Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs

In this thesis we investigate three different aspects of graph theory. Firstly, we consider interesecting systems of independent sets in graphs, and the extension of the classical theorem of Erdos, Ko and Rado to graphs. Our main results are a proof of an Erdos-Ko-Rado type theorem for a class of trees, and a class of trees which form counterexamples to a conjecture of Hurlberg and Kamat, in such a way that extends the previous counterexamples given by Baber. Secondly, we investigate perfect graphs - specifically, edge modification aspects of perfect graphs and their subclasses. We give some alternative characterisations of perfect graphs in terms of edge modification, as well as considering the possible connection of the critically perfect graphs - previously studied by Wagler - to the Strong Perfect Graph Theorem. We prove that the situation where critically perfect graphs arise has no analogue in seven different subclasses of perfect graphs (e.g. chordal, comparability graphs), and consider the connectivity of a bipartite reconfiguration-type graph associated to each of these subclasses. Thirdly, we consider a graph theoretic structure called a bireflexive graph where every vertex is both adjacent and nonadjacent to itself, and use this to characterise modular decompositions as the surjective homomorphisms of these structures. We examine some analogues of some graph theoretic notions and define a “dual” version of the reconstruction conjecture

Separability and Vertex Ordering of Graphs

Many graph optimization problems, such as finding an optimal coloring, or a largest clique, can be solved by a divide-and-conquer approach. One such well-known technique is decomposition by clique separators where a graph is decomposed into special induced subgraphs along their clique separators. While the most common practice of this method employs minimal clique separators, in this work we study other variations as well. We strive to characterize their structure and in particular the bound on the number of atoms. In fact, we strengthen the known bounds for the general clique cutset decomposition and the minimal clique separator decomposition. Graph ordering is the arrangement of a graph’s vertices according to a certain logic and is a useful tool in optimization problems. Special types of vertices are often recognized in graph classes, for instance it is well-known every chordal graph contains a simplicial vertex. Vertex-ordering, based on such properties, have originated many linear time algorithms. We propose to define a new family named SE-Class such that every graph belonging to this family inherently contains a simplicial extreme, that is a vertex which is either simplicial or has exactly two neighbors which are non-adjacent. Our family lends itself to an ordering based on simplicial extreme vertices (named SEO) which we demonstrate to be advantageous for the coloring and maximum clique problems. In addition, we examine the relation of SE-Class to the family of (Even-Hole, Kite)-free graphs and show a linear time generation of SEO for (Even-Hole, Diamond, Claw)-free graphs. We showcase the applications of those two core tools, namely clique-based decomposition and vertex ordering, on the (Even-Hole, Kite)-free family

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

Double-stranded DNA organization in bacteriophage heads: an alternative toroid-based model

Studies of the organization of double-stranded DNA within bacteriophage heads during the past four decades have produced a wealth of data. However, despite the presentation of numerous models, the true organization of DNA within phage heads remains unresolved. The observations of toroidal DNA structures in electron micrographs of phage lysates have long been cited as support for the organization of DNA in a spool-like fashion. This particular model, like all other models, has not been found to be consistent will all available data. Recently we proposed that DNA within toroidal condensates produced in vitro is organized in a manner significantly different from that suggested by the spool model. This new toroid model has allowed the development of an alternative model for DNA organization within bacteriophage heads that is consistent with a wide range of biophysical data. Here we propose that bacteriophage DNA is packaged in a toroid that is folded into a highly compact structure

Combinatorial Optimization

Combinatorial Optimization is an active research area that developed from the rich interaction among many mathematical areas, including combinatorics, graph theory, geometry, optimization, probability, theoretical computer science, and many others. It combines algorithmic and complexity analysis with a mature mathematical foundation and it yields both basic research and applications in manifold areas such as, for example, communications, economics, traffic, network design, VLSI, scheduling, production, computational biology, to name just a few. Through strong inner ties to other mathematical fields it has been contributing to and benefiting from areas such as, for example, discrete and convex geometry, convex and nonlinear optimization, algebraic and topological methods, geometry of numbers, matroids and combinatorics, and mathematical programming. Moreover, with respect to applications and algorithmic complexity, Combinatorial Optimization is an essential link between mathematics, computer science and modern applications in data science, economics, and industry

