The k-metric dimension of graphs: a general approach

Let $(X,d)$ be a metric space. A set $S\subseteq X$ is said to be a $k$-metric generator for $X$ if and only if for any pair of different points $u,v\in X$, there exist at least $k$ points $w_1,w_2,\dots w_k\in S$ such that $d(u,w_i)\mathrm{\ne }d(v,w_i),\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{i}\in \{1,\dots \mathrm{k}\}\mathrm{.}$ Let ${\mathcal{R}}_k(X)$ be the set of metric generators for $X$. The $k$-metric dimension ${\dim }_k(X)$ of $(X,d)$ is defined as $${\dim }_k(X)=\inf \{\mathrm{\mid }S\mathrm{\mid }:S\in {\mathcal{R}}_k(X)\}\mathrm{.}$$ Here, we discuss the $k$-metric dimension of $(V,d_t)$, where $V$ is the set of vertices of a simple graph $G$ and the metric $d_t:V\times V\to \mathbb{N}\cup \{0\}$ is defined by $d_t(x,y)=\min \{d(x,y),t\}$ from the geodesic distance $d$ in $G$ and a positive integer $t$. The case $t\ge D(G)$, where $D(G)$ denotes the diameter of $G$, corresponds to the original theory of $k$-metric dimension and the case $t=2$ corresponds to the theory of $k$-adjacency dimension. Furthermore, this approach allows us to extend the theory of $k$-metric dimension to the general case of non-necessarily connected graphs.