In this paper, by applying the notion of metric dimension of a graph, we introduce a unique code for each vertex of a given graph such that the length of codes be as small as possible and each vertex can be identified by its code. Indeed, for an ordered subset S = {v1, . . . , vk} of vertices in a connected graph G and a vertex v of G, the l-metric S-representation of a vertex v ∈ V (G) is the vector rGl (v|S) = (aG(v, v1), . . . , aG(v, vk)) , where aG(v, vi) = min{dG(v, vi), l}, i ∈ {1, . . . , k}. S ⊆ V (G) is a l-metric generator for G if the vertices of G have pairwise different l-metric S-representations. A l-metric generator of smallest order is a l-metric basis for G, its order being the l-metric dimension diml(G) of G. We prove that the length of codes obtained from metric dimensions for some graphs is larger than the length of codes obtained from our new definition of metric dimensions and the difference can be large unbounded.

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