In this thesis we study the isoperimetric problem on trees and graphs with bounded treewidth. Let G = (V,E) be a finite, simple and undirected graph. For let δ(S,G)= {(u,v) ε E : u ε S and v ε V – S }be the edge boundary of S. Given an integer i, 1 ≤ i ≤ | V| , let the edge isoperimetric value of G at I be defined as be(i,G)= mins v;|s|= i | δ(S,G)|. For S V, let φ(S,G) = {u ε V – S : ,such that be the vertex boundary of S. Given an integer i, 1 ≤ i ≤ | V| , let the vertex isoperimetric value of G at I be defined as bv(i,G)= The edge isoperimetric peak of G is defined as be(G) =. Similarly the vertex isoperimetric peak of G is defined as bv(G)= .The problem of determining a lower bound for the vertex isoperimetric peak in complete k-ary trees of depth d,Tdkwas recently considered in[32]. In the first part of this thesis we provide lower bounds for the edge and vertex isoperimetric peaks in complete k-ary trees which improve those in[32]. Our results are then generalized to arbitrary (rooted)trees. Let i be an integer where . For each i define the connected edge isoperimetric value and the connected vertex isoperimetric value of G at i as follows: is connected and is connected A meta-Fibonacci sequence is given by the reccurence a(n)= a(x1(n)+ a1′(n-1))+ a(x2(n)+ a2′(n -2)), where xi: Z+ → Z+ , i =1,2, is a linear function of n and ai′(j)= a(j) or ai′(j)= -a(j), for i=1,2. Sequences belonging to this class have been well studied but in general their properties remain intriguing. In the second part of the thesis we show an interesting connection between the problem of determining and certain meta-Fibonacci sequences. In the third part of the thesis we study the problem of determining be and bv algorithmically for certain special classes of graphs. Definition 0.1. A tree decomposition of a graph G = (V,E) is a pair where I is an index set, is a collection of subsets of V and T is a tree whose node set is I such that the following conditions are satisfied: (For mathematical equations pl see the pdf file)

This thesis touches three different topics in graph theory, namely, rainbow colouring, product dimension and boxicity. Rainbow colouring An edge colouring of a graph is called a rainbow colouring, if every pair of vertices is connected by atleast one path in which no two edges are coloured the same. The rainbow connection number of a graph is the minimum number of colours required to rainbow colour it. In this thesis we give upper bounds on rainbow connection number based on graph invariants like minimum degree, vertex connectivity, and radius. We also give some computational complexity results for special graph classes. Product dimension The product dimension or Prague dimension of a graph G is the smallest natural number k such that G is an induced subgraph of a direct product of k complete graphs. In this thesis, we give upper bounds on the product dimension for forests, bounded tree width graphs and graphs of bounded degeneracy. Boxicity and cubicity The boxicity (cubicity of a graph G is the smallest natural number k such that G can be represented as an intersection graph of axis-parallel rectangular boxes(axis-parallel unit cubes) in Rk .In this thesis, we study the boxicity and the cubicity of Cartesian, strong and direct products of graphs and give estimates on the boxicity and the cubicity of a product graph based on invariants of the component graphs. Separation dimension The separation dimension of a hypergraph H is the smallest natural number k for which the vertices of H can be embedded in Rk such that any two disjoint edges of H can be separated by a hyper plane normal to one of the axes. While studying the boxicity of line graphs, we noticed that a box representation of the line graph of a hypergraph has a nice geometric interpretation. Hence we introduced this new parameter and did an extensive study of the same.

Cette thèse est composée de deux parties, chacune portant sur un sous-domaine de l'optimisation combinatoire a priori distant de l'autre. Le premier thème de recherche abordé est la programmation biniveau stochastique. Se cachent derrière ce terme deux sujets de recherche relativement peu étudiés conjointement, à savoir d'un côté la programmation stochastique, et de l'autre la programmation biniveau. La programmation mathématique (PM) regroupe un ensemble de méthodes de modélisation et de résolution, pouvant être utilisées pour traiter des problèmes pratiques que se posent des décideurs. La programmation stochastique et la programmation biniveau sont deux sous-domaines de la PM, permettant chacun de modéliser un aspect particulier de ces problèmes pratiques. Nous élaborons un modèle mathématique issu d'un problème appliqué, où les aspects biniveau et stochastique sont tous deux sollicités, puis procédons à une série de transformations du modèle. Une méthode de résolution est proposée pour le PM résultant. Nous démontrons alors théoriquement et vérifions expérimentalement la convergence de cette méthode. Cet algorithme peut être utilisé pour résoudre d'autres programmes biniveaux que celui qui est proposé.Le second thème de recherche de cette thèse s'intitule "problèmes de coupe et de couverture partielles dans les graphes". Les problèmes de coupe et de couverture sont parmi les problèmes de graphe les plus étudiés du point de vue complexité et algorithmique. Nous considérons certains de ces problèmes dans une variante partielle, c'est-à-dire que la propriété de coupe ou de couverture dont il est question doit être vérifiée partiellement, selon un paramètre donné, et non plus complètement comme c'est le cas pour les problèmes originels. Précisément, les problèmes étudiés sont le problème de multicoupe partielle, de coupe multiterminale partielle, et de l'ensemble dominant partiel. Les versions sommets des ces problèmes sont également considérés. Notons que les problèmes en variante partielle généralisent les problèmes non partiels. Nous donnons des algorithmes exacts lorsque cela est possible, prouvons la NP-difficulté de certaines variantes, et fournissons des algorithmes approchés dans des cas assez généraux. / This thesis is composed of two parts, each part belonging to a sub-domain of combinatorial optimization a priori distant from the other. The first research subject is stochastic bilevel programming. This term regroups two research subject rarely studied together, namely stochastic programming on the one hand, and bilevel programming on the other hand. Mathematical Programming (MP) is a set of modelisation and resolution methods, that can be used to tackle practical problems and help take decisions. Stochastic programming and bilevel programming are two sub-domains of MP, each one of them being able to model a specific aspect of these practical problems. Starting from a practical problem, we design a mathematical model where the bilevel and stochastic aspects are used together, then apply a series of transformations to this model. A resolution method is proposed for the resulting MP. We then theoretically prove and numerically verify that this method converges. This algorithm can be used to solve other bilevel programs than the ones we study.The second research subject in this thesis is called "partial cut and cover problems in graphs". Cut and cover problems are among the most studied from the complexity and algorithmical point of view. We consider some of these problems in a partial variant, which means that the cut or cover property that is looked into must be verified partially, according to a given parameter, and not completely, as it was the case with the original problems. More precisely, the problems that we study are the partial multicut, the partial multiterminal cut, and the partial dominating set. Versions of these problems were vertices are