DE Number: 6143133
zbMATH Keywords: total \(\{k\}\)-dominating function
zbMATH Keywords: total \(\{k\}\)-domination number
zbMATH Keywords: total \(\{k\}\)-domatic number
zbMATH Keywords: minimum weight
zbMATH Keywords: Nordhaus-Gaddum type result
zbMATH Keywords: cylinder
zbMATH Keywords: poduct of cycle and path
reviewed by: William G. Brown
MaRDI profile type: Publication
Recommended article: On the \(\{k\}\)-domatic number of graphs
Recommended article: Total \(\{k\}\)-domination in special graphs
Recommended article: Q5382133
Recommended article: Q5251983
Recommended article: The total \(\{k\}\)-domatic number of a graph
Recommended article: Total and paired domination numbers of cylinders
Recommended article: The \(\{k\}\)-domatic number of a graph
Recommended article: The total {k}-domatic number of digraphs
Recommended article: Bounds for the domination number of toroidal grid graphs
Recommended article: The domination numbers of cylindrical grid graphs
description: scientific article; zbMATH DE number 6143133