On the adjacency dimension of graphs

Abstract

A generator of a metric space is a set S of points in the space with the property that every point of the space is uniquely determined by its distances from the elements of S. Given a simple graph G = (V, E), we define the distance function dG,2 : V × V → N ∪ {0}, as dG,2(x, y) = min{dG(x, y), 2}, where dG(x, y) is the length of a shortest path between x and y and N is the set of positive integers. Then (V, dG,2) is a metric space. We say that a set S ⊆ V is a k-adjacency generator for G if for every two vertices x, y ∈ V , there exist at least k vertices w1, w2, ..., w k ∈ S such that dG,2(x, wi) = dG,2(y, wi), for every i ∈ {1, ..., k}.A minimum cardinality k-adjacency generator is called a k-adjacency basis of G and its cardinality, the k-adjacency dimension of G.In this article we study the problem of finding the k-adjacency dimension of a graph. We give some necessary and sufficient conditions for the existence of a k-adjacency basis of an arbitrary graph G and we obtain general results on the k-adjacency dimension, including general bounds and closed formulae for some families of graphs. In particular, we obtain closed formulae for the k-adjacency dimension of join graphs G + H in terms of the k-adjacency dimension of G and H. These results concern the k-metric dimension, as join graphs have diameter two. As we can expect, the obtained results will become important tools for the study of the k-metric dimension of lexicographic product graphs and corona product graphs. Moreover, several results obtained in this paper need not be restricted to the metric dG,2, they can be expressed in a more general setting, for instance, by using the metric dG,t(x, y) = min{dG(x, y), t} for t ∈ N.

On the k-metric dimension of metric spaces

“…Example 5.4. The Petersen graph, which is illustrated in Figure 1, has dimension sequence (3,4,7,8,9,10, +∞, . .…”

“…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.…”

“…For instance, for the graph shown in Figure 2 the maximum value of k is four. It was shown in [8,9] that for any graph of order n this problem has time complexity of order O(n 3 ). If we consider the discrete metric space (X, d 0 ) (equivalently, a compete graph), then dim 1 (X) = |X| − 1 and dim 2 (X) = |X|.…”

“…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 k-partition dimension of graphs

“…It can be noted that if t is at least the diameter of G, then the metric d G,t is equivalent to d G . Articles made in this sense we can mention [15,19,20,24,23] and the Ph.D. thesis [13]. Particularly, in these last works the k-metric generators of graphs with metric d G,2 were called as k-adjacency generators.…”