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

On defensive alliances and line graphs

Let $\Gamma$ be a simple graph of size $m$ and degree sequence $\delta_1\ge \delta_2\ge ... \ge \delta_n$. Let ${\cal L}(\Gamma)$ denotes the line graph of $\Gamma$. The aim of this paper is to study mathematical properties of the alliance number, ${a}({\cal L}(\Gamma)$, and the global alliance number, $\gamma_{a}({\cal L}(\Gamma))$, of the line graph of a simple graph. We show that $\lceil\frac{\delta_{n}+\delta_{n-1}-1}{2}\rceil \le {a}({\cal L}(\Gamma))\le \delta_1.$ In particular, if $\Gamma$ is a $\delta$-regular graph ($\delta>0$), then $a({\cal L}(\Gamma))=\delta$, and if $\Gamma$ is a $(\delta_1,\delta_2)$-semiregular bipartite graph, then $a({\cal L}(\Gamma))=\lceil \frac{\delta_1+\delta_2-1}{2} \rceil$. As a consequence of the study we compare $a({\cal L}(\Gamma))$ and ${a}(\Gamma)$, and we characterize the graphs having $a({\cal L}(\Gamma))<4$. Moreover, we show that the global-connected alliance number of ${\cal L}(\Gamma)$ is bounded by $\gamma_{ca}({\cal L}(\Gamma)) \ge \lceil\sqrt{D(\Gamma)+m-1}-1\rceil,$ where $D(\Gamma)$ denotes the diameter of $\Gamma$, and we show that the global alliance number of ${\cal L}(\Gamma)$ is bounded by $\gamma_{a}({\cal L}(\Gamma))\geq \lceil\frac{2m}{\delta_{1}+\delta_{2}+1}\rceil$. The case of strong alliances is studied by analogy

Defensive alliances in graphs: a survey

A set $S$ of vertices of a graph $G$ is a defensive $k$-alliance in $G$ if every vertex of $S$ has at least $k$ more neighbors inside of $S$ than outside. This is primarily an expository article surveying the principal known results on defensive alliances in graph. Its seven sections are: Introduction, Computational complexity and realizability, Defensive $k$-alliance number, Boundary defensive $k$-alliances, Defensive alliances in Cartesian product graphs, Partitioning a graph into defensive $k$-alliances, and Defensive $k$-alliance free sets.Comment: 25 page