Perfectly contractile graphs and quadratic toric rings

Perfect graphs form one of the distinguished classes of finite simple graphs. In 2006, Chudnovsky, Robertson, Saymour and Thomas proved that a graph is perfect if and only if it has no odd holes and no odd antiholes as induced subgraphs, which was conjectured by Berge. We consider the class ${\mathcal A}$ of graphs that have no odd holes, no antiholes and no odd stretchers as induced subgraphs. In particular, every graph belonging to ${\mathcal A}$ is perfect. Everett and Reed conjectured that a graph belongs to ${\mathcal A}$ if and only if it is perfectly contractile. In the present paper, we discuss graphs belonging to ${\mathcal A}$ from a viewpoint of commutative algebra. In fact, we conjecture that a perfect graph $G$ belongs to ${\mathcal A}$ if and only if the toric ideal of the stable set polytope of $G$ is generated by quadratic binomials. Especially, we show that this conjecture is true for Meyniel graphs, perfectly orderable graphs, and clique separable graphs, which are perfectly contractile graphs.Comment: 10 page

The world of hereditary graph classes viewed through Truemper configurations

In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms

Topics in graph colouring and extremal graph theory

In this thesis we consider three problems related to colourings of graphs and one problem in extremal graph theory. Let $G$ be a connected graph with $n$ vertices and maximum degree $\Delta(G)$. Let $R_k(G)$ denote the graph with vertex set all proper $k$-colourings of $G$ and two $k$-colourings are joined by an edge if they differ on the colour of exactly one vertex. Our first main result states that $R_{\Delta(G)+1}(G)$ has a unique non-trivial component with diameter $O(n^2)$. This result can be viewed as a reconfigurations analogue of Brooks' Theorem and completes the study of reconfigurations of colourings of graphs with bounded maximum degree. A Kempe change is the operation of swapping some colours $a$, $b$ of a component of the subgraph induced by vertices with colour $a$ or $b$. Two colourings are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. Our second main result states that all $\Delta(G)$-colourings of a graph $G$ are Kempe equivalent unless $G$ is the complete graph or the triangular prism. This settles a conjecture of Mohar (2007). Motivated by finding an algorithmic version of a structure theorem for bull-free graphs due to Chudnovsky (2012), we consider the computational complexity of deciding if the vertices of a graph can be partitioned into two parts such that one part is triangle-free and the other part is a collection of complete graphs. We show that this problem is NP-complete when restricted to five classes of graphs (including bull-free graphs) while polynomial-time solvable for the class of cographs. Finally we consider a graph-theoretic version formulated by Holroyd, Spencer and Talbot (2007) of the famous Erd\H{o}s-Ko-Rado Theorem in extremal combinatorics and obtain some results for the class of trees

Structural solutions to maximum independent set and related problems

In this thesis, we study some fundamental problems in algorithmic graph theory. Most natural problems in this area are hard from a computational point of view. However, many applications demand that we do solve such problems, even if they are intractable. There are a number of methods in which we can try to do this: 1) We may use an approximation algorithm if we do not necessarily require the best possible solution to a problem. 2) Heuristics can be applied and work well enough to be useful for many applications. 3) We can construct randomised algorithms for which the probability of failure is very small. 4) We may parameterize the problem in some way which limits its complexity. In other cases, we may also have some information about the structure of the instances of the problem we are trying to solve. If we are lucky, we may and that we can exploit this extra structure to find efficient ways to solve our problem. The question which arises is - How far must we restrict the structure of our graph to be able to solve our problem efficiently? In this thesis we study a number of problems, such as Maximum Indepen- dent Set, Maximum Induced Matching, Stable-II, Efficient Edge Domina- tion, Vertex Colouring and Dynamic Edge-Choosability. We try to solve problems on various hereditary classes of graphs and analyse the complexity of the resulting problem, both from a classical and parameterized point of view

Structures métriques et leurs groupes d’automorphismes : reconstruction, homogénéité, moyennabilité et continuité automatique

