Local edge metric dimensions via corona products and integer linear programming

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∈E(G) and a subset S of V(G), the representation of e with respect to S={x₁,…,x_k} is the vector r_G(e|S)=(d₁,…,d_k), where d_i=d_G(e,x_i) for i∈[k]. If r_G(e|S)≠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 edim_ℓ(G), is the minimum cardinality among all local edge metric generators in G. For two non-trivial graphs G and H, we determine edim_ℓ(G⋄H) in the edge corona product G⋄H and we determine edim_ℓ(G∘H) in the corona product G⋄H. We also formulate the problem of computing edim_ℓ(G) as an integer linear programming model.