A Study on Graph Coloring and Digraph Connectivity

This dissertation focuses on coloring problems in graphs and connectivity problems in digraphs. We obtain the following advances in both directions.;1. Results in graph coloring. For integers k,r \u3e 0, a (k,r)-coloring of a graph G is a proper coloring on the vertices of G with k colors such that every vertex v of degree d( v) is adjacent to vertices with at least min{lcub}d( v),r{rcub} different colors. The r-hued chromatic number, denoted by chir(G ), is the smallest integer k for which a graph G has a (k,r)-coloring.;For a k-list assignment L to vertices of a graph G, a linear (L,r)-coloring of a graph G is a coloring c of the vertices of G such that for every vertex v of degree d(v), c(v)∈ L(v), the number of colors used by the neighbors of v is at least min{lcub}dG(v), r{rcub}, and such that for any two distinct colors i and j, every component of G[c --1({lcub}i,j{rcub})] must be a path. The linear list r-hued chromatic number of a graph G, denoted chiℓ L,r(G), is the smallest integer k such that for every k-list L, G has a linear (L,r)-coloring. Let Mad( G) denotes the maximum subgraph average degree of a graph G. We prove the following. (i) If G is a K3,3-minor free graph, then chi2(G) ≤ 5 and chi3(G) ≤ 10. Moreover, the bound of chi2( G) ≤ 5 is best possible. (ii) If G is a P4-free graph, then chir(G) ≤q chi( G) + 2(r -- 1), and this bound is best possible. (iii) If G is a P5-free bipartite graph, then chir( G) ≤ rchi(G), and this bound is best possible. (iv) If G is a P5-free graph, then chi2(G) ≤ 2chi(G), and this bound is best possible. (v) If G is a graph with maximum degree Delta, then each of the following holds. (i) If Delta ≥ 9 and Mad(G) \u3c 7/3, then chiℓL,r( G) ≤ max{lcub}lceil Delta/2 rceil + 1, r + 1{rcub}. (ii) If Delta ≥ 7 and Mad(G)\u3c 12/5, then chiℓ L,r(G)≤ max{lcub}lceil Delta/2 rceil + 2, r + 2{rcub}. (iii) If Delta ≥ 7 and Mad(G) \u3c 5/2, then chi ℓL,r(G)≤ max{lcub}lcei Delta/2 rceil + 3, r + 3{rcub}. (vi) If G is a K 4-minor free graph, then chiℓL,r( G) ≤ max{lcub}r,lceilDelta/2\rceil{rcub} + lceilDelta/2rceil + 2. (vii) Every planar graph G with maximum degree Delta has chiℓL,r(G) ≤ Delta + 7.;2. Results in digraph connectivity. For a graph G, let kappa( G), kappa\u27(G), delta(G) and tau( G) denote the connectivity, the edge-connectivity, the minimum degree and the number of edge-disjoint spanning trees of G, respectively. Let f(G) denote kappa(G), kappa\u27( G), or Delta(G), and define f¯( G) = max{lcub}f(H): H is a subgraph of G{rcub}. An edge cut X of a graph G is restricted if X does not contain all edges incident with a vertex in G. The restricted edge-connectivity of G, denoted by lambda2(G), is the minimum size of a restricted edge-cut of G. We define lambda 2(G) = max{lcub}lambda2(H): H ⊂ G{rcub}.;For a digraph D, let kappa;(D), lambda( D), delta--(D), and delta +(D) denote the strong connectivity, arc-strong connectivity, minimum in-degree, and out-degree of D, respectively. For each f ∈ {lcub}kappa,lambda, delta--, +{rcub}, define f¯(D) = max{lcub} f(H): H is a subdigraph of D{rcub}.;Catlin et al. in [Discrete Math., 309 (2009), 1033-1040] proved a characterization of kappa\u27(G) in terms of tau(G). We proved a digraph version of this characterization by showing that a digraph D is k-arc-strong if and only if for any vertex v in D, D has k-arc-disjoint spanning arborescences rooted at v. We also prove a characterization of uniformly dense digraphs analogous to the characterization of uniformly dense undirected graphs in [Discrete Applied Math., 40 (1992) 285--302]. (Abstract shortened by ProQuest.)

