‎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 ${\\\\rm edim}(G)$ 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.

Keywords