On Dimension Partitions in Discrete Metric Spaces

Let (W,d) be a metric space and S = {s1 . . . sk} an ordered list of subsets of W . The distance between p ∈ W and si ∈ S is d(p, si) = min{ d(p, q) : q ∈ si }. S is a resolving set forW if d(x, si) = d(y, si) for all si implies x = y. A metric basis is a resolving set of minimal cardinality, named the metric dimension of (W,d). The metric dimension has been extensively studied in the literature when W is a graph and S is a subset of points (classical case) or when S is a partition of W ; the latter is known as the partition dimension problem. We have recently studied the case where W is the discrete space Z for a subset of points; in this paper, we tackle the partition dimension problem for classical Minkowski distances as well as polyhedral gauges and chamfer norms in Z.