On the Metric Dimension of Infinite Graphs |
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| Institution: | 1. Faculty of civil engineering, architecture and geodesy, University of Split, Croatia;2. University of Ljubljana, FMF, 1000 Ljubljana, Slovenia;3. Faculty of Information Studies, 8000 Novo Mesto, Slovenia;1. Faculty of Electrical Engineering and Computer Science, University of Maribor, Smetanova 17, Maribor SI-2000, Slovenia;2. Faculty of Natural Sciences and Mathematics, University of Maribor, Koro?ka cesta 160, Maribor SI-2000, Slovenia;3. Departamento de Estadística e Investigación Operativa, Escuela Superior de Ingeniería, Universidad de Cádiz, Av. de la Universidad 10, Campus Universitario de Puerto Real 11519, Spain;4. Institute of Mathematics, Physics and Mechanics, Jadranska 19, Ljubljana SI-1000, Slovenia;5. Departamento de Matemáticas, Escuela Politécnica Superior de Algeciras, Universidad de Cádiz, Av. Ramón Puyol s/n, Algeciras 11202, Spain |
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| Abstract: | A set of vertices S resolves a connected graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of a graph G is the minimum cardinality of a resolving set. In this paper we undertake the metric dimension of infinite locally finite graphs, i.e., those infinite graphs such that all its vertices have finite degree. We give some necessary conditions for an infinite graph to have finite metric dimension and characterize infinite trees with finite metric dimension. We also establish some general results about the metric dimension of the Cartesian product of finite and infinite graphs, and obtain the metric dimension of the Cartesian product of several families of graphs. |
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