Minimal Connectivity

A k-connected graph such that deleting any edge / deleting any vertex / contracting any edge results in a graph which is not k-connected is called minimally / critically / contraction-critically k-connected. These three classes play a prominent role in graph connectivity theory, and we give a brief introduction with a light emphasis on reduction- and construction theorems for classes of k-connected graphs.Comment: IMADA-preprint-math, 33 page

On Pairwise Graph Connectivity

A graph on at least k+1 vertices is said to have global connectivity k if any two of its vertices are connected by k independent paths. The local connectivity of two vertices is the number of independent paths between those specific vertices. This dissertation is concerned with pairwise connectivity notions, meaning that the focus is on local connectivity relations that are required for a number of or all pairs of vertices. We give a detailed overview about how uniformly k-connected and uniformly k-edge-connected graphs are related and provide a complete constructive characterization of uniformly 3-connected graphs, complementing classical characterizations by Tutte. Besides a tight bound on the number of vertices of degree three in uniformly 3-connected graphs, we give results on how the crossing number and treewidth behaves under the constructions at hand. The second central concern is to introduce and study cut sequences of graphs. Such a sequence is the multiset of edge weights of a corresponding Gomory-Hu tree. The main result in that context is a constructive scheme that allows to generate graphs with prescribed cut sequence if that sequence satisfies a shifted variant of the classical Erdős-Gallai inequalities. A complete characterization of realizable cut sequences remains open. The third central goal is to investigate the spectral properties of matrices whose entries represent a graph's local connectivities. We explore how the spectral parameters of these matrices are related to the structure of the corresponding graphs, prove bounds on eigenvalues and related energies, which are sums of absolute values of all eigenvalues, and determine the attaining graphs. Furthermore, we show how these results translate to ultrametric distance matrices and touch on a Laplace analogue for connectivity matrices and a related isoperimetric inequality

Propriétés géométriques du nombre chromatique : polyèdres, structures et algorithmes

Computing the chromatic number and finding an optimal coloring of a perfect graph can be done efficiently, whereas it is an NP-hard problem in general. Furthermore, testing perfection can be carried- out in polynomial-time. Perfect graphs are characterized by a minimal structure of their sta- ble set polytope: the non-trivial facets are defined by clique-inequalities only. Conversely, does a similar facet-structure for the stable set polytope imply nice combinatorial and algorithmic properties of the graph ? A graph is h-perfect if its stable set polytope is completely de- scribed by non-negativity, clique and odd-circuit inequalities. Statements analogous to the results on perfection are far from being understood for h-perfection, and negative results are missing. For ex- ample, testing h-perfection and determining the chromatic number of an h-perfect graph are unsolved. Besides, no upper bound is known on the gap between the chromatic and clique numbers of an h-perfect graph. Our first main result states that the operations of t-minors keep h- perfection (this is a non-trivial extension of a result of Gerards and Shepherd on t-perfect graphs). We show that it also keeps the Integer Decomposition Property of the stable set polytope, and use this to answer a question of Shepherd on 3-colorable h-perfect graphs in the negative. The study of minimally h-imperfect graphs with respect to t-minors may yield a combinatorial co-NP characterization of h-perfection. We review the currently known examples of such graphs, study their stable set polytope and state several conjectures on their structure. On the other hand, we show that the (weighted) chromatic number of certain h-perfect graphs can be obtained efficiently by rounding-up its fractional relaxation. This is related to conjectures of Goldberg and Seymour on edge-colorings. Finally, we introduce a new parameter on the complexity of the matching polytope and use it to give an efficient and elementary al- gorithm for testing h-perfection in line-graphs.Le calcul du nombre chromatique et la détermination d'une colo- ration optimale des sommets d'un graphe sont des problèmes NP- difficiles en général. Ils peuvent cependant être résolus en temps po- lynomial dans les graphes parfaits. Par ailleurs, la perfection d'un graphe peut être décidée efficacement. Les graphes parfaits sont caractérisés par la structure de leur poly- tope des stables : les facettes non-triviales sont définies exclusivement par des inégalités de cliques. Réciproquement, une structure similaire des facettes du polytope des stables détermine-t-elle des propriétés combinatoires et algorithmiques intéressantes? Un graphe est h-parfait si les facettes non-triviales de son polytope des stables sont définies par des inégalités de cliques et de circuits impairs. On ne connaît que peu de résultats analogues au cas des graphes parfaits pour la h-perfection, et on ne sait pas si les problèmes sont NP-difficiles. Par exemple, les complexités algorithmiques de la re- connaissance des graphes h-parfaits et du calcul de leur nombre chro- matique sont toujours ouvertes. Par ailleurs, on ne dispose pas de borne sur la différence entre le nombre chromatique et la taille maxi- mum d'une clique d'un graphe h-parfait. Dans cette thèse, nous montrons tout d'abord que les opérations de t-mineurs conservent la h-perfection (ce qui fournit une extension non triviale d'un résultat de Gerards et Shepherd pour la t-perfection). De plus, nous prouvons qu'elles préservent la propriété de décompo- sition entière du polytope des stables. Nous utilisons ce résultat pour répondre négativement à une question de Shepherd sur les graphes h-parfaits 3-colorables. L'étude des graphes minimalement h-imparfaits (relativement aux t-mineurs) est liée à la recherche d'une caractérisation co-NP com- binatoire de la h-perfection. Nous faisons l'inventaire des exemples connus de tels graphes, donnons une description de leur polytope des stables et énonçons plusieurs conjectures à leur propos. D'autre part, nous montrons que le nombre chromatique (pondéré) de certains graphes h-parfaits peut être obtenu efficacement en ar- rondissant sa relaxation fractionnaire à l'entier supérieur. Ce résultat implique notamment un nouveau cas d'une conjecture de Goldberg et Seymour sur la coloration d'arêtes. Enfin, nous présentons un nouveau paramètre de graphe associé aux facettes du polytope des couplages et l'utilisons pour donner un algorithme simple et efficace de reconnaissance des graphes h- parfaits dans la classe des graphes adjoints

