If \(S=\{v_1,\ldots , v_k\}\) is an ordered subset of vertices of a connected graph G and e is an edge of G, then the vector \(r_G(e|S) = (d_G(v_1,e), \ldots , d_G(v_k,e))\) is the edge metric S-representation of e. If the vertices of G have pairwise different edge metric S-representations, then S is an edge metric generator for G. The cardinality of a smallest edge metric generator is the edge metric dimension \(\mathrm{edim}(G)\) of G. A general sharp upper bound on the edge metric dimension of hierarchical products \(G(U)\sqcap H\) is proved. Exact formula is derived for the case when \(|U| = 1\). An integer linear programming model for computing the edge metric dimension is proposed. Several examples are provided which demonstrate how these two methods can be applied to obtain the edge metric dimensions of some applicable graphs.

如果\（S = \ {v_1，\ ldots，v_k \} \）是连通图G的顶点的有序子集，而e是G的边，则向量\（r_G（e | S）=（d_G （V_1，E），\ ldots，D_G（V_K，E））\）是边缘度量小号的-表示ë。如果G的顶点具有成对的边缘度量S-表示，则S是G的边缘度量生成器。最小边缘度量产生器的基数是边缘度量尺寸\（\ mathrm {EDIM}（G）\）的ģ。证明了层次积\（G（U）\ sqcap H \）的边缘度量维度上的一般尖锐上界。对于\（| U | = 1 \）的情况，可以得出精确的公式。提出了一种用于计算边缘度量维数的整数线性规划模型。提供了几个示例，这些示例演示了如何将这两种方法应用于获得某些适用图形的边缘度量尺寸。