Local edge metric dimensions via corona products and integer linear programming

Abstract

Let G be a connected graph. The distance between two vertices u and v in G, denoted by $d_G(u,v)$, is the number of edges in a shortest path from u to v, while the distance between an edge $e = xy$ and a vertex v in G is $d_G(e,v) = \min \{d_G(x,v),d_G(y,v)\}$. For an edge $e \in E(G)$ and a subset S of V(G), the representation of e with respect to $S=\{x_1,\ldots ,x_k\}$ is the vector $r_G(e|S) =(d_1,\ldots ,d_k)$, where $d_i=d_G(e,x_i)$ for $i \in [k]$. If $r_G(e|S)\ne r_G(f|S)$ for every two adjacent edges e and f of G, then S is called a local edge metric generator for G. The local edge metric dimension of G, denoted by $\mathrm{edim_\ell }(G)$, is the minimum cardinality among all local edge metric generators in G. For two non-trivial graphs G and H, we determine $\mathrm{edim_\ell }(G \diamond H)$ in the edge corona product $G \diamond H$ and we determine $\mathrm{edim_\ell }(G\circ H)$ in the corona product $G\diamond H$. We also formulate the problem of computing $\mathrm{edim_\ell }(G)$ as an integer linear programming model.