Protecting a Graph with Mobile Guards

Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve

Guarding Networks Through Heterogeneous Mobile Guards

In this article, the issue of guarding multi-agent systems against a sequence of intruder attacks through mobile heterogeneous guards (guards with different ranges) is discussed. The article makes use of graph theoretic abstractions of such systems in which agents are the nodes of a graph and edges represent interconnections between agents. Guards represent specialized mobile agents on specific nodes with capabilities to successfully detect and respond to an attack within their guarding range. Using this abstraction, the article addresses the problem in the context of eternal security problem in graphs. Eternal security refers to securing all the nodes in a graph against an infinite sequence of intruder attacks by a certain minimum number of guards. This paper makes use of heterogeneous guards and addresses all the components of the eternal security problem including the number of guards, their deployment and movement strategies. In the proposed solution, a graph is decomposed into clusters and a guard with appropriate range is then assigned to each cluster. These guards ensure that all nodes within their corresponding cluster are being protected at all times, thereby achieving the eternal security in the graph.Comment: American Control Conference, Chicago, IL, 201

Um problema de dominação eterna : classes de grafos, métodos de resolução e perspectiva prática

Orientadores: Cid Carvalho de Souza, Orlando LeeTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: O problema do conjunto dominante m-eterno é um problema de otimização em grafos que tem sido muito estudado nos últimos anos e para o qual se têm listado aplicações em vários domínios. O objetivo é determinar o número mínimo de guardas que consigam defender eternamente ataques nos vértices de um grafo; denominamos este número o índice de dominação m-eterna do grafo. Nesta tese, estudamos o problema do conjunto dominante m-eterno: lidamos com aspectos de natureza teórica e prática e abordamos o problema restrito a classes especícas de grafos e no caso geral. Examinamos o problema do conjunto dominante m-eterno com respeito a duas classes de grafos: os grafos de Cayley e os conhecidos grafos de intervalo próprios. Primeiramente, mostramos ser inválido um resultado sobre os grafos de Cayley presente na literatura, provamos que o resultado é válido para uma subclasse destes grafos e apresentamos outros achados. Em segundo lugar, fazemos descobertas em relação aos grafos de intervalo próprios, incluindo que, para estes grafos, o índice de dominação m-eterna é igual à cardinalidade máxima de um conjunto independente e, por consequência, o índice de dominação m-eterna pode ser computado em tempo linear. Tratamos de uma questão que é fundamental para aplicações práticas do problema do conjunto dominante m-eterno, mas que tem recebido relativamente pouca atenção. Para tanto, introduzimos dois métodos heurísticos, nos quais formulamos e resolvemos modelos de programação inteira e por restrições para computar limitantes ao índice de dominação m-eterna. Realizamos um vasto experimento para analisar o desempenho destes métodos. Neste processo, geramos um benchmark contendo 750 instâncias e efetuamos uma avaliação prática de limitantes ao índice de dominação m-eterna disponíveis na literatura. Por m, propomos e implementamos um algoritmo exato para o problema do conjunto dominante m-eterno e contribuímos para o entendimento da sua complexidade: provamos que a versão de decisão do problema é NP-difícil. Pelo que temos conhecimento, o algoritmo proposto foi o primeiro método exato a ser desenvolvido e implementado para o problema do conjunto dominante m-eternoAbstract: The m-eternal dominating set problem is a graph-protection optimization problem that has been an active research topic in the recent years and reported to have applications in various domains. It asks for the minimum number of guards that can eternally defend attacks on the vertices of a graph; this number is called the m-eternal domination number of the graph. In this thesis, we study the m-eternal dominating set problem by dealing with aspects of theoretical and practical nature and tackling the problem restricted to specic classes of graphs and in the general case. We examine the m-eternal dominating set problem for two classes of graphs: Cayley graphs and the well-known proper interval graphs. First, we disprove a published result on the m-eternal domination number of Cayley graphs, show that the result is valid for a subclass of these graphs, and report further ndings. Secondly, we present several discoveries regarding proper interval graphs, including that, for these graphs, the m- eternal domination number equals the maximum size of an independent set and, as a consequence, the m-eternal domination number can be computed in linear time. We address an issue that is fundamental to practical applications of the m-eternal dominating set problem but that has received relatively little attention. To this end, we introduce two heuristic methods, in which we propose and solve integer and constraint programming models to compute bounds on the m-eternal domination number. By performing an extensive experiment to validate the features of these methods, we generate a 750-instance benchmark and carry out a practical evaluation of bounds for the m-eternal domination number available in the literature. Finally, we propose and implement an exact algorithm for the m-eternal dominating set problem and contribute to the knowledge on its complexity: we prove that the decision version of the problem is NP-hard. As far as we know, the proposed algorithm was the first developed and implemented exact method for the m-eternal dominating set problemDoutoradoCiência da ComputaçãoDoutor em Ciência da Computação141964/2013-8CAPESCNP

On the Complexity of Co-secure Dominating Set Problem

A set $D \subseteq V$ of a graph $G=(V, E)$ is a dominating set of $G$ if every vertex $v\in V\setminus D$ is adjacent to at least one vertex in $D.$ A set $S \subseteq V$ is a co-secure dominating set (CSDS) of a graph $G$ if $S$ is a dominating set of $G$ and for each vertex $u \in S$ there exists a vertex $v \in V\setminus S$ such that $uv \in E$ and $(S\setminus \{u\}) \cup \{v\}$ is a dominating set of $G$. The minimum cardinality of a co-secure dominating set of $G$ is the co-secure domination number and it is denoted by $\gamma_{cs}(G)$. Given a graph $G=(V, E)$, the minimum co-secure dominating set problem (Min Co-secure Dom) is to find a co-secure dominating set of minimum cardinality. In this paper, we strengthen the inapproximability result of Min Co-secure Dom for general graphs by showing that this problem can not be approximated within a factor of $(1- \epsilon)\ln |V|$ for perfect elimination bipartite graphs and star convex bipartite graphs unless P=NP. On the positive side, we show that Min Co-secure Dom can be approximated within a factor of $O(\ln |V|)$ for any graph $G$ with $\delta(G)\geq 2$. For $3$-regular and $4$-regular graphs, we show that Min Co-secure Dom is approximable within a factor of $\dfrac{8}{3}$ and $\dfrac{10}{3}$, respectively. Furthermore, we prove that Min Co-secure Dom is APX-complete for $3$-regular graphs.Comment: 12 pages, 2 figure

Secure domination number of kk-subdivision of graphs

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is the domination number of $G$. A dominating set $D$ is called a secure dominating set of $G$, if for every $u\in V-D$, there exists a vertex $v\in D$ such that $uv \in E$ and $D-\{v\}\cup\{u\}$ is a dominating set of $G$. The cardinality of a smallest secure dominating set of $G$, denoted by $\gamma_s(G)$, is the secure domination number of $G$. For any $k \in \mathbb{N}$, the $k$-subdivision of $G$ is a simple graph $G^{\frac{1}{k}}$ which is constructed by replacing each edge of $G$ with a path of length $k$. In this paper, we study the secure domination number of $k$-subdivision of $G$.Comment: 10 Pages, 8 Figure

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view