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 $\gamma_{k}^{a}(\Gamma)$

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

Open k-monopolies in graphs: complexity and related concepts

Closed monopolies in graphs have a quite long range of applications in several problems related to overcoming failures, since they frequently have some common approaches around the notion of majorities, for instance to consensus problems, diagnosis problems or voting systems. We introduce here open $k$-monopolies in graphs which are closely related to different parameters in graphs. Given a graph $G=(V,E)$ and $X\subseteq V$, if $\delta_X(v)$ is the number of neighbors $v$ has in $X$, $k$ is an integer and $t$ is a positive integer, then we establish in this article a connection between the following three concepts: - Given a nonempty set $M\subseteq V$ a vertex $v$ of $G$ is said to be $k$-controlled by $M$ if $\delta_M(v)\ge \frac{\delta_V(v)}{2}+k$. The set $M$ is called an open $k$-monopoly for $G$ if it $k$-controls every vertex $v$ of $G$. - A function $f: V\rightarrow \{-1,1\}$ is called a signed total $t$-dominating function for $G$ if $f(N(v))=\sum_{v\in N(v)}f(v)\geq t$ for all $v\in V$. - A nonempty set $S\subseteq V$ is a global (defensive and offensive) $k$-alliance in $G$ if $\delta_S(v)\ge \delta_{V-S}(v)+k$ holds for every $v\in V$. In this article we prove that the problem of computing the minimum cardinality of an open $0$-monopoly in a graph is NP-complete even restricted to bipartite or chordal graphs. In addition we present some general bounds for the minimum cardinality of open $k$-monopolies and we derive some exact values.Comment: 18 pages, Discrete Mathematics & Theoretical Computer Science (2016

Partitioning A Graph In Alliances And Its Application To Data Clustering

Any reasonably large group of individuals, families, states, and parties exhibits the phenomenon of subgroup formations within the group such that the members of each group have a strong connection or bonding between each other. The reasons of the formation of these subgroups that we call alliances differ in different situations, such as, kinship and friendship (in the case of individuals), common economic interests (for both individuals and states), common political interests, and geographical proximity. This structure of alliances is not only prevalent in social networks, but it is also an important characteristic of similarity networks of natural and unnatural objects. (A similarity network defines the links between two objects based on their similarities). Discovery of such structure in a data set is called clustering or unsupervised learning and the ability to do it automatically is desirable for many applications in the areas of pattern recognition, computer vision, artificial intelligence, behavioral and social sciences, life sciences, earth sciences, medicine, and information theory. In this dissertation, we study a graph theoretical model of alliances where an alliance of the vertices of a graph is a set of vertices in the graph, such that every vertex in the set is adjacent to equal or more vertices inside the set than the vertices outside it. We study the problem of partitioning a graph into alliances and identify classes of graphs that have such a partition. We present results on the relationship between the existence of such a partition and other well known graph parameters, such as connectivity, subgraph structure, and degrees of vertices. We also present results on the computational complexity of finding such a partition. An alliance cover set is a set of vertices in a graph that contains at least one vertex from every alliance of the graph. The complement of an alliance cover set is an alliance free set, that is, a set that does not contain any alliance as a subset. We study the properties of these sets and present tight bounds on their cardinalities. In addition, we also characterize the graphs that can be partitioned into alliance free and alliance cover sets. Finally, we present an approximate algorithm to discover alliances in a given graph. At each step, the algorithm finds a partition of the vertices into two alliances such that the alliances are strongest among all such partitions. The strength of an alliance is defined as a real number p, such that every vertex in the alliance has at least p times more neighbors in the set than its total number of neighbors in the graph). We evaluate the performance of the proposed algorithm on standard data sets

Alliance free sets in Cartesian product graphs

Let $G=(V,E)$ be a graph. For a non-empty subset of vertices $S\subseteq V$, and vertex $v\in V$, let $\delta_S(v)=|\{u\in S:uv\in E\}|$ denote the cardinality of the set of neighbors of $v$ in $S$, and let $\bar{S}=V-S$. Consider the following condition: {equation}\label{alliancecondition} \delta_S(v)\ge \delta_{\bar{S}}(v)+k, \{equation} which states that a vertex $v$ has at least $k$ more neighbors in $S$ than it has in $\bar{S}$. A set $S\subseteq V$ that satisfies Condition (\ref{alliancecondition}) for every vertex $v \in S$ is called a \emph{defensive} $k$-\emph{alliance}; for every vertex $v$ in the neighborhood of $S$ is called an \emph{offensive} $k$-\emph{alliance}. A subset of vertices $S\subseteq V$, is a \emph{powerful} $k$-\emph{alliance} if it is both a defensive $k$-alliance and an offensive $(k +2)$-alliance. Moreover, a subset $X\subset V$ is a defensive (an offensive or a powerful) $k$-alliance free set if $X$ does not contain any defensive (offensive or powerful, respectively) $k$-alliance. In this article we study the relationships between defensive (offensive, powerful) $k$-alliance free sets in Cartesian product graphs and defensive (offensive, powerful) $k$-alliance free sets in the factor graphs