On the Metric Dimension of Infinite Graphs

Details On the Metric Dimension of Infinite Graphs On the Metric Dimension of Infinite Graphs

2009-04-30

arxiv.org/abs/0904.4826v1

Mathematics

Combinatorics

17 pages, 17 figures, 3 tables, 16 references

Scientific paper

A set of vertices $S$ \emph{resolves} a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The \emph{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.

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