On the adjacency dimension of graphs

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 $d_{G,2}:V\times V\rightarrow \mathbb{N}\cup \{0\}$, as $d_{G,2}(x,y)=\min\{d_G(x,y),2\},$ where $d_G(x,y)$ is the length of a shortest path between $x$ and $y$ and $\mathbb{N}$ is the set of positive integers. Then $(V,d_{G,2 })$ is a metric space. We say that a set $S\subseteq V$ is a $k$-adjacency generator for $G$ if for every two vertices $x,y\in V$, there exist at least $k$ vertices $w_1,w_2,...,w_k\in S$ such that $$d_{G,2}(x,w_i)\ne d_{G,2}(y,w_i),\; \mbox{for every}\; i\in \{1,...,k\}.$$ 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.