Online choosability of graphs

We study several problems in graph coloring. In list coloring, each vertex $v$ has a set $L(v)$ of available colors and must be assigned a color from this set so that adjacent vertices receive distinct colors; such a coloring is an $L$-coloring, and we then say that $G$ is $L$-colorable. Given a graph $G$ and a function $f:V(G)\to\N$, we say that $G$ is $f$-choosable if $G$ is $L$-colorable for any list assignment $L$ such that $|L(v)|\ge f(v)$ for all $v\in V(G)$. When $f(v)=k$ for all $v$ and $G$ is $f$-choosable, we say that $G$ is $k$-choosable. The least $k$ such that $G$ is $k$-choosable is the choice number, denoted $\ch(G)$. We focus on an online version of this problem, which is modeled by the Lister/Painter game. The game is played on a graph in which every vertex has a positive number of tokens. In each round, Lister marks a nonempty subset $M$ of uncolored vertices, removing one token at each marked vertex. Painter responds by selecting a subset $D$ of $M$ that forms an independent set in $G$. A color distinct from those used on previous rounds is given to all vertices in $D$. Lister wins by marking a vertex that has no tokens, and Painter wins by coloring all vertices in $G$. When Painter has a winning strategy, we say that $G$ is $f$-paintable. If $f(v)=k$ for all $v$ and $G$ is $f$-paintable, then we say that $G$ is $k$-paintable. The least $k$ such that $G$ is $k$-paintable is the paint number, denoted \pa(G). In Chapter 2, we develop useful tools for studying the Lister/Painter game. We study the paintability of graph joins and of complete bipartite graphs. In particular, \pa(K_{k,r})\le k if and only if $r<k^k$. In Chapter 3, we study the Lister/Painter game with the added restriction that the proper coloring produced by Painter must also satisfy some property $\mathcal{P}$. The main result of Chapter 3 provides a general method to give a winning strategy for Painter when a strategy for the list coloring problem is already known. One example of a property $\mathcal{P}$ is that of having an $r$-dynamic coloring, where a proper coloring is $r$-dynamic if each vertex $v$ has at least $\min\set{r,d(v)}$ distinct colors in its neighborhood. For any graph $G$ and any $r$, we give upper bounds on how many tokens are necessary for Painter to produce an $r$-dynamic coloring of $G$. The upper bounds are in terms of $r$ and the genus of a surface on which $G$ embeds. In Chapter 4, we study a version of the Lister/Painter game in which Painter must assign $m$ colors to each vertex so that adjacent vertices receive disjoint color sets. We characterize the graphs in which $2m$ tokens is sufficient to produce such a coloring. We strengthen Brooks' Theorem as well as Thomassen's result that planar graphs are 5-choosable. In Chapter 5, we study sum-paintability. The sum-paint number of a graph $G$, denoted \spa(G), is the least $\sum f(v)$ over all $f$ such that $G$ is $f$-paintable. We prove the easy upper bound: \spa(G)\le|V(G)|+|E(G)|. When \spa(G)=|V(G)|+|E(G)|, we say that $G$ is sp-greedy. We determine the sum-paintability of generalized theta-graphs. The generalized theta-graph $\Theta_{\ell_1,\dots,\ell_k}$ consists of two vertices joined by $k$ paths of lengths \VEC \ell1k. We conjecture that outerplanar graphs are sp-greedy and prove several partial results toward this conjecture. In Chapter 6, we study what happens when Painter is allowed to allocate tokens as Lister marks vertices. The slow-coloring game is played by Lister and Painter on a graph $G$. Lister marks a nonempty set of uncolored vertices and scores 1 point for each marked vertex. Painter colors all vertices in an independent subset of the marked vertices with a color distinct from those used previously in the game. The game ends when all vertices have been colored. The sum-color cost of a graph $G$, denoted \scc(G), is the maximum score Lister can guarantee in the slow-coloring game on $G$. We prove several general lower and upper bounds for \scc(G). In more detail, we study trees and prove sharp upper and lower bounds over all trees with $n$ vertices. We give a formula to determine \scc(G) exactly when $\alpha(G)\le2$. Separately, we prove that \scc(G)=\spa(G) if and only if $G$ is a disjoint union of cliques. Lastly, we give lower and upper bounds on \scc(K_{r,s})

