Random planar graphs have been the subject of much recent work. Many basic properties of the standard uniform random planar graph $P_{n}$, by which we mean a graph chosen uniformly at random from the set of all planar graphs with vertex set $ { 1,2, ldots, n }$, are now known, and variations on this standard random graph are also attracting interest. Prominent among the work on $P_{n}$ have been asymptotic results for the probability that $P_{n}$ will be connected or contain given components/ subgraphs. Such progress has been achieved through a combination of counting arguments cite{mcd} and a generating function approach cite{gim}. More recently, attention has turned to $P_{n,m}$, the graph taken uniformly at random from the set of all planar graphs on ${ 1,2, ldots, n }$ with exactly $m(n)$ edges (this can be thought of as a uniform random planar graph with a constraint on the average degree). In cite{ger} and cite{gim}, the case when $m(n) =~!lfloor qn floor$ for fixed $q in (1,3)$ has been investigated, and results obtained for the events that $P_{n, lfloor qn floor}$ will be connected and that $P_{n, lfloor qn floor}$ will contain given subgraphs. In Part I of this thesis, we use elementary counting arguments to extend the current knowledge of $P_{n,m}$. We investigate the probability that $P_{n,m}$ will contain given components, the probability that $P_{n,m}$ will contain given subgraphs, and the probability that $P_{n,m}$ will be connected, all for general $m(n)$, and show that there is different behaviour depending on which `region' the ratio $rac{m(n)}{n}$ falls into. In Part II, we investigate the same three topics for a uniform random planar graph with constraints on the maximum and minimum degrees.

Suppose G is a graph with n vertices and m edges. Let n¡¬ be the maximum number of vertices in an induced bipartite subgraph of G and let m¡¬ be the maximum number of edges in a spanning bipartite subgraph of G. Then b(G) = m¡¬/m is called the bipartite density of G, and b∗(G) = n¡¬/n is called the bipartite ratio of G. It is proved in [18] that if G is a 2-connected triangle-free subcubic graph, then apart from seven exceptional graphs, we have b(G) ≥ 17/21. If G is a 2-connected triangle-free subcubic graph, then b∗(G) ≥ 5/7 provided that G is not the Petersen graph and not the dodecahedron. These two results are consequences of a more technical result which is proved by induction: If G is a 2-connected triangle-free subcubic graph with minimum degree 2, then G has an induced bipartite subgraph H with |V (H)| ≥ (5n3 + 6n2 + ǫ(G))/7, where ni = ni(G) are the number of degree i vertices of G, and ǫ(G) ∈ {−2,−1, 0, 1}. To determine ǫ(G), four classes of graphs G1, G2, G3 and F-cycles are onstructed. For G ∈ Gi, we have ǫ(G) = −3 + i and for an F-cycle G, we have ǫ(G) = 0. Otherwise, ǫ(G) = 1. To construct these graph classes, eleven graph operations are used. This thesis studies the structural property of graphs in G1, G2, G3. First of all, a computer algorithm is used to generate all the graphs in Gi for i = 1, 2, 3. Let P be the set of 2-edge connected subcubic triangle-free planar graphs with minimum degree 2. Let G¡¬ 1 be the set of graphs in P with all faces of degree 5, G¡¬2 the set of graphs in P with all faces of degree 5 except that one face has degree 7, and G¡¬3 the set of graphs in P with all faces of degree 5 except that either two faces are of degree 7 or one face is of degree 9. By checking the graphs generated by the computer algorithm, it is easy to see that Gi ⊆ G¡¬i for i = 1, 2, 3. The main results of this thesis are that for i = 1, 2, Gi = G¡¬i and G¡¬3 = G3 ¡åR, where R is a set of nine F-cycles.

