Neste trabalho, aplicamos o algoritmo Breadth-First Search para encontrar o tamanho de uma componente conectada no grafo aleatório de Erdos-Rényi. Uma cadeia de Markov é obtida deste procedimento. Apresentamos alguns resultados bem conhecidos sobre o comportamento dessa cadeia de Markov. Combinamos alguns destes resultados para obter uma proposição sobre a probabilidade da componente atingir um determinado tamanho e um resultado de convergência do estado da cadeia neste instante. Posteriormente, aplicamos o teorema de convergência de Darling (2002) a sequência de cadeias de Markov reescaladas e indexadas por N, o número de vértices do grafo, para mostrar que as trajetórias dessas cadeias convergem uniformemente em probabilidade para a solução de uma equação diferencial ordinária. Deste resultado segue a bem conhecida lei fraca dos grandes números para a componente gigante do grafo aleatório de Erdos-Rényi, no caso supercrítico. Além disso, obtemos o limite do fluído para um modelo epidêmico que é uma extensão daquele proposto em Kurtz et al. (2008). / In this work, we apply the Breadth-First Search algorithm to find the size of a connected component of the Erdos-Rényi random graph. A Markov chain is obtained of this procedure. We present some well-known results about the behavior of this Markov chain, and combine some of these results to obtain a proposition about the probability that the component reaches a certain size and a convergence result about the state of the chain at that time. Next, we apply the convergence theorem of Darling (2002) to the sequence of rescaled Markov chains indexed by N, the number of vertices of the graph, to show that the trajectories of these chains converge uniformly in probability to the solution of an ordinary dierential equation. From the latter result follows the well-known weak law of large numbers of the giant component of the Erdos-Renyi random graph, in the supercritical case. Moreover, we obtain the uid limit for an epidemic model which is an extension of that proposed in Kurtz et al. (2008).

Algorithms will be domonstrated for how to embed complete bipartite graphs onto 2xn type grids, where the imimum grid cutwidth is attained.

Après une présentation générale des cartes planaires, nous définissons les polyèdres en coin, étudiés par Eppstein et Mumford. Nous en venons rapidement à introduire les triangulations en coin, qui sont les cartes duales des squelettes des polyèdres en coin, et en donnons quelques propriétés. Nous proposons un algorithme de réalisation de polyèdres en coin de complexité linéaire. Pour cela, l'étude des triangulations en coin conduit à des problèmes d'énumération. Une méthode classique, connue depuis Tutte, donne le résultat voulu en faisant intervenir la série des nombres de Catalan. La recherche d'une explication combinatoire à la présence des nombres de Catalan a rendu souhaitable l'utilisation d'autres méthodes, fondées sur des découpages et des recollements de morceaux de triangulations en coin. Ainsi apparaît la famille des triangulations en amande, qui est une nouvelle représentation des nombres de Catalan, qui est en bijection directe avec la famille des arbres binaires, et qui complète notre algorithme de réalisation de polyèdres en coin. Nous apportons enfin une conclusion à ces travaux en tentant de généraliser nos méthodes à des cartes dont les faces sont de degré fixé, mais quelconque. / After a general presentation of planar maps, we define corner polyhedra, studied by Eppstein and Mumford. We soon introduce corner triangulations, that are dual maps of the skeletons of corner polyhedra, and we give some properties of them.We offer a linear time algorithm to realize corner polyhedra. For that, the study of corner triangulations leads to enumeration problems. A classic method, known from Tutte, gives the wanted result, making the series of Catalan numbers appearing. The research for a combinatorial explanation of the presence of Catalan numbers induces the use of other methods, based on cuttings and gluings of some parts of corner triangulations. Thus appears the family of almond triangulations, that is a new representation of Catalan numbers, in bijection with the binary trees family, and that completes our corner polyhedra realization algorithm. We eventually give a conclusion to these works, trying to generalize our methods to maps whose faces have an any fixed degree.

The foreseen massive deployment of the internet of things (IoT) is expected to suffer from high security risks. This mainly results from the difficulty to monitor and cure the IoT devices in such large-scale deployment. In this thesis, we propose a spatial random deployment of special nodes (firewalls) which can detect and cure infected nodes within certain radius. An important concern is to add sufficient number of firewalls to make an epidemics finite and, hence, prevent malware outbreak over the whole network. The problem will be analyzed using percolation theory. Namely, we derive an upperbound for the critical intensity of spatial firewalls which guarantees prevention of large-scale network epidemics, regardless of the intensity of regular nodes. Using tools from percolation theory, we analyze the proposed solution and show the conditions required to ensure its efficiency.

Philosophiae Doctor - PhD / The history of the pursuit of uncolourable cubic graphs dates back more than a century. This pursuit has evolved from the slow discovery of individual uncolourable cubic graphs such as the famous Petersen graph and the Blanusa snarks, to discovering in nite classes of uncolourable cubic graphs such as the Louphekine and Goldberg snarks, to investigating parameters which measure the uncolourability of cubic graphs. These parameters include resistance, oddness and weak oddness, ow resistance, among others. In this thesis, we consider current ideas and problems regarding the uncolourability of cubic graphs, centering around these parameters. We introduce new ideas regarding the structural complexity of these graphs in question. In particular, we consider their 3-critical subgraphs, speci cally in relation to resistance. We further introduce new parameters which measure the uncolourability of cubic graphs, speci cally relating to their 3-critical subgraphs and various types of cubic graph reductions. This is also done with a view to identifying further problems of interest. This thesis also presents solutions and partial solutions to long-standing open conjectures relating in particular to oddness, weak oddness and resistance.

