The Simultaneous Metric Dimension of Families Composed by Lexicographic Product Graphs

Abstract

Let G be a graph family defined on a common (labeled) vertex set V . A set S ⊆ V is said to be a simultaneous metric generator for G if for every G ∈ G and every pair of different vertices u, v ∈ V there exists s ∈ S such that dG(s, u) = dG(s, v), where dG denotes the geodesic distance. A simultaneous adjacency generator for G is a simultaneous metric generator under the metric dG,2(x, y) = min{dG(x, y), 2}. A minimum cardinality simultaneous metric (adjacency) generator for G is a simultaneous metric (adjacency) basis, and its cardinality the simultaneous metric (adjacency) dimension of G. Based on the simultaneous adjacency dimension, we study the simultaneous metric dimension of families composed by lexicographic product graphs.

Simultaneous Resolvability in Families of Corona Product Graphs

“…It was also shown in [18] that if G is graph family defined on a common vertex set V , such that for every pair of different vertices u, v ∈ V there exists a graph G ∈ G where u and v are twins, then Sd A (G) = |V | − 1. In particular, any family G containing a complete graph or an empty graph satisfies Sd A (G) = |V | − 1.…”

“…and E w is the set of all edges having at least one vertex in B. It was shown in [18] that B G w ∼ = B G ′ w for any f ∈ S(B) and any graph G ′ ∈ G B,f (G). We define the graph family G B (G), associated to B, as…”

“…Lemma 11. [18] Let G be a connected graph. If D(G) ≥ 6 or G ∈ {P n , C n } for n ≥ 7, or G is a graph of girth g(G) ≥ 5 and minimum degree δ(G) ≥ 3 then for every adjacency generator B for G and every v ∈ V (G), B ⊆ N G (v).…”

“…The following result, presented in [18], characterizes a large number of families of the form G + H whose simultaneous adjacency bases are formed by the union of simultaneous adjacency bases of G and H.…”

“…A smallest simultaneous metric generator for G is a simultaneous metric basis of G, and its cardinality the simultaneous metric dimension of G, is denoted by Sd(G) or explicitly by Sd(G 1 , G 2 , ..., G k ). By analogy, we defined in [18] the concept of simultaneous adjacency generator for G, simultaneous adjacency basis of G and the simultaneous adjacency dimension of G, denoted by Sd A (G) or explicitly by Sd A (G 1 , G 2 , ..., G t ). For instance, the set {1, 3, 6, 7, 8} is a simultaneous adjacency basis of the family G = {G 1 , G 2 , G 3 } shown in Figure 1, while the set {1, 6, 7, 8} is a simultaneous metric basis, so Sd A (G) = 5 and Sd(G) = 4.…”