This dissertation studies the colored Jones polynomial of knots and links, colored by representations of simple Lie algebras, and the stability of its coefficients. Chapter 1 provides an explicit formula for the second plethysm of an arbitrary representation of sl3. This allows for an explicit formula for the colored Jones polynomial of the trefoil, and more generally, for T(2,n) torus knots. This formula for the sl3 colored Jones polynomial of T(2,n)$ torus knots makes it possible to verify the Degree Conjecture for those knots, to efficiently compute the sl3 Witten-Reshetikhin-Turaev invariants of the Poincare sphere, and to guess a Groebner basis for the recursion ideal of the sl3 colored Jones polynomial of the trefoil. Chapter 2 gives a formulation of a stability conjecture for the coefficients of the colored Jones polynomial of a knot, colored by irreducible representations in a fixed ray of a simple Lie algebra. The conjecture is verified for all torus knots and all simple Lie algebras of rank 2. Chapter 3 supplies an efficient method to compute those q-series that come from planar graphs (i.e., reduced Tait graphs of alternating links) and compute several terms of those series for all graphs with at most 8 edges. In addition, a graph-theory proof of a theorem of Dasbach-Lin which identifies the coefficient of q^k in those series for k=0,1,2 in terms of polynomials on the number of vertices, edges and triangles of the graph is given. Chapter 4 provides a study of the structure of the stable coefficients of the Jones polynomial of an alternating link.The first four stable coefficients are identified with polynomial invariants of a (reduced) Tait graph of the link projection. A free polynomial algebra of invariants of graphs whose elements give invariants of alternating links is introduced which strictly refines the first four stable coefficients. It is conjectured that all stable coefficients are elements of this algebra, and experimental evidence for the fifth and sixth stable coefficient is given. The results are illustrated in tables of all alternating links with at most 10 crossings and all irreducible planar graphs with at most 6 vertices.

Matching is one of the most fundamental algorithmic graph problems. Many variants of matching problems have been studied on different classes of graphs, the one of special interest to us being the Maximum Cardinality Bipartite Matching in Planar Graphs. In this work, we present a novel sparsification based approach for computing maximum/perfect bipartite matching in planar graphs. The overall complexity of our algorithm is O(n^6/5 log² n) where n is the number of vertices in the graph, bettering the O(n^3/2) time achieved independently by Hopcroft-Karp algorithm and by Lipton and Tarjan divide and conquer approach using planar separators. Our algorithm combines the best of both these standard algorithms along with our sparsification technique and rich planar graph properties to achieve the speed up. Our algorithm is not the fastest, with the existence of O(n log³ n) algorithm based on max-flow reduction. / MS

Effective visualization of graphs is a powerful tool to help understand the relationships among the graph's underlying objects and to interact with them. Several styles for drawing graphs have emerged over the last three decades. Polyline drawing is a widely used style for drawing graphs, where each node is mapped to a distinct point in the plane and each edge is mapped to a polygonal chain between their corresponding nodes. Some common optimization criteria for such a drawing are defined in terms of area requirement, number of bends per edge, angular resolution, number of distinct line segments, edge crossings, and number of planar layers. In this thesis we develop algorithms for drawing graphs that optimize different aesthetic qualities of the drawing. Our algorithms seek to simultaneously optimize multiple drawing aesthetics, reveal potential trade-offs among them, and improve many previous graph drawing algorithms. We start by exploring probable trade-offs in the context of planar graphs. We prove that every $n$-vertex planar triangulation $G$ with maximum degree $\Delta$ can be drawn with at most $2n+t-3$ segments and $O(8^t \cdot \Delta^{2t})$ area, where $t$ is the number of leaves in a Schnyder tree of $G$. We then show that one can improve the area by allowing the edges to have bends. Since compact drawings often suffer from bad angular resolution, we seek to compute polyline drawings with better angular resolution. We develop a polyline drawing algorithm that is simple and intuitive, yet implies significant improvement over known results. At this point we move our attention to drawing nonplanar graphs. We prove that every thickness-$t$ graph can be drawn on $t$ planar layers with $\min\{O(2^{t/2} \cdot n^{1-1/\beta}), 2.25n +O(1)\}$ bends per edge, where $\beta = 2^{\lceil (t-2)/2 \rceil }$. Previously, the bend complexity, i.e., the number of bends per edge, was not known to be sublinear for $t>2$. We then examine the case when the number of available layers is restricted. The layers may now contain edge crossings. We develop a technique to draw complete graphs on two layers, which improves previous upper bounds on the number of edge crossings in such drawings. / October 2016

The famous Four Color Theorem states that any planar graph can be properly colored using at most four colors. However, if we want to properly color the square of a planar graph (or alternatively, color the graph using distinct colors on vertices at distance up to two from each other), we will always require at least \Delta + 1 colors, where \Delta is the maximum degree in the graph. For all \Delta, Wegner constructed planar graphs (even without 3-cycles) that require about \frac{3}{2} \Delta colors for such a coloring. To prove a stronger upper bound, we consider only planar graphs that contain no 4-cycles and no 5-cycles (but which may contain 3-cycles). Zhu, Lu, Wang, and Chen showed that for a graph G in this class with \Delta \ge 9, we can color G^2 using no more than \Delta + 5 colors. In this thesis we improve this result, showing that for a planar graph G with maximum degree \Delta \ge 32 having no 4-cycles and no 5-cycles, at most \Delta + 3 colors are needed to properly color G^2. Our approach uses the discharging method, and the result extends to list-coloring and other related coloring concepts as well.

