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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
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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.

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José Cáceres, Carmen Hernando, Mercé Mora, Ignacio M. Pelayo, María L. Puertas, Carlos Seara, David R. Wood. 2006-03-02. On the Metric Dimension of Cartesian Products of Graphs. https://doi.org/10.1137/050641867

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