Local edge metric dimensions via corona products and integer linear programming

期刊论文
发表日期：

2024年8月4日

查看全文

Fateme Amini,
Michael A. Henning,
Mostafa Tavakoli

摘要

Let G be a connected graph. The distance between two vertices u and v in G, denoted by $d_G(u,v)$ $d_{G} (u, v)$ , is the number of edges in a shortest path from u to v, while the distance between an edge $e = xy$ $e = x y$ and a vertex v in G is $d_G(e,v) = \min \{d_G(x,v),d_G(y,v)\}$ $d_{G} (e, v) = min {d_{G} (x, v), d_{G} (y, v)}$ . For an edge $e \in E(G)$ $e \in E (G)$ and a subset S of V(G), the representation of e with respect to $S=\{x_1,\ldots ,x_k\}$ $S = {x_{1}, \dots, x_{k}}$ is the vector $r_G(e|S) =(d_1,\ldots ,d_k)$ $r_{G} (e | S) = (d_{1}, \dots, d_{k})$ , where $d_i=d_G(e,x_i)$ $d_{i} = d_{G} (e, x_{i})$ for $i \in [k]$ $i \in [k]$ . If $r_G(e|S)\ne r_G(f|S)$ $r_{G} (e | S) \neq r_{G} (f | S)$ 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 $\mathrm{edim_\ell }(G)$ ${edim}_{ℓ} (G)$ , is the minimum cardinality among all local edge metric generators in G. For two non-trivial graphs G and H, we determine $\mathrm{edim_\ell }(G \diamond H)$ ${edim}_{ℓ} (G ⋄ H)$ in the edge corona product $G \diamond H$ $G ⋄ H$ and we determine $\mathrm{edim_\ell }(G\circ H)$ ${edim}_{ℓ} (G \circ H)$ in the corona product $G\diamond H$ $G ⋄ H$ . We also formulate the problem of computing $\mathrm{edim_\ell }(G)$ ${edim}_{ℓ} (G)$ as an integer linear programming model.