The metric dimension of the lexicographic product of graphs

Strong resolving partitions for strong product graphs and Cartesian product graphs

“…The parameters related to metric dimension have not escaped to this. For instance, the metric dimensions of Cartesian product graphs, strong product graphs, lexicographic product graphs, corona product graphs and rooted product graphs were studied in [1,9,21,24,29] and [13], respectively. The strong metric dimensions of Cartesian product graph, strong product graph, corona product graphs and rooted product graphs were studied in [14][15][16]23] and [17], respectively.…”

The Local Metric Dimension of the Lexicographic Product of Graphs

“…A set S ⊂ V of vertices in a graph G = (V, E) is said to be an adjacency generator for G if for every two vertices x, y ∈ V − S there exists s ∈ S such that s is adjacent to exactly one of x and y. A minimum cardinality adjacency generator is called an adjacency basis of G, and its cardinality the adjacency dimension of G, denoted by adim(G) [15]. The concepts of local adjacency generator, local adjacency basis and local adjacency dimension are defined by analogy, and the local adjacency dimension of a graph G is denoted by adim l (G).…”

The Simultaneous Local Metric Dimension of Graph Families

“…Applications of the metric dimension to the navigation of robots in networks are discussed in [4] and applications to chemistry in [5,6]. This invariant was studied further in a number of other papers including, for instance [7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

“…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, denoted by dim l (G). Additionally, Jannesari and Omoomi [16] introduced the concept of adjacency resolving sets as a result of considering the two-distance in V(G), which is defined as d G,2 (u, v) = min{d G (u, v), 2} for any two vertices u, v ∈ V(G). A set of vertices S such that any pair of vertices of V(G) is distinguished by an element s in S considering the two-distance in V(G) is called an adjacency generator for G. If we only ask S to distinguish the pairs of adjacent vertices, we call S a local adjacency generator.…”