Relationships Between the 2-Metric Dimension and the 2-Adjacency Dimension in the Lexicographic Product of Graphs

• Published in: Graphs and combinatorics Vol. 32; no. 6; pp. 2367 - 2392
• Main Authors: Estrada-Moreno, A., Yero, I. G., Rodríguez-Velázquez, J. A.
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01.11.2016
• Subjects: Combinatorial analysis; Combinatorics; Engineering Design; Generators; Graphs; Mathematics; Mathematics and Statistics; Original Paper; Shortest-path problems; Texts; 05C76; 05C12; 2-Adjacency dimension; Lexicographic product graphs; 2-Metric dimension
• ISSN: 0911-0119; 1435-5914
• DOI: 10.1007/s00373-016-1736-5

Given a connected simple graph G = ( V ( G ) , E ( G ) ) , a set S ⊆ V ( G ) is said to be a 2-metric generator for G if and only if for any pair of different vertices u , v ∈ V ( G ) , there exist at least two vertices w 1 , w 2 ∈ S such that d G ( u , w i ) ≠ d G ( v , w i ) , for every i ∈ { 1 , 2 } , where d G ( x , y ) is the length of a shortest path between x and y . The minimum cardinality of a 2-metric generator is the 2-metric dimension of G , denoted by dim 2 ( G ) . The metric d G , 2 : V ( G ) × V ( G ) ⟼ N ∪ { 0 } is defined as d G , 2 ( x , y ) = min { d G ( x , y ) , 2 } . Now, a set S ⊆ V ( G ) is a 2-adjacency generator for G , if for every two vertices x , y ∈ V ( G ) there exist at least two vertices w 1 , w 2 ∈ S , such that d G , 2 ( x , w i ) ≠ d G , 2 ( y , w i ) for every i ∈ { 1 , 2 } . The minimum cardinality of a 2-adjacency generator is the 2-adjacency dimension of G , denoted by adim 2 ( G ) . In this article, we obtain closed formulae for the 2-metric dimension of the lexicographic product G ∘ H of two graphs G and H . Specifically, we show that dim 2 ( G ∘ H ) = n · adim 2 ( H ) + f ( G , H ) , where f ( G , H ) ≥ 0 , and determine all the possible values of f ( G ,  H ).