This thesis focuses on the study of Polish groups seen as automorphism groups of metric structures. The observation that every non-archimedean Polish group is the automorphism group of an ultrahomogeneous countable structure has indeed led to fruitful interactions between group theory and model theory. In the framework of metric model theory, introduced by Ben Yaacov, Henson and Usvyastov, this correspondence has been extended to all Polish groups by Melleray. In this thesis, we study various facets of this correspondence. The relationship between a structure and its automorphism group is particularly close in the setting of ℵ0-categorical structures. Indeed, the Ahlbrandt-Ziegler reconstruction theorem allows one to recover an ℵ0-categorical structure, up to bi-interpretability, from its automorphism group. In a joint work with Itai Ben Yaacov, we generalize this result to separably categorical metric structures. Besides, ultrahomogeneous countable structures have the advantage of being completely determined by their finitely generated substructures. In particular, this enabled Moore to give a combinatorial characterization of amenability for nonarchimedean Polish groups. We extend this characterization to all Polish groups and we deduce that amenability is a Gδ condition. Still in a reconstruction perspective, we are interested in the automatic continuity property for Polish groups. Sabok and Malicki introduced conditions of a combinatorial nature on an ultrahomogeneous metric structure that imply the automatic continuity property for its automorphism group. We show that these conditions carry to countable powers, which leads to the groups Aut(μ)N, U(l2)N and Iso(U)N satisfying the automatic continuity property. Those conditions are a weakening of the property of having ample generics. In a joint work with Francois Le Maitre, we exhibit the first examples of connected groups with ample generics, which answers a question of Kechris and Rosendal. Finally, in a joint work with Isabel Muller and Aristotelis Panagiotopoulos, we study the relative homogeneity of substructures in an ultrahomogeneous countable structure. We characterize it completely by a property of the types over the substructures: being determined by a finite setCette thèse porte sur l'étude des groupes polonais vus comme groupes d'automorphismes de structures métriques. L'observation que tout groupe polonais non archimédien est le groupe d'automorphismes d'une structure dénombrable ultra homogène a en effet mené à des interactions fructueuses entre la théorie des groupes et la théorie des modèles. Dans le cadre de la théorie des modèles métriques, introduite par Ben Yaacov, Henson et Usvyatsov, cette correspondance a été étendue par Melleray à tous les groupes polonais. Dans cette thèse, nous étudions diverses facettes de cette correspondance. Le lien entre une structure et son groupe d automorphismes est particulièrement étroit dans le cadre des structures ℵ0-categoriques. En effet, le théorème de reconstruction d'Ahlbrandt-Ziegler permet de retrouver une structure ℵ0-categorique, à bi-interprètabilité près, à partir de son groupe d'automorphismes. Dans un travail en commun avec Itai Ben Yaacov, nous généralisons ce résultat aux structures métriques separablement catégoriques. Les structures dénombrables ultra homogènes ont de plus l avantage d'être complètement déterminées par leurs sous-structures finiment engendrées. Cela a notamment permis a Moore de donner une caractérisation combinatoire de la moyennabilité des groupes polonais non archimédiens. Nous étendons cette caractérisation à tous les groupes polonais et nous en déduisons que la moyennabilite est une condition Gδ. Toujours dans une optique de reconstruction, nous nous intéressons à la propriété de continuité automatique pour les groupes polonais. Sabok et Malicki ont introduit des conditions de nature combinatoire sur une structure métrique ultra homogène qui impliquent la propriété de continuité automatique pour son groupe d'automorphismes. Nous montrons que ces conditions passent à la puissance dénombrable, ce qui a pour conséquence que les groupes Aut(μ)N, U(l2)N et Iso(U)N satisfont la propriété de continuité automatique. Ces conditions sont un affaiblissement du fait d'avoir des amples génériques. Dans un travail en commun avec Francois Le Maitre, nous exhibons les premiers exemples de groupes connexes qui ont des amples génériques, ce qui répond à une question de Kechris et Rosenda

