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 $G\,\square\,H$. We prove that the metric dimension of $G\,\square\,G$ 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 $G\,\square\,H$ 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 $G\,\square\,G$ is unbounded.

https://users.monash.edu.au/~davidwo/papers/MetricDimension-arXiv.pdf

On the Metric Dimension of Cartesian Products of Graphs

The base size of a permutation group, and the metric dimension of a graph, are two of a number of related parameters of groups, graphs, coherent configurations and association schemes. They have been repeatedly re-defined with different terminology in various different areas, including computational group theory and the graph isomorphism problem. We survey results on these parameters in their many incarnations, and propose a consistent terminology for them. We also present some new results, including on the base sizes of wreath products in the product action, and on the metric dimension of Johnson and Kneser graphs.

/pdf/base-size-metric-dimension-and-other-invariants-of-groups-ndmpltrqpe.pdf

Base size, metric dimension and other invariants of groups and graphs

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.

A set of vertices $S$ resolves a connected 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 $G$. Let ${\cal G}_{\beta,D}$ be the set of graphs with metric dimension $\beta$ and diameter $D$. It is well-known that the minimum order of a graph in ${\cal G}_{\beta,D}$ is exactly $\beta+D$. The first contribution of this paper is to characterise the graphs in ${\cal G}_{\beta,D}$ with order $\beta+D$ for all values of $\beta$ and $D$. Such a characterisation was previously only known for $D\leq2$ or $\beta\leq1$. The second contribution is to determine the maximum order of a graph in ${\cal G}_{\beta,D}$ for all values of $D$ and $\beta$. Only a weak upper bound was previously known.

/pdf/extremal-graph-theory-for-metric-dimension-and-diameter-270ct32xot.pdf

Ortrud R. Oellerman

Papers

The metric dimension and metric independence of a graph