Metric graphic sets

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For an ordered subset W = {w1, w2, . . . , wk} of vertices in a connected graph G and a vertex v of G, the metric representation of v with respect to W is the k-vector r(v|W) = (d(v, w1), d(v, w2), . . . , d(v, wk)), where d(v, wi) is the distance of the vertices v and wi in G. The set W is called a resolving set of G if r(u|W) = r(v|W) implies u = v. The metric dimension of G, denoted by β(G), is the minimum cardinality of a resolving set of G, and a resolving set of G with cardinality equal to its metric dimension is called a metric basis of G. A set T of vectors is called a positive lattice set if all the coordinates in each vector of T are positive integers. A positive lattice set T consisting of n k-vectors is called a metric graphic set if there exists a simple connected graph G of order n + k with β(G) = k such that T = {r(ui|S) : ui ∈ V (G)\S, 1 ≤ i ≤ n} for some metric basis S = {s1, s2, . . . , sk} of G. If such G exists, then we say G is a metric graphic realization of T. In this paper, we introduce the concept of metric graphic sets anchored on the concept of metric dimension and provide some characterizations. We also give necessary and sufficient conditions for any positive lattice set consisting of 2 k-vectors to be a metric graphic set. We provide an upper bound for the sum of all the coordinates of any metric graphic set and enumerate some properties of positive lattice sets consisting of n 2-vectors that are not metric graphic sets.