On the Metric Dimension of Cartesian Products of Graphs
Combinatorics
2007-06-13 v3
Abstract
A set S of vertices in a graph G resolves G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. This paper studies the metric dimension of cartesian products G*H. We prove that the metric dimension of G*G is tied in a strong sense to the minimum order of a so-called doubly resolving set in G. Using bounds on the order of doubly resolving sets, we establish bounds on G*H for many examples of G and H. One of our main results is a family of graphs G with bounded metric dimension for which the metric dimension of G*G is unbounded.
Keywords
Cite
@article{arxiv.math/0507527,
title = {On the Metric Dimension of Cartesian Products of Graphs},
author = {José Cáceres and Carmen Hernando and Mercé Mora and Ignacio M. Pelayo and María L. Puertas and Carlos Seara and David R. Wood},
journal= {arXiv preprint arXiv:math/0507527},
year = {2007}
}