For a nontrivial connected graph G, let c : V (G) → N be a vertex coloring of G where adjacent vertices may be colored the same and let V 1 , V 2 ,. .. , V k be the resulting color classes. For a vertex v of G, the metric color code of v is the k-vector code(v) = (d(v, V 1), d(v, V 2), · · · , d(v, V k)), where d(v, V i) is the minimum distance between v and a vertex in V i. If code(u) = code(v...

In a graph G = (V, E), a vertex v ∈ V is said to distinguish two vertices x and y if dG(v, x) 6= dG(v, y). A set S ⊆ V is said to be a local metric generator for G if any pair of adjacent vertices of G is distinguished by some element of S. A minimum local metric generator is called a local metric basis and its cardinality the local metric dimension of G. A set S ⊆ V is said to be a simultaneou...