On the metric representation of the vertices of a graph

The version of record is available online at: http://dx.doi.org/10.1007/s40840-023-01582-3
The metric representation of a vertex u in a connected graph G respect to an ordered vertex subset is the vector of distances . A vertex subset W is a resolving set of G if , for every with . Thus, a resolving set with n elements provides a set of metric representation vectors 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 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 , we characterize the sets that are uniquely realizable as the set of metric representation vectors of a graph G with respect to a resolving set W.
This work was partially supported by the grants of the Spanish Ministry of Science and Innovation PID2021-123278OB-I00 and PID2019-104129GB-I00, both funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”, and Gen. Cat. DGR 2021-SGR-00266.
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