How to Draw a Graph

This dissertation solves two problems relating to the structure of graphs. The first of these is motivated by Kuratowski's Theorem, perhaps the most famous result in graph theory. This theorem states the $K\sb5$ and K$\sb{3,3}$ are the only non-planar graphs for which both $G\\ e$, the deletion of the edge e, and G/e, the contraction of the edge e, are planar for all edges e of G. We characterize the non-planar graphs for which $G\\ e$ or G/e is planar for all edges e of G. The second problem we solve is motivated by Tutte's wheels theorem. An immediate consequence of this theorem is that the wheel graphs are the basic building blocks for the collection of simple, 3-connected graphs. Therefore it is of interest to examine the structure of the graphs that do not have a minor isomorphic to the k-spoked wheel, $W\sb{k}$, for small values of k. Dirac determined that the graphs having no $W\sb3$-minor are the series-parallel networks. It follows easily from Tutte's wheels theorem is that $W\sb3$ is the only graph that has a $W\sb3$-minor and no $W\sb4$-minor. Oxley characterized the graphs that have a $W\sb4$-minor and no $W\sb5$-minor. We characterize the planar graphs that have a $W\sb5$-minor and no $W\sb6$-minor. We also determine the best-possible upper bound on the number of edges of such a graph.

This paper presents an efficient approach for modeling chain derives using multibody dynamics formulation based on the spatial algebra. The recursive nonlinear dynamic equations of motion are formulated using spatial Cartesian coordinates and joint variables to form an augmented set of differential-algebraic equations. The spatial algebra is used to express the kinematic and dynamic equations leading to consistent and compact set of equations. The connectivity graph is used to derive the system connectivity matrix based on the system topological relations. The connectivity matrix is used to eliminate the Cartesian quantities and to project the forces and inertia into the joint subspace. This approach will result in a minimum set of equation and can avoid iteratively solving the system of differential and algebraic equations to satisfy the constraint equations. In order to accurately capture the full dynamics of the chain links, each link in the chain is modeled as rigid body with full 6 degrees of freedom. To avoid singularities in closed loop configurations, the chain drive is considered a kinematically decoupled subsystem and the interaction between the links and other system components is modeled using force elements. The out-of-plane misalignment between the sprockets can be easily modeled using a compliant force element to model the joints between each two adjacent links. The nonlinear three dimensional contact forces between the chain links and the sprockets are modeled using elastic spring-damper element and accounts for the sliding friction. The proposed approach can be used to model complex drive chain, bicycle chain as well as conveyance systems. Results show that realistic behavior of the chain as well as out-of-plane vibration can be easily captured using the presented approach. The proposed approach for chain drive subsystem could be easily appended to any other multibody simulation system.

Neuronal connectivity graphs can give valuable insight into the structure of the brain and patterns of functional activity. In this paper we describe a connectivity visualization technique using contrasting styles to emphasize the differences between pairs of graphs. Our proposed technique enables interactive exploration of connectivity graphs at the various levels of the hierarchical structure of the brain — atlas regions, brain lobes and cerebral hemispheres. We demonstrate our methods on data obtained from matched pairs of subjects in a Parkinson's disease (PD) study. Control group data is displayed in a smooth, organic style with continuous transitions between views at different levels of the brain hierarchy using edge bundling and a new node bundling technique. Data from the PD group is displayed with glitch and noise effects inspired by PD symptoms, highlighting the differences between each subject and their matched pair from the control group. We conclude by describing avenues for further evaluation of this preliminary work.

We introduce the notion of a connectivity graph —an abstract representation of the topology of concurrently interacting entities, which allows us to encapsulate generic principles of reasoning about deadlock freedom . Connectivity graphs are parametric in their vertices (representing entities like threads and channels) and their edges (representing references between entities) with labels (representing interaction protocols). We prove deadlock and memory leak freedom in the style of progress and preservation and use separation logic as a meta theoretic tool to treat connectivity graph edges and labels substructurally. To prove preservation locally, we distill generic separation logic rules for local graph transformations that preserve acyclicity of the connectivity graph. To prove global progress locally, we introduce a waiting induction principle for acyclic connectivity graphs. We mechanize our results in Coq, and instantiate our method with a higher-order binary session-typed language to obtain the first mechanized proof of deadlock and leak freedom.

Parallel MRI (pMRI) reconstruction techniques are commonly used to reduce scan time by undersampling the k-space data. GRAPPA, a k-space based pMRI technique, is widely used clinically because of its robustness. In GRAPPA, the missing k-space data are estimated by solving a set of linear equations; however, this set of equations does not take advantage of the correlations within the missing k-space data. All k-space data in a neighborhood acquired from a phased-array coil are correlated. The correlation can be estimated easily as a self-constraint condition, and formulated as an extra set of linear equations to improve the performance of GRAPPA. The authors propose a modified k-space based pMRI technique called self-constraint GRAPPA (SC-GRAPPA) which combines the linear equations of GRAPPA with these extra equations to solve for the missing k-space data. Since SC-GRAPPA utilizes a least-squares solution of the linear equations, it has a closed-form solution that does not require an iterative solver. The SC-GRAPPA equation was derived by incorporating GRAPPA as a prior estimate. SC-GRAPPA was tested in a uniform phantom and two normal volunteers. MR real-time cardiac cine images with acceleration rate 5 and 6 were reconstructed using GRAPPA and SC-GRAPPA. SC-GRAPPA showed a significantly lower artifact level, and a greater than 10% overall signal-to-noise ratio (SNR) gain over GRAPPA, with more significant SNR gain observed in low-SNR regions of the images. SC-GRAPPA offers improved pMRI reconstruction, and is expected to benefit clinical imaging applications in the future.

