Powerful alliances in graphs

AbstractFor a graph G=(V,E), a non-empty set S⊆V is a defensive alliance if for every vertex v in S, v has at most one more neighbor in V−S than it has in S, and S is an offensive alliance if for every v∈V−S that has a neighbor in S, v has more neighbors in S than in V−S. A powerful alliance is both defensive and offensive. We initiate the study of powerful alliances in graphs

A New Self-Stabilizing Maximal Matching Algorithm

The maximal matching problem has received considerable attention in the self-stabilizing community. Previous work has given different self-stabilizing algorithms that solves the problem for both the adversarial and fair distributed daemon, the sequential adversarial daemon, as well as the synchronous daemon. In the following we present a single self-stabilizing algorithm for this problem that unites all of these algorithms in that it stabilizes in the same number of moves as the previous best algorithms for the sequential adversarial, the distributed fair, and the synchronous daemon. In addition, the algorithm improves the previous best moves complexities for the distributed adversarial daemon from O(n^2) and O(delta m) to O(m) where n is the number of processes, m is thenumber of edges, and delta is the maximum degree in the graph

Self-stabilizing algorithms for Connected Vertex Cover and Clique decomposition problems

In many wireless networks, there is no fixed physical backbone nor centralized network management. The nodes of such a network have to self-organize in order to maintain a virtual backbone used to route messages. Moreover, any node of the network can be a priori at the origin of a malicious attack. Thus, in one hand the backbone must be fault-tolerant and in other hand it can be useful to monitor all network communications to identify an attack as soon as possible. We are interested in the minimum \emph{Connected Vertex Cover} problem, a generalization of the classical minimum Vertex Cover problem, which allows to obtain a connected backbone. Recently, Delbot et al.~\cite{DelbotLP13} proposed a new centralized algorithm with a constant approximation ratio of $2$ for this problem. In this paper, we propose a distributed and self-stabilizing version of their algorithm with the same approximation guarantee. To the best knowledge of the authors, it is the first distributed and fault-tolerant algorithm for this problem. The approach followed to solve the considered problem is based on the construction of a connected minimal clique partition. Therefore, we also design the first distributed self-stabilizing algorithm for this problem, which is of independent interest

An Introduction to Temporal Graphs: An Algorithmic Perspective

A \emph{temporal graph} is, informally speaking, a graph that changes with time. When time is discrete and only the relationships between the participating entities may change and not the entities themselves, a temporal graph may be viewed as a sequence $G_1,G_2\ldots,G_l$ of static graphs over the same (static) set of nodes $V$. Though static graphs have been extensively studied, for their temporal generalization we are still far from having a concrete set of structural and algorithmic principles. Recent research shows that many graph properties and problems become radically different and usually substantially more difficult when an extra time dimension in added to them. Moreover, there is already a rich and rapidly growing set of modern systems and applications that can be naturally modeled and studied via temporal graphs. This, further motivates the need for the development of a temporal extension of graph theory. We survey here recent results on temporal graphs and temporal graph problems that have appeared in the Computer Science community

Nearly Perfect Sets in Graphs

In a graph G = (V; E), a set of vertices S is nearly perfect if every vertex in V \Gamma S is adjacent to at most one vertex in S. Nearly perfect sets are closely related to 2-packings of graphs, strongly stable sets, dominating sets and efficient dominating sets. We say a nearly perfect set S is 1-minimal if for every vertex u in S, the set S \Gamma fug is not nearly perfect. Similarly, a nearly perfect set S is 1-maximal if for every vertex u in V \Gamma S, S [ fug is not a nearly perfect set. Lastly, we define n p (G) to be the minimum cardinality of a 1-maximal nearly perfect set, and N p (G) to be the maximum cardinality of a 1-minimal nearly perfect set. In this paper we calculate these parameters for some classes of graphs. We show that the decision problem for n p (G) is NP-complete; we give a linear algorithm for determining n p (T ) for any tree T ; and we show that N p (G) can be calculated for any graph G in polynomial time. 1 Introduction Let G = (V; E) be a graph. We say..