On the Metric Representation of the Vertices of a Graph

The metric representation of a vertex u in a connected graph G respect to an ordered vertex subset W={ω _1, … , ω _n}⊂ V(G) is the vector of distances r(u| W)=(d(u,ω _1), … , d(u,ω _n)) . A vertex subset W is a resolving set of G if r(u| W) r(v| W) , for every u,v∈ V(G) with u v . Thus, a resolving set with n elements provides a set of metric representation vectors S⊂ℤ^n with cardinal equal to the order of the graph. In this paper, we address the reverse point of view; that is, we characterize the finite subsets S⊂ℤ^n that are realizable as the set of metric representation vectors of a graph G with respect to some resolving set W. We also explore the role that the strong product of paths plays in this context. Moreover, in the case n=2 , we characterize the sets S⊂ℤ^2 that are uniquely realizable as the set of metric representation vectors of a graph G with respect to a resolving set W.