is a local metric generator for G if every two adjacent vertices of G are distinguished by some vertex of S. A local metric generator with the minimum cardinality is called a local metric basis for G and its cardinality, the local metric dimension of G. In this paper, we study the problem of finding exact values for the local metric dimension of corona product of graphs.

For an ordered subset W = {w 1 , w 2 , . . . w k } of vertices and a vertex u in a connected graph G, the representation of u with respect to W is the ordered k-tuple r(u|W, where d(x, y) represents the distance between the vertices x and y. The set W is a local metric generator for G if every two adjacent vertices of G have distinct representations. A minimum local metric generator is called a local metric basis for G and its cardinality the local metric dimension of G. We show that the computation of the local metric dimension of a graph with cut vertices is reduced to the computation of the local metric dimension of the so-called primary subgraphs. The main results are applied to specific constructions including bouquets of graphs, rooted product graphs, corona product graphs, block graphs and chain of graphs.

A vertex v is said to distinguish two other vertices x and y of a nontrivial connected graph G if the distance from v to x is different from the distance from v to y. A set S ⊆ V (G) is a local metric set for G if every two adjacent vertices of G are distinguished by some vertex of S. A local metric set with minimum cardinality is called a local metric basis for G and its cardinality, the local metric dimension of G, denoted by dim l (G). In this paper we present tight bounds for the local metric dimension of subgraphamalgamation of graphs with special emphasis in the case of subgraphs which are isometric embeddings.

The metric dimension is quite a well-studied graph parameter. Recently, the adjacency dimension and the local metric dimension have been introduced and studied. In this paper, we give a general formula for the local metric dimension of the lexicographic product G • H of a connected graph G of order n and a family H composed by n graphs. We show that the local metric dimension of G • H can be expressed in terms of the true twin equivalence classes of G and the local adjacency dimension of the graphs in H.

Abstract:In a graph G = (V, E), a vertex v ∈ V is said to distinguish two vertices x and y if 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 simultaneous local metric generator for a graph family G = {G 1 , G 2 , . . . , G k }, defined on a common vertex set, if it is a local metric generator for every graph of the family. A minimum simultaneous local metric generator is called a simultaneous local metric basis and its cardinality the simultaneous local metric dimension of G. We study the properties of simultaneous local metric generators and bases, obtain closed formulae or tight bounds for the simultaneous local metric dimension of several graph families and analyze the complexity of computing this parameter.