Image synthesis based on a model of human vision

Modern computer graphics systems are able to construct renderings of such high quality that viewers are deceived into regarding the images as coming from a photographic source. Large amounts of computing resources are expended in this rendering process, using complex mathematical models of lighting and shading. However, psychophysical experiments have revealed that viewers only regard certain informative regions within a presented image. Furthermore, it has been shown that these visually important regions contain low-level visual feature differences that attract the attention of the viewer. This thesis will present a new approach to image synthesis that exploits these experimental findings by modulating the spatial quality of image regions by their visual importance. Efficiency gains are therefore reaped, without sacrificing much of the perceived quality of the image. Two tasks must be undertaken to achieve this goal. Firstly, the design of an appropriate region-based model of visual importance, and secondly, the modification of progressive rendering techniques to effect an importance-based rendering approach. A rule-based fuzzy logic model is presented that computes, using spatial feature differences, the relative visual importance of regions in an image. This model improves upon previous work by incorporating threshold effects induced by global feature difference distributions and by using texture concentration measures. A modified approach to progressive ray-tracing is also presented. This new approach uses the visual importance model to guide the progressive refinement of an image. In addition, this concept of visual importance has been incorporated into supersampling, texture mapping and computer animation techniques. Experimental results are presented, illustrating the efficiency gains reaped from using this method of progressive rendering. This visual importance-based rendering approach is expected to have applications in the entertainment industry, where image fidelity may be sacrificed for efficiency purposes, as long as the overall visual impression of the scene is maintained. Different aspects of the approach should find many other applications in image compression, image retrieval, progressive data transmission and active robotic vision

Text Segmentation in Web Images Using Colour Perception and Topological Features

The research presented in this thesis addresses the problem of Text Segmentation in Web images. Text is routinely created in image form (headers, banners etc.) on Web pages, as an attempt to overcome the stylistic limitations of HTML. This text however, has a potentially high semantic value in terms of indexing and searching for the corresponding Web pages. As current search engine technology does not allow for text extraction and recognition in images, the text in image form is ignored. Moreover, it is desirable to obtain a uniform representation of all visible text of a Web page (for applications such as voice browsing or automated content analysis). This thesis presents two methods for text segmentation in Web images using colour perception and topological features. The nature of Web images and the implicit problems to text segmentation are described, and a study is performed to assess the magnitude of the problem and establish the need for automated text segmentation methods. Two segmentation methods are subsequently presented: the Split-and-Merge segmentation method and the Fuzzy segmentation method. Although approached in a distinctly different way in each method, the safe assumption that a human being should be able to read the text in any given Web Image is the foundation of both methods’ reasoning. This anthropocentric character of the methods along with the use of topological features of connected components, comprise the underlying working principles of the methods. An approach for classifying the connected components resulting from the segmentation methods as either characters or parts of the background is also presented

Solving hard subgraph problems in parallel

This thesis improves the state of the art in exact, practical algorithms for finding subgraphs. We study maximum clique, subgraph isomorphism, and maximum common subgraph problems. These are widely applicable: within computing science, subgraph problems arise in document clustering, computer vision, the design of communication protocols, model checking, compiler code generation, malware detection, cryptography, and robotics; beyond, applications occur in biochemistry, electrical engineering, mathematics, law enforcement, fraud detection, fault diagnosis, manufacturing, and sociology. We therefore consider both the ``pure'' forms of these problems, and variants with labels and other domain-specific constraints. Although subgraph-finding should theoretically be hard, the constraint-based search algorithms we discuss can easily solve real-world instances involving graphs with thousands of vertices, and millions of edges. We therefore ask: is it possible to generate ``really hard'' instances for these problems, and if so, what can we learn? By extending research into combinatorial phase transition phenomena, we develop a better understanding of branching heuristics, as well as highlighting a serious flaw in the design of graph database systems. This thesis also demonstrates how to exploit two of the kinds of parallelism offered by current computer hardware. Bit parallelism allows us to carry out operations on whole sets of vertices in a single instruction---this is largely routine. Thread parallelism, to make use of the multiple cores offered by all modern processors, is more complex. We suggest three desirable performance characteristics that we would like when introducing thread parallelism: lack of risk (parallel cannot be exponentially slower than sequential), scalability (adding more processing cores cannot make runtimes worse), and reproducibility (the same instance on the same hardware will take roughly the same time every time it is run). We then detail the difficulties in guaranteeing these characteristics when using modern algorithmic techniques. Besides ensuring that parallelism cannot make things worse, we also increase the likelihood of it making things better. We compare randomised work stealing to new tailored strategies, and perform experiments to identify the factors contributing to good speedups. We show that whilst load balancing is difficult, the primary factor influencing the results is the interaction between branching heuristics and parallelism. By using parallelism to explicitly offset the commitment made to weak early branching choices, we obtain parallel subgraph solvers which are substantially and consistently better than the best sequential algorithms

