Given a simple connected undirected graph G = (V, E), for u, v ∈ V d(u, v) denotes the distance between u and v in G, i.e. the length of a shortest u−v path. A vertex x of the graph G is said to resolve two vertices u and v of G if d(x, u)= d(x, v). An ordered vertex set R = {x1, x2, ... , xk} of G is a resolving set of G if every two distinct vertices of G are resolved by some vertex of R. A metric basis of G is a resolving set of the minimum cardinality. The metric dimension of G, denoted by (G), is the cardinality of its metric basis.

Similarly, for a given connected graph G, a vertex w ∈ V and an edge uv ∈ E, the distance between the vertex w and the edge uv is dened as d(W, uv) = min{d(W, u), d(W, v)}. A vertex w ∈ V resolves two edges e1 and e2 (e1, e2 ∈ E), if d(W, e1)= d(W, e2). A set S of vertices in a connected graph G is an edge metric generator for G if every two edges of G are resolved by some vertex of S. The smallest cardinality of an edge metric generator of G is called the edge metric dimension and is denoted by E(G). An edge metric basis for G is an edge metric generator of G with cardinality E(G). Given an edge e ∈ E and an ordered vertex set S = {x1, x2, ... , xk}, the k-touple r(e, S) = (d(e, x1), d(e, x2), ... , d(e, xk)) is called the edge metric representation of e with respect to S.