On b-colorings and b-continuity of graphs

A b-coloring of G is a proper vertex coloring such that there is a vertex in each color class, which is adjacent to at least one vertex in every other color class. Such a vertex is called a color-dominating vertex. The b-chromatic number of G is the largest k such that there is a b-coloring of G by k colors. Moreover, if for every integer k, between chromatic number and b-chromatic number, there exists a b-coloring of G by k colors, then G is b-continuous. Determining the b-chromatic number of a graph G and the decision whether the given graph G is b-continuous or not is NP-hard. Therefore, it is interesting to find new results on b-colorings and b-continuity for special graphs. In this thesis, for several graph classes some exact values as well as bounds of the b-chromatic number were ascertained. Among all we considered graphs whose independence number, clique number, or minimum degree is close to its order as well as bipartite graphs. The investigation of bipartite graphs was based on considering of the so-called bicomplement which is used to determine the b-chromatic number of special bipartite graphs, in particular those whose bicomplement has a simple structure. Then we studied some graphs whose b-chromatic number is close to its t-degree. At last, the b-continuity of some graphs is studied, for example, for graphs whose b-chromatic number was already established in this thesis. In particular, we could prove that Halin graphs are b-continuous.:Contents 1 Introduction 2 Preliminaries 2.1 Basic terminology 2.2 Colorings of graphs 2.2.1 Vertex colorings 2.2.2 a-colorings 3 b-colorings 3.1 General bounds on the b-chromatic number 3.2 Exact values of the b-chromatic number for special graphs 3.2.1 Graphs with maximum degree at most 2 3.2.2 Graphs with independence number close to its order 3.2.3 Graphs with minimum degree close to its order 3.2.4 Graphs G with independence number plus clique number at most number of vertices 3.2.5 Further known results for special graphs 3.3 Bipartite graphs 3.3.1 General bounds on the b-chromatic number for bipartite graphs 3.3.2 The bicomplement 3.3.3 Bicomplements with simple structure 3.4 Graphs with b-chromatic number close to its t-degree 3.4.1 Regular graphs 3.4.2 Trees and Cacti 3.4.3 Halin graphs 4 b-continuity 4.1 b-spectrum of special graphs 4.2 b-continuous graph classes 4.2.1 Known b-continuous graph classes 4.2.2 Halin graphs 4.3 Further graph properties concerning b-colorings 4.3.1 b-monotonicity 4.3.2 b-perfectness 5 Conclusion Bibliograph

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Complexity of Maximum Cut on Interval Graphs

We resolve the longstanding open problem concerning the computational complexity of Max Cut on interval graphs by showing that it is NP-complete

On vertex independence number of uniform hypergraphs

Abstract Let H be an r-uniform hypergraph with r ≥ 2 and let α(H) be its vertex independence number. In the paper bounds of α(H) are given for different uniform hypergraphs: if H has no isolated vertex, then in terms of the degrees, and for triangle-free linear H in terms of the order and average degree.</jats:p

Kernelization and Enumeration: New Approaches to Solving Hard Problems

NP-Hardness is a well-known theory to identify the hardness of computational problems. It is believed that NP-Hard problems are unlikely to admit polynomial-time algorithms. However since many NP-Hard problems are of practical significance, different approaches are proposed to solve them: Approximation algorithms, randomized algorithms and heuristic algorithms. None of the approaches meet the practical needs. Recently parameterized computation and complexity has attracted a lot of attention and been a fruitful branch of the study of efficient algorithms. By taking advantage of the moderate value of parameters in many practical instances, we can design efficient algorithms for the NP-Hard problems in practice. In this dissertation, we discuss a new approach to design efficient parameterized algorithms, kernelization. The motivation is that instances of small size are easier to solve. Roughly speaking, kernelization is a preprocess on the input instances and is able to significantly reduce their sizes. We present a 2k kernel for the cluster editing problem, which improves the previous best kernel of size 4k; We also present a linear kernel of size 7k 2d for the d-cluster editing problem, which is the first linear kernel for the problem. The kernelization algorithm is simple and easy to implement. We propose a quadratic kernel for the pseudo-achromatic number problem. This implies that the problem is tractable in term of parameterized complexity. We also study the general problem, the vertex grouping problem and prove it is intractable in term of parameterized complexity. In practice, many problems seek a set of good solutions instead of a good solution. Motivated by this, we present the framework to study enumerability in term of parameterized complexity. We study three popular techniques for the design of parameterized algorithms, and show that combining with effective enumeration techniques, they could be transferred to design efficient enumeration algorithms

