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.