A Note on Outer-Independent 2-Rainbow Domination in Graphs

Let G be a graph with vertex set V(G) and f:V(G)→{∅,{1},{2},{1,2}} be a function. We say that f is an outer-independent 2-rainbow dominating function on G if the following two conditions hold: (i)V∅={x∈V(G):f(x)=∅} is an independent set of G. (ii)∪u∈N(v)f(u)={1,2} for every vertex v∈V∅. The outer-independent 2-rainbow domination number of G, denoted by γoir2(G), is the minimum weight ω(f)=∑x∈V(G)|f(x)| among all outer-independent 2-rainbow dominating functions f on G. In this note, we obtain new results on the previous domination parameter. Some of our results are tight bounds which improve the well-known bounds β(G)≤γoir2(G)≤2β(G), where β(G) denotes the vertex cover number of G. Finally, we study the outer-independent 2-rainbow domination number of the join, lexicographic, and corona product graphs. In particular, we show that, for these three product graphs, the parameter achieves equality in the lower bound of the previous inequality chain

Distances and Domination in Graphs

This book presents a compendium of the 10 articles published in the recent Special Issue “Distance and Domination in Graphs”. The works appearing herein deal with several topics on graph theory that relate to the metric and dominating properties of graphs. The topics of the gathered publications deal with some new open lines of investigations that cover not only graphs, but also digraphs. Different variations in dominating sets or resolving sets are appearing, and a review on some networks’ curvatures is also present

Outer-independent total Roman domination in graphs

Given a graph $G$ with vertex set $V$, a function $f:V\rightarrow \{0,1,2\}$ is an outer-independent total Roman dominating function on $G$ if \begin{itemize} \item every vertex $v\in V$ for which $f(v)=0$ is adjacent to at least one vertex $u\in V$ such that $f(u)=2$, \item every vertex $x\in V$ for which $f(x)\ge 1$ is adjacent to at least one vertex $y\in V$ such that $f(y)\ge 1$, and \item any two different vertices $a,b$ for which $f(a)=f(b)=0$ are not adjacent. \end{itemize} The minimum weight $\omega(f)=\sum_{w\in V}f(w)$ of any outer-independent total Roman dominating function on $G$ is the outer-independent total Roman domination number, $\gamma_{oitR}(G)$, of $G$. In this article, we introduce the concepts above and begin with the study of its combinatorial and computational properties. For instance, we give several closed relationships between this parameter and other ones related with domination in graphs. We prove that computing $\gamma_{oitR}$ of a graph $G$ is an NP-hard problem. In addition, we present some closed formulae for $\gamma_{oitR}(G)$ in the cases $G$ represents some special families of graphs