k-metric resolvability in graphs

The k-Metric Dimension of a Unicyclic Graph

“…However, the particular case of k = 2 had also been previously defined in [4]. Recent studies on the k-metric dimension of a graph can be consulted in [1,[5][6][7][8][9]. Independently of the aforementioned articles, k-metric dimension was studied in [10][11][12] with a computer science oriented approach.…”

On the k-metric dimension of metric spaces

“…The vertex set V of a graph G supports a natural graph metric d, where d(u, v) is the smallest number of edges that can be used to join u to v. Some basic results on the k-metric dimension of a graph have recently been obtained in [7][8][9][10][11]. Moreover, it was shown in [20] that the problem of computing the k-metric dimension of a graph is NP-complete.…”

“…The theory was developed further in 2013 for general metric spaces [1]. More recently, the theory of metric dimension has been generalised, again in the context of graph theory, to the notion of a k-metric dimension, where k is any positive integer, and where the case k = 1 corresponds to the original theory [7][8][9][10][11]. Here we develop the idea of the k-metric dimension both in graph theory and in metric spaces.…”

On the adjacency dimension of graphs

“…The concept of k-metric generator introduced by Estrada-Moreno, Yero and Rodríguez-Velázquez [4,6], is a natural extension of the concept of metric generator. A set S ⊆ V is said to be a k-metric generator for G if and only if any pair of vertices of G is distinguished by at least k elements of S, i.e., for any pair of different vertices u, v ∈ V , there exist at least k vertices w 1 , w 2 , ..., w k ∈ S such that d G (u, w i ) = d G (v, w i ), for every i ∈ {1, ..., k}.…”