Affiliations: [a] Université Clermont-Auvergne, CNRS, Mines de Saint-Étienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont-Ferrand, France. [email protected]
| [b] Department of Mathematics, Universidad de Cádiz, Av. Ramón Puyol s/n, 11202 Algeciras, Spain. [email protected]
Correspondence:
[†]
Address for correspondence: Department of Mathematics and Statistics, 20014 Turun Yliopisto, Finland
Note: [*] Also affiliated at: Department of Mathematics and Statistics, University of Turku, FI-20014, Finland
Abstract: This investigation is firstly focused into showing that two metric parameters represent the same object in graph theory. That is, we prove that the multiset resolving sets and the ID-colorings of graphs are the same thing. We also consider some computational and combinatorial problems of the multiset dimension, or equivalently, the ID-number of graphs. We prove that the decision problem concerning finding the multiset dimension of graphs is NP-complete. We consider the multiset dimension of king grids and prove that it is bounded above by 4. We also give a characterization of the strong product graphs with one factor being a complete graph, and whose multiset dimension is not infinite.
Keywords: multiset resolving set, multiset dimension, ID-colorings, ID-number, king grids
