Décomposition et Domination dans les Graphes
Fairouz Beggas · HAL (Le Centre pour la Communication Scientifique Directe) · 2017
Graph theory is considered as a field exploring a large variety of proof techniques in discrete mathematics. Thus, the various problems treated in this theory have applications in a lot of other scientific fields such as computer science, physics, sociology, game theory, etc. In this thesis, three major problems are considered: the multidecomposition of multigraphs, the [1, 2]- domination and the edge monitoring. The fact that these three problems are of different nature allowed us to explore several proof techniques in this thesis. The first part of this thesis deals with a popular aspect of research in graph theory called graph decomposition. Intuitively, a decomposition into subgraphs allows us to describe the original graph with a set of copies of these subgraphs. In this part, we give a particular interest to the multidecomposition of a complete multigraph into edge disjoint stars and cycles. Thus, we investigate the problem of (Sk, Ck)-multidecomposition of the complete multigraph and give necessary and sufficient conditions for such a multidecomposition to exist. The second and third parts are the most important parts in terms of effort and spent time. They are devoted to problems related to domination in graphs. The original domination problem is to find a minimum set of vertices such that every vertex outside the dominating set is adjacent to at least one vertex from the dominating set. Many variants of theoretical and practical interest have been studied in the literature. The second studied problem is called the [i, j]-domination in graphs. This problem was introduced by Chellali et al. in 2013. In addition to the properties of domination, this variant has the particularity that each non-dominating vertex should be adjacent to at least i dominating vertices but also to at most j of them. We particularly focus on the [1, 2]-domination. It has been shown that the problem remains NP-complete. We are interested to study this problem on a particular graph namely the generalized Petersen graph. This graph was introduced by Watkins and has a lot of interesting properties. Moreover, several graph theoretical parameters have been studied on this graph class because of it unique structure. In addition, a study of the [1, 2]-total domination is also proposed at the end of this part. The last problem is a new variant called edge monitoring problem and was introduced by Dong et al. in 2008. It consists to find a set of vertices that monitors (dominates) the edge set of a graph such as a vertex monitors an edge if it forms a triangle with it i.e. it dominates both extremities of the edge. An edge can be monitored by one or more vertices. Three variants of the problem are considered in this part namely the edge monitoring, uniform edge monitoring and weighted edge monitoring. The essence of this problem lies on its combinatorial aspect and its range of applications in networks; especially wireless sensor networks. This problem is known to be NP-hard. Given the complexity of this kind of problems, we are first interested by a theoretical study: variants of the problem, bounds, characterizations, etc. We give more in depth studies of the problem for several graph classes