Dans cette thèse, nous présentons trois problèmes concernant les graphes planaires.Nous travaillons tout d'abord sur les dessins planaires non-alignés, c'est-à-dire des dessins planaires de graphes sur une grille sans que deux sommets se trouvent sur la même ligne ou la même colonne.Nous caractérisons les graphes planaires possédant un tel dessin sur une grille de taille $n times n$, et nous présentons deux algorithmes générant un dessin planaire non-aligné avec arêtes brisées sur cette grille pour tout graphe planaire, avec $n-3$ ou $min(frac{2n-3}{5},$ $#{text{triangles s{'e}parateurs}}+1)$ brisures au total.Nous proposons également deux algorithmes dessinant un dessin planaire non-aligné sur des grilles d'aire $O(n^4)$. Nous donnons des résultats spécifiques concernant les graphes 4-connexes et de type triangle-emboîté.Le second sujet de cette thèse est la domination de puissance dans les graphes planaires. Nous exhibons une famille de graphes ayant un nombre de domination de puissance $gamma_P$ au moins égal à $frac{n}{6}$. Nous montrons aussi que pour tout graphe planaire maximal $G$ à $n geq 6$ sommets, $gamma_P(G) leq frac{n-2}{4}$. Enfin, nous étudions les grilles triangulaires $T_k$ à bord hexagonal de dimension $k$ et nous montrons que $frac{k}{3} - frac{1}{6} leq gamma_P(T_k) leq lceil frac{k}{3} rceil$.Nous étudions également l'énumération des orientations planaires Eulériennes. Nous proposons une nouvelle décomposition de ces cartes. En considérant les orientations des dernières $2k-1$ arêtes autour de la racine, nous définissons des sous- et sur-ensembles des orientations planaires Eulériennes paramétrés par $k$.Pour chaque classe, nous proposons un système d'équations fonctionnelles définissant leur série génératrice, et nous prouvons que celle-ci est toujours algébrique. Nous montrons ainsi que la constance de croissance des orientations planaires Eulériennes est entre 11.56 et 13.005. / In this thesis, we present results on three different problems concerning planar graphs.We first give some new results on planar non-aligned drawings, i.e. planar grid drawings where vertices are all on different rows and columns.We show that not every planar graph has a non-aligned drawing on an $n times n$-grid, but we present two algorithms generating a non-aligned polyline drawings on such a grid requiring either $n-3$ or $min(frac{2n-3}{5},$ $#{text{separating triangles}}+1)$ bends in total.Concerning non-minimal grids, we give two algorithms drawing a planar non-aligned drawing on grids with area of order $n^4$. We also give specific results for 4-connected graphs and nested-triangle graphs.The second topic is power domination in planar graphs. We present a family of graphs with power dominating number $gamma_P$ at least $frac{n}{6}$. We then prove that for every maximal planar graph $G$ of order $n$, $gamma_P(G) leq frac{n-2}{4}$, and we give a constructive algorithm.We also prove that for triangular grids $T_k$ of dimension $k$ with hexagonal-shape border, $frac{k}{3} - frac{1}{6} leq gamma_P(T_k) leq lceil frac{k}{3} rceil$.Finally, we focus on the enumeration of planar Eulerian orientations. After proposing a new decomposition for these maps, we define subsets and supersets of planar Eulerian orientations with parameter $k$, generated by looking at the orientations of the last $2k-1$ edges around the root vertex.For each set, we give a system of functional equations defining its generating function, and we prove that it is always algebraic.This way, we show that the growth rate of planar Eulerian orientations is between 11.56 and 13.005.

