Computing the local metric dimension of a graph from the local metric dimension of primary subgraphs

الأصل · EN

For an ordered subset W = {w₁, w₂, wₖ} 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)=(d(v,w₁), d(v,w₂),, d(v,wₖ)), 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.