Nous introduisons d'abord la classe des graphes $k$-dégénérés qui est souvent utilisée pour modéliser des grands graphes épars issus du monde réel. Nous proposons de nouveaux algorithmes d'énumération pour ces graphes. En particulier, nous construisons un algorithme énumérant tous les cycles simples de tailles fixés dans ces graphes, en temps optimal.Nous proposons aussi un algorithme dont la complexité dépend de la taille de la solution pour le problème d'énumération des cliques maximales de ces graphes. Dans un second temps nous considérons les graphes en tant que systèmes distribués et nous nous intéressons à des questions liées à la notion de couplage lorsqu’aucune supposition n’est faite sur l'état initial du système, qui peut donc être correct ou incorrect. Dans ce cadre nous proposons un algorithme retournant une deux tiers approximation du couplage maximum.Nous proposons aussi un algorithme retournant un couplage maximal quand les communications sont restreintes de telle manière à simuler le paradigme du passage de message. Le troisième objet d'étude n'est pas directement lié à l'algorithmique de graphe, bien que quelques techniques classiques de ce domaine soient utilisées pour obtenir certains de nos résultats.Nous introduisons et étudions certaines familles de polytopes, appelées Zonotopes Primitifs, qui peuvent être décrits comme la somme de Minkowski de vecteurs primitifs. Nous prouvons certaines propriétés combinatoires de ces polytopes et illustrons la connexion avec le plus grand diamètre possible de l'enveloppe convexe de points à coordonnées entières à valeurs dans$[k]$, en dimension $d$. Dans un second temps,nous étudions des paramètres de petites instances de Zonotopes Primitifs, tels que leur nombre de sommets, entre autres. / We start by studying the class of $k$-degenerate graphs which are often used to model sparse real-world graphs. We focus one numeration questions for these graphs. That is,we try and provide algorithms which must output, without duplication, all the occurrences of some input subgraph. We investigate the questions of finding all cycles of some givensize and all maximal cliques in the graph. Ourtwo contributions are a worst-case output sizeoptimal algorithm for fixed-size cycleenumeration and an output sensitive algorithmfor maximal clique enumeration for this restricted class of graphs. In a second part weconsider graphs in a distributed manner. Weinvestigate questions related to finding matchings of the network, when no assumptionis made on the initial state of the system. Thesealgorithms are often referred to as selfstabilizing.Our two main contributions are analgorithm returning an approximation of themaximum matching and a new algorithm formaximal matching when communication simulates message passing. Finally, weintroduce and investigate some special families of polytopes, namely primitive zonotopes,which can be described as the Minkowski sumof short primitive vectors. We highlight connections with the largest possible diameter ofthe convex hull of a set of points in dimension d whose coordinates are integers between 0 and k.Our main contributions are new lower bounds for this diameter question as well as descriptions of small instances of these objects.

Cette thèse étudie la détection de communautés dans le contexte des réseaux sociaux. Un réseau social peut être modélisé par un graphe dans lequel les sommets représentent les membres et les arêtes représentent les relations entre les membres. En particulier, j'étudie quatre différentes définitions de communauté. D'abord, une structure en communautés peut être définie par une partition des sommets telle que tout sommet a une plus grande proportion de voisins dans sa partie que dans toute autre partie. Cette définition peut être adaptée pour l'étude d'une seule communauté. Ensuite, une communauté peut être vue comme un sous graphe tel que tout couple de sommets sont à distance 2 dans ce sous graphe. Enfin, dans le contexte des sites de rencontre, je propose d'étudier une définition de communauté potentielle dans le sens où les membres de la communauté ne se connaissent pas, mais sont liés par des connaissances communes. Pour ces trois définitions, j'étudie la complexité computationnelle et l'approximation de problèmes liés à l'existence ou la recherche de telles communautés dans les graphes. / This thesis deals with community detection in the context of social networks. A social network can be modeled by a graph in which vertices represent members, and edges represent relationships. In particular, I study four different definitions of a community. First, a community structure can be defined as a partition of the vertices such that each vertex has a greater proportion of neighbors in its part than in any other part. This definition can be adapted in order to study only one community. Then, a community can be viewed as a subgraph in which every two vertices are at distance 2 in this subgraph. Finally, in the context of online meetup services, I investigate a definition for potential communities in which members do not know each other but are related by their common neighbors. In regard to these proposed definitions, I study computational complexity and approximation within problems that either relate to the existence of such communities or to finding them in graphs.

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Peg solitaire is a game in which pegs are placed in every hole but one and the player jumps over pegs along rows or columns to remove them. Usually, the goal of the player is to leave only one peg. In a 2011 paper, this game is generalized to graphs. In this thesis, we consider a variation of peg solitaire on graphs in which pegs can be removed either by jumping them or merging them together. To motivate this, we survey some of the previous papers in the literature. We then determine the solvability of several classes of graphs including stars and double stars, caterpillars, trees of small diameter, particularly four and five, and articulated caterpillars. We conclude this thesis with several open problems related to this study.

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