Efficient location of resources in cylindrical networks
type: Item
instance of: scholarly article
title: Efficient location of resources in cylindrical networks
zbMATH Open document ID: 1391.90367
author: José Juan Carreño
author: José Antonio Martínez
author: M. L. Puertas
published in: Symmetry
publication date: 2018-06-25T00:00:00Z
review text: Summary: The location of resources in a network satisfying some optimization property is a classical combinatorial problem that can be modeled and solved by using graphs. Key tools in this problem are the domination-type properties, which have been defined and widely studied in different types of graph models, such as undirected and directed graphs, finite and infinite graphs, simple graphs and hypergraphs. When the required optimization property is that every node of the network must have access to exactly one node with the desired resource, the appropriate models are the efficient dominating sets. However, the existence of these vertex sets is not guaranteed in every graph, so relaxing some conditions is necessary to ensure the existence of some kind of dominating sets, as efficient as possible, in a larger number of graphs. In this paper, we study independent \([1, 2]\)-sets, a generalization of efficient dominating sets defined by \textit{M. Chellali} et al. [Australas. J. Comb. 59, 144--156 (2014; Zbl 1296.05148)], in the case of cylindrical networks. It is known that efficient dominating sets exist in very special cases of cylinders, but the particular symmetry of these graphs will allow us to provide regular patterns that guarantee the existence of independent \([1, 2]\)-sets in every cylinder, except in one single case, and to compute exact values of the optimal parameter, the independent \([1, 2]\)-number, in cylinders of selected sizes.
Mathematics Subject Classification ID: 90B80
Mathematics Subject Classification ID: 90B10
Mathematics Subject Classification ID: 90C35
Mathematics Subject Classification ID: 05C69
zbMATH DE Number: 6893780
zbMATH Keywords: Cartesian product of graphs
zbMATH Keywords: efficient domination
zbMATH Keywords: tropical matrix algebra
describes a project that uses: CSparse
MaRDI profile type: Publication
full work available at URL: sym10010024
OpenAlex ID: W2784053604
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DOI: 10.3390/SYM10010024
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description: scientific article; zbMATH DE number 6893780