The Fluid Dynamics of Heart Development: The effect of morphology on flow at several stages

Proper cardiogenesis requires a delicate balance between genetic and environmental (epigenetic) signals, and mechanical forces. While many cellular biologists and geneticists have extensively studied heart morphogenesis using various experimental techniques, only a few scientists have begun using mathematical modeling as a tool for studying cardiogenic events. Hemodynamic processes, such as vortex formation, are important in the generation of shear at the endothelial surface layer and strains at the epithelial layer, which aid in proper morphology and functionality. The purpose of this thesis is to study the underlying fluid dynamics in various stages on heart development, in particular, the morphogenic stages when the heart is a linear heart tube as well as during the onset of ventricular trabeculation. Previous mathematical models of the linear heart tube stage have focused on mechanisms of valveless pumping, whether dynamic suction pumping (impedance pumping) or peristalsis; however, they all have neglected hematocrit. The impact of blood cells was examined by fluid-structure interaction simulations, via the immersed boundary method. Moreover, electrophysiology models were incorporated into an immersed boundary framework, and bifurcations within the morphospace were studied that give rise to a spectrum of pumping regimes, with peristaltic-like waves of contraction and impedance pumping at the extremes. Lastly, effects of resonant pumping, damping, and boundary inertial effects (added mass) were studied for dynamic suction pumping. The other stage of heart development considered here is during the onset of ventricular trabeculation. This occurs after the heart has undergone the cardiac looping stage and now is a multi-chambered pumping system with primitive endocardial cushions, which act as precursors to valve leaflets. Trabeculation introduces complex morphology onto the inner lining of the endocardium in the ventricle. This transition of a smooth endocardium to one with complex geometry, may have significant effect on the intracardial fluid dynamics and stress distribution within emrbyonic hearts. Previous studies have not included these geometric perturbations along the ventricular endocardium. The role of trabeculae on intracardial (and intertrabecular) flows was studied using two different mathematical models implemented within an immersed boundary framework. It is shown that the trabecular geometry and number density have a significant effect on such flows. Furthermore this thesis also focused attention to the creation of software for scientists and engineers to perform fluid-structure interaction simulations at an accelerated rate, in user-friendly environments for beginner programmers, e.g., MATLAB or Python 3.5. The software, IB2d, performs fully coupled fluid-structure interaction problems using Charles Peskin's immersed boundary method. IB2d is capable of running a vast range of biomechanics models and contains multiple options for constructing material properties of the fiber structure, advection-diffusion of a chemical gradient, muscle mechanics models, Boussinesq approximations, and artificial forcing to drive boundaries with a preferred motion. The software currently contains over 50 examples, ranging from rubber-bands oscillating to flow past a cylinder to a simple aneurysm model to falling spheres in a chemical gradient to jellyfish locomotion to a heart tube pumping coupled with electrophysiology, muscle, and calcium dynamics modelsDoctor of Philosoph

Three-in-a-Tree in Near Linear Time

The three-in-a-tree problem is to determine if a simple undirected graph contains an induced subgraph which is a tree connecting three given vertices. Based on a beautiful characterization that is proved in more than twenty pages, Chudnovsky and Seymour [Combinatorica 2010] gave the previously only known polynomial-time algorithm, running in $O(mn^2)$ time, to solve the three-in-a-tree problem on an $n$-vertex $m$-edge graph. Their three-in-a-tree algorithm has become a critical subroutine in several state-of-the-art graph recognition and detection algorithms. In this paper we solve the three-in-a-tree problem in $\tilde{O}(m)$ time, leading to improved algorithms for recognizing perfect graphs and detecting thetas, pyramids, beetles, and odd and even holes. Our result is based on a new and more constructive characterization than that of Chudnovsky and Seymour. Our new characterization is stronger than the original, and our proof implies a new simpler proof for the original characterization. The improved characterization gains the first factor $n$ in speed. The remaining improvement is based on dynamic graph algorithms.Comment: 46 pages, 12 figures, accepted to STOC 202