On the Edge Resolvability of Double Generalized Petersen Graphs

For a connected graph G = (V(G), E(G)), let v ∈ V(G) be a vertex and e = uw ∈ E(G) be an edge. The distance between the vertex v and the edge e is given by dG(e, v) = min{dG(u, v), dG(w, v)}. A vertex w ∈ V(G) distinguishes two edges e1, e2 ∈ E(G) if dG(w, e1) ≠ dG(w, e2). A well‐known graph invariant related to resolvability of graph edges, namely, the edge resolving set, is studied for a family of 3‐regular graphs. A set S of vertices in a connected graph G is an edge metric generator for G if every two edges of G are distinguished by some vertex of S. The smallest cardinality of an edge metric generator for G is called the edge metric dimension and is denoted by βe(G). As a main result, we investigate the minimum number of vertices which works as the edge metric generator of double generalized Petersen graphs, DGP(n, 1). We have proved that βe(DGP(n,1)) = 4 when n ≡ 0,1,3(mod4) and βe(DGP(n, 1)) = 3 when n ≡ 2(mod4).