The traditional one-component transmission error of parallel-axis helical gears is generalized to a three-component transmission error which characterizes the composite displacement in the plane-of-contact resulting from arbitrary small deviations in the positions of both gears of a meshing pair from the positions of their rigid perfect involute counterparts. A set of linear algebraic equations is derived for the contribution to the three generalized transmission error components arising from elastic deformations of the teeth and gear bodies and deviations of the tooth running surfaces from equispaced perfect involute surfaces. It is shown how to combine this set of equations with the generalized transmission error definition and the equations of motion of a gear system to predict the dynamic response of gear elements in the system. For the case of negligible gearbody and bearing/bearing support inertial forces, an additional set of algebraic equations that includes the effects of bearing flexibility and misalignment is derived. Combining the solution of this set of equations with the above-mentioned generalized transmission error equations yields the three-component generalized static transmission error.

Let G be a graph, and let t 0 be a real number. Then G is t-tough if t!(G − S) jSj for all S V (G) with !(G − S) > 1, where !(G − S) denotes the number of components of G − S. The toughness of G, denoted by (G), is the maximum value of t for which G is t-tough (taking (Kn) = 1 for all n 1). G is minimally t-tough if (G) = t and (H) < t for every proper spanning subgraph H of G. We discuss how the toughness of (spanning) subgraphs of G and related graphs depends on (G), we give some sucient (degree) conditions implying (G) t, and we study which subdivisions of 2-connected graphs have minimally 2-tough squares.

The Clar number of a (hydro)carbon molecule, introduced by Clar (The aromatic sextet, 1972), is the maximum number of mutually disjoint resonant hexagons in the molecule. Calculating the Clar number can be formulated as an optimization problem on 2-connected plane graphs. Namely, it is the maximum number of mutually disjoint even faces a perfect matching can simultaneously alternate on. It was proved by Abeledo and Atkinson (Linear Algebra Appl 420(2):441–448, 2007) that the Clar number can be computed in polynomial time if the plane graph has even faces only. We prove that calculating the Clar number in general 2-connected plane graphs is \(\mathsf {NP}\)-hard. We also prove \(\mathsf {NP}\)-hardness of the maximum independent set problem for 2-connected plane graphs with odd faces only, which may be of independent interest. Finally, we give an exact algorithm that determines the Clar number of a given 2-connected plane graph. The algorithm has a polynomial running time if the length of the shortest odd join in the planar dual graph is fixed, which gives an efficient algorithm for some fullerene classes, such as carbon nanotubes.

We have used the elements of graph theory to describe NMR cross‐relaxation networks in macromolecules and to analyze different experiments used for topological editing. We propose a new experiment, block decoupled NOESY (B.D. NOESY), that splits the cross‐relaxation network of a macromolecular system into two noninteracting subdomains. Splitting is achieved by arranging for the effective z‐components of the magnetization in one subdomain, as viewed from the other subdomain, to be zero. The B.D. NOESY experiment, in favorable cases, can usefully simplify the analysis of cross‐relaxation spectra by removing certain ambiguities. We demonstrate the method by separating the aromatic resonances from the rest of the cross‐relaxation network of a small globular protein: turkey ovomucoid third domain, Mr 6000. The resulting spectrum provides a clearer picture of cross‐relaxation pathways that involve only aromatic or only aliphatic spins. By comparing the B.D. NOESY spectrum with the normal NOESY spectrum, we were able to identify cross peaks that contain contributions from indirect magnetization transfer (spin diffusion) mediated by the aromatic side chains. In the terminology of graph theory, these experiments decompose topological networks of cross‐relaxation into two subgraphs, the join of which generates the original graph.

The problem of calculating the reflected and transmitted fields as a function of scan angle in an infinite phased array of parallel-plate waveguides is studied on an analytical basis. Two infinite sets of simultaneous linear equations are derived for the scattered fields internal and external to the waveguides. These sets of equations are attacked by a newly developed modified residue calculus method. Application of the results to the problem of designing rectangular waveguide arrays with improved impedance performance over a wide range of H -plane scan angles is discussed. It is indicated that the present analysis may serve as a first step toward investigating the more general problem of phased arrays scanning at an arbitrary angle.

An approximate inverse preconditioner and its implementation for conjugate gradient method

A Direct Discrete-time IDA-PBC Design Method for a Class of Underactuated Hamiltonian Systems

An international conference on large sparse sets of simultaneous linear equations was held at St. Catherine's College, Oxford, from 5th to 8th April, 1970. It was organized by the Institute of Mathematics and its Applications, and it brought together many specialists from widely differing fields, all dealing with the problem of solving large sets of linear equations.

Design and analysis of new complex zeroing neural network for a set of dynamic complex linear equations

Freuder and Elfe (1996) introduced Neighborhood Inverse Consistency (NIC) as a strong local consistency property for binary CSPs. While enforcing NIC can significantly filter the variables domains, the proposed algorithm is too costly to be used on dense graphs or for lookahead during search. In this paper, we introduce and characterize Relational Neighborhood Inverse Consistency (RNIC) as a local consistency property that operates on the dual graph of a non-binary CSP. We describe and characterize a practical algorithm for enforcing it. We argue that defining RNIC on the dual graph unveils unsuspected opportunities to reduce the computational cost of our algorithm and increase its filtering effectiveness. We show how to achieve those effects by modifying the topology of the dual graph, yielding new variations the RNIC property. We also introduce an adaptive strategy to automatically select the appropriate property to enforce given the connectivity of the dual graph. We integrate the resulting techniques as full lookahead strategies in a backtrack search procedure for solving CSPs, and demonstrate the effectiveness of our approach for solving known difficult benchmark problems.