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

• Published in: International journal of computer mathematics Vol. 92; no. 4; pp. 686 - 693
• Main Authors: Rodriguez-Velazquez, Juan A, Garcia Gomez, Carlos, Barragan-Ramirez, Gabriel A
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 03.04.2015; Taylor & Francis Ltd
• Subjects: Blocking; Chains; Computation; Construction specifications; corona graphs; Generators; Graph theory; Graphs; local metric dimension; Measurement; metric dimension; primary subgraphs; Representations; rooted product graphs
• ISSN: 0020-7160; 1029-0265
• DOI: 10.1080/00207160.2014.918608

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)=(d(u, w 1 ), d(u, w 2 ), ... , d(u, w k )), 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.