Perceptually Modulated Level of Detail for Virtual Environments

Institute for Computing Systems ArchitectureThis thesis presents a generic and principled solution for optimising the visual complexity of any arbitrary computer-generated virtual environment (VE). This is performed with the ultimate goal of reducing the inherent latencies of current virtual reality (VR) technology. Effectively, we wish to remove extraneous detail from an environment which the user cannot perceive, and thus modulate the graphical complexity of a VE with little or no perceptual artifacts. The work proceeds by investigating contemporary models and theories of visual perception and then applying these to the field of real-time computer graphics. Subsequently, a technique is devised to assess the perceptual content of a computer-generated image in terms of spatial frequency (c/deg), and a model of contrast sensitivity is formulated to describe a user's ability to perceive detail under various conditions in terms of this metric. This allows us to base the level of detail (LOD) of each object in a VE on a measure of the degree of spatial detail which the user can perceive at any instant (taking into consideration the size of an object, its angular velocity, and the degree to which it exists in the peripheral field). Additionally, a generic polygon simplification framework is presented to complement the use of perceptually modulated LOD. The efficient implementation of this perceptual model is discussed and a prototype system is evaluated through a suite of experiments. These include a number of low-level psychophysical studies (to evaluate the accuracy of the model), a task performance study (to evaluate the effects of the model on the user), and an analysis of system performance gain (to evaluate the effects of the model on the system). The results show that for the test application chosen, the frame rate of the simulation was manifestly improved (by four to five-fold) with no perceivable drop in image fidelity. As a result, users were able to perform the given wayfinding task more proficiently and rapidly. Finally, conclusions are drawn on the application and utility of perceptually-based optimisations; both in reference to this work, and in the wider context

