The aim of this paper is to obtain closed formulas for the perfect domination number, the Roman domination number and the perfect Roman domination number of lexicographic product graphs. We show that these formulas can be obtained relatively easily for the case of the first two parameters. The picture is quite different when it concerns the perfect Roman domination number. In this case, we obtain general bounds and then we give sufficient and/or necessary conditions for the bounds to be achieved. We also discuss the case of perfect Roman graphs and we characterize the lexicographic product graphs where the perfect Roman domination number equals the Roman domination number.

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 $d_t$.

A vertex cover of a graph G = ( V, E) is a set X ⊂ V such that each edge of G is incident to at least one vertex of X. The vertex cover number τ( G) is the minimum cardinality of a vertex cover of G. A dominating set D ⊆ V is a weakly connected dominating set of G if the subgraph G[ D] w = ( N[ D], E w ) weakly induced by D, is connected, where E w is the set of all edges having at least one vertex in D. The weakly connected domination number γ w ( G) of G is the minimum cardinality among all weakly connected dominating sets of G. In this article we characterize the graphs where γ w ( G) = τ( G). In particular, we focus our attention on bipartite graphs, regular graphs, unicyclic graphs, block graphs and corona graphs.

A vertex w of a connected graph G strongly resolves two vertices u, v ∈ V ( G), if there exists some shortest u – w path containing v or some shortest v – w path containing u. A set S of vertices is a strong metric generator for G if every pair of vertices of G is strongly resolved by some vertex of S. The smallest cardinality of a strong metric generator for G is called the strong metric dimension of G. In this paper we obtain several tight bounds or closed formulae for the strong metric dimension of the Cartesian sum of graphs in terms of the strong metric dimension, clique number or twins-free clique number of its factor graphs.