On the Metric Dimension of Cartesian Products of Graphs

José Cáceres, Carmen Hernando, Mercé Mora, Ignacio M. Pelayo, María L. Puertas, Carlos Seara, David R. Wood

Published 2005-07-26, updated 2006-03-02Version 3

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.