On the metric dimension of infinite graphs

Authors:
J. CáCeres;C. Hernando;M. Mora;I. M. Pelayo;M. L. Puertas
Affiliations:
Universidad de Almería, Almería, Spain;Universitat Politècnica de Catalunya, Barcelona, Spain;Universitat Politècnica de Catalunya, Barcelona, Spain;Universitat Politècnica de Catalunya, Barcelona, Spain;Universidad de Almería, Almería, Spain
Venue:
Discrete Applied Mathematics
Year:
2012

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Abstract

A set of vertices Sresolves a 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 study the metric dimension of infinite graphs such that all its vertices have finite degree. We give necessary conditions for those graphs to have finite metric dimension and characterize infinite trees with finite metric dimension. We also establish some results about the metric dimension of the cartesian product of finite and infinite graphs, and give the metric dimension of the cartesian product of several families of graphs.