On the metric dimension of Cartesian powers of a graph
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A set of vertices S resolves a graph if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of a graph is the minimum cardinality of a resolving set of the graph. Fix a connected graph G on q > 2 vertices, and let M be the distance matrix of G. We prove that if there exists w ∈Z^q such that ∑_i w_i = 0 and the vector Mw, after sorting its coordinates, is an arithmetic progression with nonzero common difference, then the metric dimension of the Cartesian product of n copies of G is (2+o(1))n/_q n. In the special case that G is a complete graph, our results close the gap between the lower bound attributed to Erdős and Rényi and the upper bound by Chvátal. The main tool is the Möbius function of a certain partially ordered set on N.
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