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.

翻译：本研究首先聚焦于论证图论中两个度量参数代表同一对象，即证明图的多重集分辨集与ID-染色是等价的。我们进一步考虑了多重集维数（等价于图的ID-数）相关的若干计算与组合问题。通过证明，判定图的多重集维数的决策问题属于NP完全问题。针对国王网格图的多重集维数，我们证明了其上确界不超过$4$。此外，给出了其中一个因子为完全图的强乘积图的多重集维数非无穷的刻画条件。