Accurate Docking is Achieved by Decoupling Systematic Sampling from Scoring

This dissertation discusses two main projects from my thesis work. The first project focuses on the development of a small molecule docking program, SKATE, for drug discovery. The second project focuses on the critical analysis of the thermal stability of a mini-protein, FSD-1. SKATE is a novel approach to small molecule docking. It removes any inter-dependence between sampling and scoring to improve docking accuracy. SKATE systematically and exhaustively samples a ligand\u27s conformational, rotational and translational degrees of freedom, as constrained by a receptor pocket, to find sterically allowed poses. A total of 266 ligands were re-docked to their respective receptors to assess SKATE\u27s performance. The results show that SKATE was able to sample poses within 2 Angstrom RMSD of the native structure for 97% of the cases. The best performing scoring function was able to rank a pose that is within 2 Angstrom RMSD of the native structure as the top-scoring pose for 83% of the cases. Compared to published data, SKATE has a higher self-docking accuracy rate than or is at least comparable to GOLD, Glide, MolDock and Surflex. The cross-docking accuracy of SKATE was assessed by docking 83 ligands to their respective receptors. The cross-docking results were comparable to those in published methods. Mini-proteins that contain fewer than 50 amino acids often serve as model systems for studying protein folding because their small size makes long time-scale simulations possible. However, not all mini-proteins are created equal. The stability and structure of FSD-1, a 28-residue mini-protein that adopts the Beta Beta Alpha zinc-finger motif independent of zinc binding, was investigated using circular dichroism: CD), differential scanning calorimetry: DSC), and replica-exchange molecular dynamics: REMD). FSD-1\u27s broad melting transition, similar to that of a helix-to-coil transition, was observed in CD, DSC, and REMD experiments. The N-terminal -hairpin was found to be flexible. FSD-1\u27s apparent melting temperature of 41 degrees C may be a reflection of the melting of its alpha-helical segment instead of the entire protein. Thus, FSD-1\u27s status as a model system for studying protein folding should be reconsidered despite its attractiveness for being small in size and it was designed to contain essential helix, sheet, and turn secondary structures. An electronic copy of this dissertation is available online at www.ccb.wustl.edu/~jafen

Network-centric methods for heterogeneous multiagent systems

We present tools for a network topology based characterization of heterogeneity in multiagent systems, thereby providing a framework for the analysis and design of heterogeneous multiagent networks from a network structure view-point. In heterogeneous networks, agents with a diverse set of resources coordinate with each other. Coordination among different agents and the structure of the underlying network topology have significant impacts on the overall behavior and functionality of the system. Using constructs from graph theory, a qualitative as well as a quantitative analysis is performed to examine an inter-relationship between the network topology and the distribution of agents with various capabilities in heterogeneous networks. Our goal is to allow agents maximally exploit heterogeneous resources available within the network through local interactions, thus exploring a promise heterogeneous networks hold to accomplish complicated tasks by leveraging upon the assorted capabilities of agents. For a reliable operations of such systems, the issue of security against intrusions and malicious agents is also addressed. We provide a scheme to secure a network against a sequence of intruder attacks through a set of heterogeneous guards. Moreover, robustness of networked systems against noise corruption and structural changes in the underlying network topology is also examined.Ph.D

Airborne Directional Networking: Topology Control Protocol Design

This research identifies and evaluates the impact of several architectural design choices in relation to airborne networking in contested environments related to autonomous topology control. Using simulation, we evaluate topology reconfiguration effectiveness using classical performance metrics for different point-to-point communication architectures. Our attention is focused on the design choices which have the greatest impact on reliability, scalability, and performance. In this work, we discuss the impact of several practical considerations of airborne networking in contested environments related to autonomous topology control modeling. Using simulation, we derive multiple classical performance metrics to evaluate topology reconfiguration effectiveness for different point-to-point communication architecture attributes for the purpose of qualifying protocol design elements

Design study for Thermal Infrared Multispectral Scanner (TIMS)

The feasibility of dividing the 8-12 micrometer thermal infrared wavelength region into six spectral bands by an airborne line scanner system was investigated. By combining an existing scanner design with a 6 band spectrometer, a system for the remote sensing of Earth resources was developed. The elements in the spectrometer include an off axis reflective collimator, a reflective diffraction grating, a triplet germanium imaging lens, a photoconductive mercury cadmium telluride sensor array, and the mechanical assembly to hold these parts and maintain their optical alignment across a broad temperature range. The existing scanner design was modified to accept the new spectrometer and two field filling thermal reference sources