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表示同一对象。具体而言，我们证明了图的多重集解析集与ID着色实为同一概念。我们还探讨了多重集维数（即图的ID数）相关的若干计算与组合问题。我们证明了关于确定图的多重集维数的判定问题是NP完全的。我们研究了王网格的多重集维数，并证明其上界为4。此外，我们刻画了以完全图为一个因子的强积图，并给出了其多重集维数不为无穷的特征条件。