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

Triangle density and contractibility

Let G be a noncomplete k -connected graph such that the graphs obtained from contracting any edge in G are not k-connected, and let t(G) denote the number of triangles in G. Thomassen proved t(G) ≥ 1, which was later improved by Mader to t(G) ≥ 1/3|V(G)|. Here we show t(G) ≥ 2/3|V(G)| (which is best possible in general). Furthermore it is proved that, for k ≥ 4, a k-connected graph without two disjoint triangles must contain an edge not contained in a triangle whose contraction yields a k-connected graph. As an application, for k ≥ 4 every k-connected graph G admits two disjoint induced cycles C1, C2 such that G - V(C1) and G - V(C2) are (k - 3)-connected

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

Genome Assembly Techniques

Since the publication of the human genome in 2001, the price and the time of DNA sequencing have dropped dramatically. The genome of many more species have since been sequenced, and genome sequencing is an ever more important tool for biologists. This trend will likely revolutionize biology and medicine in the near future where the genome sequence of each individual person, instead of a model genome for the human, becomes readily accessible. Nevertheless, genome assembly remains a challenging computational problem, even more so with second generation sequencing technologies which generate a greater amount of data and make the assembly process more complex. Research to quickly, cheaply and accurately assemble the increasing amount of DNA sequenced is of great practical importance. In the first part of this thesis, we present two software developed to improve genome assemblies. First, Jellyfish is a fast k-mer counter, capable of handling large data sets. k-mer frequencies are central to many tasks in genome assembly (e.g. for error correction, finding read overlaps) and other study of the genome (e.g. finding highly repeated sequences such as transposons). Second, Chromosome Builder is a scaffolder and contig placement software. It aims at improving the accuracy of genome assembly. In the second part of this thesis we explore several problems dealing with graphs. The theory of graphs can be used to solve many computational problems. For example, the genome assembly problem can be represented as finding an Eulerian path in a de Bruijn graph. The physical interactions between proteins (PPI network), or between transcription factors and genes (regulatory networks), are naturally expressed as graphs. First, we introduce the concept of "exactly 3-edge-connected" graphs. These graphs have only a remote biological motivation but are interesting in their own right. Second, we study the reconstruction of ancestral network which aims at inferring the state of ancestral species' biological networks based on the networks of current species

Aspects of pediatric hemodynamics. A study of young children undergoing corrective heart surgery.

AbstractBackground: Assessing critically ill patients is demanding because their clinical signs are not always easy to interpret. This is particularly true for pediatric patients, due to their small size and robust circulatory compensatorymechanism. Physicians working in pediatric intensive care settings seldom use invasive monitoring, due to possible complications and the risks involved. Instead, they rely on simple vital parameters and blood gases during their clinical work. However, a number of studies have shown that this clinical approach can provide inaccurate data when dealing with critically ill patients. There are very few monitors currently available that have been fully validated for estimating hemodynamic parameters in young children. A less invasive hemodynamic monitor could help physicians make clinical decisions and improve future levels of care, by providing more reliable information on the perioperative hemodynamic status of pediatric patients, without additional risks.Purpose: The primary aim of the studies in this thesis was to look at different aspects of pediatric hemodynamics, by comparing the novel hemodynamic COstatus monitor, with earlier reference methods. In our studies we analyzed: 1) aggrement and precision of cardiac output, 2) detection and estimation of of intracardiac shunts and 3) estimation of different blood volumes. A secondary aim of this thesis was to establish reference values of hemodynamic values in young children using COstatus, and comparing idexing of same values with body surface area and body weight.Methods: Children (under 15 kilograms) undergoing corrective heart surgery at Lund Children´s Hospital were enrolled in our studies.Results:Paper I – Estimation of cardiac outputThe COstatus provided excellent precision and agreement in estimating cardiac output in young children, compared with perivascular flow probe placed around the ascending aorta.Paper II – Estimation of intracardiac shunts. The COstatus detected intracardiac shunts to the same extent as the “gold standard” echocardiography. However, it slightly underestimated the degrees of the shunts in small and moderate shunts when it was compared to two other reference methods, namely perivascular ultrasonic flow probes (placed around the pulmonary truncus and ascending aorta) and the oximetric shunt equation (using arterial and venous blood gases).Paper III – Normalization of hemodynamic parameters. Body weight produced a better normalization of hemodynamic parameters than body surface area in young children.Paper IV – Estimation of oxygen uptake. Indirect calorimetry seemed to overestimate oxygen uptake in youngchildren, compared to the reverse Fick method.Paper V – Estimation of body surface area. Commonly used body surface area formulas disagreed in youngchildren, Mosteller formula came closest to the mean body surface area.Conclusions: COstatus is accurate, precise and less invasive than earlier reference methods and might enable future cardiac output comparison studies in the intensive care setting. COstatus detects shunts accurately but algorithms for shunt size estimations might be overly cautious. Caution is advised regarding the use of indirect calorimetry and direct Fick method in cardiac output comparison studies

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

Development and Evaluation of an Impedance Cardiographic System to Measure Cardiac Output and Other Cardiac Parameters, 1 July 1968 - 30 June 1969

Impedance cardiographic system to measure cardiac output and cardiovascular function