A linear algorithm for finding a minimum dominating set in a cactus

AbstractA dominating set in a graph G = (V,E) is a set of vertices D such that every vertex in V−D removal results in a disconnected graph. A block in a graph G is a maximal connected subgraph of G having no cutvertices. A cactus is a graph in which each block is either an edge or a cycle. In this paper we present a linear time algorithm for finding a minimum order dominating set in a cactus

New resolvability parameters of graphs

In this paper we introduce two concepts related to resolvability and the metric dimension of graphs. The kth dimension of a graph G is the maximum cardinality of a subset of vertices of G that is resolved by a set S of order k. Some first results are obtained. A pair of vertices u, v is totally resolved by a third vertex x if (Formula presented.) A total resolving set in G is a set S such that each pair of vertices of G is totally resolved be a vertex in S. The total metric dimension of a graph is the minimum cardinality of a total resolving set. We determine the total metric dimension of paths, cycles, and grids, and of the 3-cube, and the Petersen graph

The 2-dimension of a tree

Let x and y be two distinct vertices in a connected graph G. The x, ylocation of a vertex w is

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

Introduction to the design and analysis of algorithms

xi, 371 p.; 21 cm