Development of a natural food colorant from Passiflora edulis Sims epicarp

Food industry is dedicated in the continuous improvement of products safety and quality, satisfying the consumer’s requirements. Thus, this sector has invested in the development of new solutions to decrease the amount of used artificial additives and whenever possible, replacing them by more innocuous natural counterparts. Colorants are one of the most important additives in terms of marketing, since their presence influences consumer’s perceptions, choices and preferences, interfering in market success or failure of food products. In this context, there are many fruits in which their bio-residue can be used to extract colorant ingredientes. Passiflora edulis, known as passion fruit, is native from Brazil and much appreciated by the cosumers. The purple passion fruit variety (Passiflora edulis Sims) is the most common variety and is cultivated for juice and pulp production. However, the largest portion of the fruit is constituted by the epicarp, resulting in a high quantity of bio-waste production. This part of the fruit has a dark purple colour, being a rich source of pigmented compounds, namely anthocyanins, which could be used as a source of natural ingredients with colourant potential. In this work, the extraction methodology of anthocyanins from passion fruit epicarp was optimized through a heat assisted extraction (HAE), by applying a response surface methodology combining different independent variables of the process. Furthermore, the bioactive properties (antioxidant, antimicrobial and cytotoxic activities) of the optimized extract and also of the extract obtained by a conventional extraction method (maceration using 80:20 v/v of acidified ethanol/water at room temperature) were compared. The anthocyanins profile was determined by HPLC-DAD-ESI/MS, using 520 nm as the preference wavelength and operating in positive mode. The antioxidant activity was evaluated by means of TBARS and OxHLIA methodologies; the antimicrobial activity was determined using the microdilution method in Gram-positive and Gram-negative bacteria, and in fungi; and finally, the cytotoxic properties were studied in four human tumor cell lines: HeLa (cervical carcinoma), HepG2 (hepatocellular carcinoma), MCF-7 (breast adenocarcinoma) and NCI-H460 (lung carcinoma); and in a non-tumor cell line: PLP2 (primary culture of pig liver cells). The HAE optimal conditions were established as t= 37.5 min, T= 20 ºC, S= 0 % (ethanol) using 50 g/L of solid/liquid ratio, conducting to an extraction yield of 37%, with a total anthocyanins’ content of 9.02 mg of anthocyanins (A) per g of extract (dried extracted residue-basis) and 3.35 mg of A per g of dried epicarp basis. Regarding the bioactive properties, the optimized extract rich in anthocyanin compounds was highlighted with higher antioxidant, antimicrobial and cytotoxic potential in comparison to the extract obtained by the conventional method. Overall, this study has a great interest in the food sector, because it highlights the colorant and bioactive potential of this natural ingredient, while enhancing the valorization of a specific bio-residue, widely produced by the food sector.A indústria alimentar tem se dedicado à melhoria contínua da segurança e qualidade dos produtos, satisfazendo os requisitos do consumidor. Assim, este setor tem investido no desenvolvimento de novas soluções para diminuir a quantidade de aditivos artificiais usados e, sempre que possível, substituí-los por equivalentes naturais mais inócuos. Os corantes são um dos aditivos mais importantes em termos de marketing, uma vez que a sua presença influencia as percepções, escolhas e preferências do consumidor, interferindo no sucesso ou no fracasso dos produtos alimetares no mercado. Neste contexto, existem muitos frutos cujos seus subprodutos podem ser utilizados para extrair ingredientes corantes. Passiflora edulis, conhecida como maracujá, é nativa do Brasil e muito apreciada pelo cosumidor. A variedade de maracujá roxo (Passiflora edulis Sims) é a mais comum e é cultivada para produção de sumo e polpa. Contudo, a maior parte da fruta é constituída pelo epicarpo, resultando numa elevada quantidade de produção de bio-resíduos. Esta parte do fruto tem uma cor púrpura escura, sendo uma fonte rica em compostos corantes, nomeadamente antocianinas, sendo uma fonte de ingredientes com potencial corante. Neste trabalho, a metodologia de extração de antocianinas a partir do epicarpo de maracujá foi otimizada através da extração assistida por calor (HAE), aplicando uma metodologia de superfície de resposta, combinando diferentes variáveis independentes do processo. Além disso, foram comparadas as propriedades bioativas (atividades antioxidante, antimicrobiana e citotóxicas) do extrato otimizado e também do extrato obtido pelo método de extração convencional (maceração usando 80:20 v/v de etanol/água acidificada, à temperatura ambiente). O perfil de antocianinas foi determinado por HPLC-DAD-ESI/MS, usando 520 nm como o comprimento de onda preferencial e operando em modo positivo. A atividade antioxidante foi avaliada através das metodologias TBARS e OxHLIA; a atividade antimicrobiana foi determinada usando o método de microdiluição em bactérias Gram-positivas e Gram-negativas, e em fungos; e, finalmente, a citotoxicidade foi estudada utilizando quatro linhas celulares tumorais humanas: HeLa (adenocarcinoma cervical), HepG2 (carcinoma hepatocelular), MCF-7 (adenocarcinoma de mama) e NCI-H460 (carcinoma de pulmão); e numa cultura de células não tumorais: PLP2 (cultura primária de células de fígado de porco). As condições ótimas de HAE foram estabelecidas com t = 37,5 min, T = 20 ºC, S = 0% (etanol) usando 50 g/L de razão sólido/líquido, conduzindo a um rendimento de extração de 37%, com um teor total de antocianinas de 9,02 mg de antocianinas (A) por g de extracto (resíduo-base de extrato seco) e 3,35 mg de A por g de resíduo (base de epicarpo seco). Quanto às propriedades bioativas, o extrato otimizado rico em compostos antociânicos destacou-se com alto potencial antioxidante, antimicrobiano e citotóxico, comparativamente com extrato obtido pelo método convencional. No geral, este estudo tem um grande interesse na área alimentar, pois destaca o potencial corante e bioativo de um ingrediente natural, ao mesmo tempo reforçando a valorização de um subproduto específico, amplamente produzido pelo setor alimentar.Dupla diplomação com a Université Libre de Tuni

Natural Sciences in Archaeology and Cultural Heritage

A Special Issue of the international journal Sustainability under the section Sustainability of Culture & Heritage has been made, entitled Natural Sciences in Archaeology and Cultural Heritage. The bridge between science/technology and the humanities (archaeology, anthropology, history of art, and cultural heritage) has formed a well-established interdisciplinary subject with several sub-disciplines; it is growing exponentially, spurred by the fast development of technology in other fields (space exploration, medical, military, and industrial applications). On the other hand, art and culture struggle to survive due to neglect, lack of funding, or the dangers of events such as natural disasters and war. This volume strengthens and exerts the documentation of the sustainability of the issue that arises from the outcome of resulting research and the application of such a duality link. The sustainable dimension emerges from society, education, and economics through the impact of cultural growth, all of which produce a balanced society, in which prosperity, harmony, and development are merged at a sustainable local/regional/national/social level. A wide range of subjects linking the applied natural sciences with archaeology and the cultural heritage of innovative research and applications are presented in this volume

Variability in coloured titanium surfaces for jewellery

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