Global offensive kk-alliances in digraphs

In this paper, we initiate the study of global offensive $k$-alliances in digraphs. Given a digraph $D=(V(D),A(D))$, a global offensive $k$-alliance in a digraph $D$ is a subset $S\subseteq V(D)$ such that every vertex outside of $S$ has at least one in-neighbor from $S$ and also at least $k$ more in-neighbors from $S$ than from outside of $S$, by assuming $k$ is an integer lying between two minus the maximum in-degree of $D$ and the maximum in-degree of $D$. The global offensive $k$-alliance number $\gamma_{k}^{o}(D)$ is the minimum cardinality among all global offensive $k$-alliances in $D$. In this article we begin the study of the global offensive $k$-alliance number of digraphs. For instance, we prove that finding the global offensive $k$-alliance number of digraphs $D$ is an NP-hard problem for any value $k\in \{2-\Delta^-(D),\dots,\Delta^-(D)\}$ and that it remains NP-complete even when restricted to bipartite digraphs when we consider the non-negative values of $k$ given in the interval above. Based on these facts, lower bounds on $\gamma_{k}^{o}(D)$ with characterizations of all digraphs attaining the bounds are given in this work. We also bound this parameter for bipartite digraphs from above. For the particular case $k=1$, an immediate result from the definition shows that $\gamma(D)\leq \gamma_{1}^{o}(D)$ for all digraphs $D$, in which $\gamma(D)$ stands for the domination number of $D$. We show that these two digraph parameters are the same for some infinite families of digraphs like rooted trees and contrafunctional digraphs. Moreover, we show that the difference between $\gamma_{1}^{o}(D)$ and $\gamma(D)$ can be arbitrary large for directed trees and connected functional digraphs

Further results on outer independent 22-rainbow dominating functions of graphs

Let $G=(V(G),E(G))$ be a graph. A function $f:V(G)\rightarrow \mathbb{P}(\{1,2\})$ is a $2$-rainbow dominating function if for every vertex $v$ with $f(v)=\emptyset$, f\big{(}N(v)\big{)}=\{1,2\}. An outer-independent $2$-rainbow dominating function (OI$2$RD function) of $G$ is a $2$-rainbow dominating function $f$ for which the set of all $v\in V(G)$ with $f(v)=\emptyset$ is independent. The outer independent $2$-rainbow domination number (OI$2$RD number) $\gamma_{oir2}(G)$ is the minimum weight of an OI$2$RD function of $G$. In this paper, we first prove that $n/2$ is a lower bound on the OI$2$RD number of a connected claw-free graph of order $n$ and characterize all such graphs for which the equality holds, solving an open problem given in an earlier paper. In addition, a study of this parameter for some graph products is carried out. In particular, we give a closed (resp. an exact) formula for the OI$2$RD number of rooted (resp. corona) product graphs and prove upper bounds on this parameter for the Cartesian product and direct product of two graphs