Computational Complexity And Algorithms For Dirty Data Evaluation And Repairing

In this dissertation, we study the dirty data evaluation and repairing problem in relational database. Dirty data is usually inconsistent, inaccurate, incomplete and stale. Existing methods and theories of consistency describe using integrity constraints, such as data dependencies. However, integrity constraints are good at detection but not at evaluating the degree of data inconsistency and cannot guide the data repairing. This dissertation first studies the computational complexity of and algorithms for the database inconsistency evaluation. We define and use the minimum tuple deletion to evaluate the database inconsistency. For such minimum tuple deletion problem, we study the relationship between the size of rule set and its computational complexity. We show that the minimum tuple deletion problem is NP-hard to approximate the minimum tuple deletion within 17/16 if given three functional dependencies and four attributes involved. A near optimal approximated algorithm for computing the minimum tuple deletion is proposed with a ratio of 2 − 1/2r , where r is the number of given functional dependencies. To guide the data repairing, this dissertation also investigates the data repairing method by using query feedbacks, formally studies two decision problems, functional dependency restricted deletion and insertion propagation problem, corresponding to the feedbacks of deletion and insertion. A comprehensive analysis on both combined and data complexity of the cases is provided by considering different relational operators and feedback types. We have identified the intractable and tractable cases to picture the complexity hierarchy of these problems, and provided the efficient algorithm on these tractable cases. Two improvements are proposed, one focuses on figuring out the minimum vertex cover in conflict graph to improve the upper bound of tuple deletion problem, and the other one is a better dichotomy for deletion and insertion propagation problems at the absence of functional dependencies from the point of respectively considering data, combined and parameterized complexities

The Role of Clustered Organization and Generation of Mixed Properties in Macaque V2

Throughout the mammalian cortex, neurons of similar response characteristics group together into topographic functional domains. The genesis and role of this organization remains in question, but it has been proposed to affect the mixed properties of neurons. These types of neurons possess multiple receptive field preferences, such as a cell responding to a color and an oriented stimulus. To examine the functionality of clustered organization and their effect in generation of neurons possessing mixed properties, this dissertation examined the secondary visual cortex (V2) of the Macaca fasicularis. This particular cortex is comprised of domains organized according to distinct visual stimulus components, specifically clusters of neurons partitioned by color and orientation preferences within a close proximity. In the first series of experiments (Chapter 3), a computer model of a cortical area based upon macaque V2 investigated the effect of clusters of like-preferring neurons on the probability of two different preference terminals synapsing on a particular cell. These results indicate that presence of at least one cluster significantly increases the probability of multiple preferences arriving at a neuron. The second series of experiments (Chapter 4) used single unit electrophysiology to investigate the temporal properties of V2 neurons in response to achromatic and colored oriented stimuli. With the addition of color to the stimulus, an increase in latency, an increase to the time point of the maximum rate of firing, and a decreased initial-phase response with a sustained later-phase response were observed. These studies indicate that functional clusters of neurons significantly increase the joint probability of the co-localization of differing preference terminals, potentially yielding neurons with mixed preferences through these intra-areal connections. Furthermore, the temporal characteristics of V2 neurons, as seen in observed latency and time of maximum spiking, support this idea of domain-enhanced intra-areal integration

A study of pupil response components in human vision

The overall aim of the research described in this thesis was to investigate the basic mechanisms of the pupil of the eye in relation to human vision and brain function. It also evaluates the potential application of new research techniques to clinical studies that involve assessment of the visual function. Pupil response components were investigated in normal subjects and in patients with damaged visual pathways. A series of experiments were carried out to investigate the pupil response to periodic modulation of several stimulation parameters such as: luminance contrast, stimulus size, spatial and temporal frequency content, and colour. Much larger responses were found for square-wave as compared to sinusoidal luminance modulation. A model with two populations of neurones (sustained and transient) was developed to explain the non-linear combination of two response components in the light reflex. In contrast to these findings, responses to isoluminant coloured stimuli or sinusoidal gratings whose spatial average luminance is equal to that of the background do not depend of the temporal wave-form of the stimulation. Studies in patients with lesions to specific areas in the brain suggest that these responses are caused by a transient weakening of the steady central sympathetic inhibition to parasympathetic neurones innervating the sphincter muscle as a result of cortical processing of specific stimulus attributes such as colour and spatial structure. Pupil measurements in patients suffering from demyelinating neurological disorders such as multiple sclerosis and optic neuritis also confirm the existence of distinct pupil response components and reveal selective loss to chromatic and luminance pathways. The results indicate a preferential damage to thinner axons which are thought to predominantly mediate the chromatic responses. These studies suggest that the use of modern pupillometric techniques in neuro-ophthalmology can yield useful information on the extent of the damage and the progression of disease in lesions of the optic nerve.EThOS - Electronic Theses Online ServiceGBUnited Kingdo