A set D of vertices of a graph G is a double dominating set of G if |N[v]∩D|≥2 for every v∈V(G), where N[v] represents the closed neighbourhood of v. The double domination number of G is the minimum cardinality among all double dominating sets of G. In this article, we show that if G and H are graphs with no isolated vertex, then for any vertex v∈V(H) there are six possible expressions, in terms of domination parameters of the factor graphs, for the double domination number of the rooted product graph G∘vH. Additionally, we characterize the graphs G and H that satisfy each of these expressions.

Abstract 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 $w_1,w_2, \ldots w_k\in S$ such that $d(u,w_i)\ne d(v,w_i),\; \textrm{for all}\; i\in \{1, \ldots k\}.$ 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 $$\begin{equation*}\dim_k(X)=\inf\{|S|:\, S\in \mathcal{R}_k(X)\}.\end{equation*}$$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 $d_t:V\times V\rightarrow \mathbb{N}\cup \{0\}$ 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. Finally, we analyse the computational complexity of determining the $k$-metric dimension of $(V,d_t)$ for the metric $d_t$.