Edge metric dimensions via hierarchical product and integer linear programming

Abstract

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.