The endothelium solves problems that endothelial cells do not know exist

The endothelium is the single layer of cells that lines the entire cardiovascular system and that regulates vascular tone and blood-tissue exchange, recruits blood cells, modulates blood clotting and determines the formation of new blood vessels. To control each function, the endothelium uses a remarkable sensory capability to continuously monitor vanishingly small changes in the concentration of many simultaneously arriving extracellular activators that each provide cues to physiological state. Here, we suggest that the extraordinary sensory capabilities of the endothelium does not come from single cells but from the combined activity of a large number of endothelial cells. Each cell has a limited, but distinctive, sensory capacity and shares information with neighbours so that sensing is distributed among cells. Communication of information among connected cells provides a system-level sensing substantially greater than the capabilities of any single cell and, as a collective, the endothelium solves sensory problems too complex for any single cell

Locally finite graphs with ends: A topological approach, I. Basic theory

AbstractThis paper is the first of three parts of a comprehensive survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. The first two parts of the survey together provide a suitable entry point to this field for new readers; they are available in combined form from the ArXiv [18]. They are complemented by a third part [28], which looks at the theory from an algebraic-topological point of view.The topological approach indicated above has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. While the second part of this survey [19] will concentrate on those applications, this first part explores the new theory as such: it introduces the basic concepts and facts, describes some of the proof techniques that have emerged over the past 10 years (as well as some of the pitfalls these proofs have in stall for the naive explorer), and establishes connections to neighbouring fields such as algebraic topology and infinite matroids. Numerous open problems are suggested

Removable Edges in 4-Connected Graphs

Research on structural characterizations of graphs is a very popular topic in graph theory. The concepts of contractible edges and removable edges of graphs are powerful tools to study the structure of graphs and to prove properties of graphs by induction. \ud In 1998, Yin gave a convenient method to construct 4-connected graphs by using the existence of removable edges and contractible edges. He showed that a 4-connected graph can be obtained from a 2-cyclic graph by the following four operations: (i) adding edges, (ii) splitting vertices, (iii) adding vertices and removing edges, and (iv) extending vertices. Based on the above operations, we gave the following definition of removable edges in 4-connected graphs. Definition: Let $G$ be a 4-connected graph. For an edge $e$ of $G$, we perform the following operations on $G$: First, delete the edge $e$ from $G$, resulting in the graph $G-e$; Second, for each vertex $x$ of degree 3 in $G-e$, delete $x$ from $G-e$ and then completely connect the 3 neighbors of $x$ by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by $G \ominus e$. If $G\ominus e$ is still 4-connected, then the edge $e$ is called "removable"; otherwise, $e$ is called "unremovable". In the thesis we get the following results: (1) we study how many removable edges may exist in a cycle of a 4-connected graph, and we give examples to show that our results are in some sense the best possible. (2) We obtain results on removable edges in a longest cycle of a 4-connected graph. We also show that for a 4-connected graph $G$ of minimum degree at least 5 or girth at least 4, any edge of $G$ is removable or contractible. (3) We study the distribution of removable edges on a Hamilton cycle of a 4-connected graph, and show that our results cannot be improved in some sense. (4) We prove that every 4-connected graph of order at least six except 2-cyclic graph with order 6 has at least $(4|G|+16)/7$ removable edges. We also give a structural characterization of 4-connected graphs for which the lower bound is sharp. (5) We study how many removable edges there are in a spanning tree of a 4-connected graph and how many removable edges exist outside a cycle of a 4-connected graph. We also give examples to show that our results can not be improved in some sense

A Study of Arc Strong Connectivity of Digraphs

My dissertation research was motivated by Matula and his study of a quantity he called the strength of a graph G, kappa\u27( G) = max{lcub}kappa\u27(H) : H G{rcub}. (Abstract shortened by ProQuest.)