On the metric dimension of Cartesian products of graphs

A set of vertices S resolves a graph 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.