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：本研究首先聚焦于证明图论中两个度量参数代表同一对象，即我们证明图的多重集分辨集与ID染色本质相同。同时探讨多重集维数（等价于图的ID数）相关的计算与组合问题。我们证明关于寻找图的多重集维数的判定问题是NP完全的。针对皇冠网格图的多重集维数，证明其上界不超过4。最后给出具有完全图因子的强积图（当其多重集维数有限时）的特征刻画。