Cette thèse porte sur deux sujets d’algorithmique des graphes. Le premier sujet est l’inférence de réseaux. Quelle est la complexité pour déterminer un graphe inconnu à partir de requêtes de plus court chemin entre ses sommets ? Nous supposons que le graphe est de degré borné. Dans le problème de reconstruction, le but est de reconstruire le graphe ; tandis que dans le problème de vérification, le but est de vérifier qu’un graphe donné est correct. Nous développons des algorithmes probabilistes utilisant une décomposition en cellules de Voronoi. Ensuite, nous analysons des algorithmes de type glouton, et montrons qu’ils sont quasi-optimaux. Nous étudions aussi ces problèmes sur des familles particulières de graphes, démontrons des bornes inférieures, et étudions la reconstruction approximative. Le deuxième sujet est l’étude de deux problèmes d’optimisation sur les graphes planaires. Dans le problème de classification par corrélations, l’entrée est un graphe pondéré, où chaque arête a une étiquette h+i ou h-i, indiquant si ses extrémités sont ou non dans la même catégorie. Le but est de trouver une partition des sommets en catégories qui respecte au mieux les étiquettes. Dans le problème d’augmentation 2-arête-connexe, l’entrée est un graphe pondéré et un sous-ensemble R des arêtes. Le but est de trouver un sous-ensemble S des arêtes de poids minimum, tel que pour chaque arête de R, ses extrémités sont dans une composante 2-arête-connexe de l’union de R et S. Pour les graphes planaires, nous réduisons le premier problème au deuxième et montrons que les deux problèmes, bien que NP-durs, ont un schéma d’approximation en temps polynomial. Nous utilisons la technique récente de décomposition en briques. / This thesis focuses on two topics of graph algorithms. The first topic is network inference. How efficiently can we find an unknown graph using shortest path queries between its vertices? We assume that the graph has bounded degree. In the reconstruction problem, the goal is to find the graph; and in the verification problem, the goal is to check whether a given graph is correct. We provide randomized algorithms based on a Voronoi cell decomposition. Next, we analyze greedy algorithms, and show that they are near-optimal. We also study the problems on special graph classes, prove lower bounds, and study the approximate reconstruction. The second topic is optimization in planar graphs. We study two problems. In the correlation clustering problem, the input is a weighted graph, where every edge has a label of h+i or h−i, indicating whether its endpoints are in the same category or in different categories. The goal is to find a partition of the vertices into categories that tries to respect the labels. In the two-edge-connected augmentation problem, the input is a weighted graph and a subset R of edges. The goal is to produce a minimum-weight subset S of edges, such that for every edge in R, its endpoints are two-edge-connected in the union of R and S. For planar graphs, we reduce correlation clustering to two-edge-connected augmentation, and show that both problems, although they are NP-hard, have a polynomial-time approximation scheme. We build on the brick decomposition technique developed recently.

In this thesis we will focus on a problem of length bounded cut, also known as L-bounded cut. We are going to show a combinatorial algorithm for finding a minimal L-bounded cut on graphs with bounded treewidth based on dynamic programming. Then we going to show that this algorithm can also be used for finding minimal L-bounded cut on plannar graphs. We are also going to look at problem of (dG(s, t) + 1)-bounded cut. This problem is known to be NP-hard for general graphs. But it is an open problem whether this problem is also NP-hard on plannar graphs with special vertices on the outer face. We will try to outline a way, which might lead to showing that this problem is solvable in a polynomial time.

Cette thèse présente des résultats obtenus dans deux domaines : la théorie des langages, et la théorie des graphes. En théorie des langages, on s’intéresse à des problèmes de caractérisation de classes de langages réguliers. Le problème générique consiste à déterminer si un langage régulier donné peut être défini dans un certain formalisme. Les méthodes actuelles font intervenir un problème plus général appelé séparation. On présente ici deux types de contributions : une généralisation d’un résultat de décidabilité au cadre des langages de mots infinis, ainsi que des bornes inférieures pour la complexité du problème de séparation. En théorie des graphes, on considère le problème classique de coloration de graphes, où on cherche à attribuer des couleurs aux sommets d’un graphe de sorte que les sommets adjacents reçoivent des couleurs différentes, le but étant d’utiliser le moins de couleurs possible. Dans le cas des graphes peu denses, la méthode de déchargement est un atout majeur. Elle a notamment joué un rôle décisif dans la preuve du théorème des quatre couleurs. Cette méthode peut être vue comme une construction non conventionnelle d’un schéma de preuve par induction, spécifique à la classe de graphes et à la propriété considérées, et où la validité du schéma est rarement immédiate. On utilise des variantes de la méthode de déchargement pour étudier deux types de problèmes de coloration. / In this thesis, we present results obtained in two fields: formal language theory and graph theory. In formal language theory, we consider some problems of characterization of classes of regular languages. The generic problem consists in determining whether a given regular language can be defined in a fixed formalism. The current approaches use a more general problem called separation. We present here two types of contributions: a generalization of a decidability result to the setting of infinite words, together with lower bounds for the complexity of the separation problem. In graph theory, we consider the classical problem of graph coloring, where we assign colors to vertices of a graph in such a way that two adjacent vertices receive different colors. The goal is to use the fewest colors. When the graphs are sparse, a crucial tool for this is the discharging method. It is most notably decisive in the proof of the Four-Color Theorem. This method can be seen as an unconventional construction of an inductive proof scheme, specific to the considered problem and graph class, where arguing the validity of the scheme is rarely immediate. We use variants of the discharging method to study two types of coloring problems.