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

Global defensive k-alliances in graphs

Let $\Gamma=(V,E)$ be a simple graph. For a nonempty set $X\subseteq V$, and a vertex $v\in V$, $\delta_{X}(v)$ denotes the number of neighbors $v$ has in $X$. A nonempty set $S\subseteq V$ is a \emph{defensive $k$-alliance} in $\Gamma=(V,E)$ if $\delta_S(v)\ge \delta_{\bar{S}}(v)+k,$ $\forall v\in S.$ A defensive $k$-alliance $S$ is called \emph{global} if it forms a dominating set. The \emph{global defensive $k$-alliance number} of $\Gamma$, denoted by $\gamma_{k}^{a}(\Gamma)$, is the minimum cardinality of a defensive $k$-alliance in $\Gamma$. We study the mathematical properties of $\gamma_{k}^{a}(\Gamma)$

Quorum Colorings of Graphs

Let $G = (V,E)$ be a graph. A partition $\pi = \{V_1, V_2, \ldots, V_k \}$ of the vertices $V$ of $G$ into $k$ {\it color classes} $V_i$, with $1 \leq i \leq k$, is called a {\it quorum coloring} if for every vertex $v \in V$, at least half of the vertices in the closed neighborhood $N[v]$ of $v$ have the same color as $v$. In this paper we introduce the study of quorum colorings of graphs and show that they are closely related to the concept of defensive alliances in graphs. Moreover, we determine the maximum quorum coloring of a hypercube

Global defensive k-alliances in directed graphs: combinatorial and computational issues

In this paper we define the global defensive k-alliance (number) in a digraph D, and give several bounds on this parameter with characterizations of all digraphs attaining the bounds. In particular, for the case k = -1, we give a lower (an upper) bound on this parameter for directed trees (rooted trees). Moreover, the characterization of all directed trees (rooted trees) for which the equality holds is given. Finally, we show that the problem of finding the global defensive k-alliance number of a digraph is NP-hard for any suitable non-negative value of k, and in contrast with it, we also show that finding a minimum global defensive (-1)-alliance for any rooted tree is polynomial-time solvable

Offensive alliances in cubic graphs

An offensive alliance in a graph $\Gamma=(V,E)$ is a set of vertices $S\subset V$ where for every vertex $v$ in its boundary it holds that the majority of vertices in $v$'s closed neighborhood are in $S$. In the case of strong offensive alliance, strict majority is required. An alliance $S$ is called global if it affects every vertex in $V\backslash S$, that is, $S$ is a dominating set of $\Gamma$. The global offensive alliance number $\gamma_o(\Gamma)$ (respectively, global strong offensive alliance number $\gamma_{\hat{o}}(\Gamma)$) is the minimum cardinality of a global offensive (respectively, global strong offensive) alliance in $\Gamma$. If $\Gamma$ has global independent offensive alliances, then the \emph{global independent offensive alliance number} $\gamma_i(\Gamma)$ is the minimum cardinality among all independent global offensive alliances of $\Gamma$. In this paper we study mathematical properties of the global (strong) alliance number of cubic graphs. For instance, we show that for all connected cubic graph of order $n$, $$\frac{2n}{5}\le \gamma_i(\Gamma)\le \frac{n}{2}\le \gamma_{\hat{o}}(\Gamma)\le \frac{3n}{4} \le \gamma_{\hat{o}}({\cal L}(\Gamma))=\gamma_{o}({\cal L}(\Gamma))\le n,$$ where ${\cal L}(\Gamma)$ denotes the line graph of $\Gamma$. All the above bounds are tight