Orientador: Alice Kimie Miwa Libardi / Banca: Thiago de Melo / Banca: Flávia Souza Machado da Silva / Resumo: O objetivo principal deste trabalho é obter um resultado sobre separação de superficies por grafos. A Homologia Relativa é a principal ferramenta usada, obtendo uma versão particular da Dualidade de Lefschetz. Para a elaboração desta dissertação foram estudados: grafos, homologia simplicial, homologia relativa e grafos em superficies. O estudo foi baseado em grande parte no livro Graphs, Surfaces and Homology de P. J. Giblin / Abstract: The main goal of this work is to get a result on separation of surfaces by graphs. The Relative Homology is the principal tool used and we get a particular version of Lefschetz duality. For the preparation of this dissertation we studied: graphs, simplicial homology, relative homology and graphs on surfaces. The study was based on the book Graphs, Surfaces and Homology of P. J. Giblin / Mestre

Orientador: Patrícia Helena Araújo da Silva Nogueira / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-17T08:03:36Z (GMT). No. of bitstreams: 1 Staib_FredericoFontes_M.pdf: 14016255 bytes, checksum: 4ec6112a82029b5e16c1c450779bf803 (MD5) Previous issue date: 2010 / Resumo: Neste trabalho, estudamos a teoria das bases de Gröbner e sua aplicação ao problema da k-coloração de grafos, estabelecendo assim uma interessante conexão entre a álgebra abstrata e a matemática discreta. Fazemos também uma abordagem de caráter lúdico, traduzindo o passatempo chamado Sudoku em um problema de 9-coloração e utilizando a teoria apresentada para resolvê-lo através das bases de Gröbner / Abstract: In the present work, we study the Gröbner basis theory and its application on the graph k-coloring problem, establishing an interesting relation between abstract algebra and discrete mathematics. We make a ludic approach, translating the puzzle called Sudoku to a 9-coloring problem and using the given theory to solve it by the Gröbner basis / Mestrado / Algebra / Mestre em Matemática

Orientador: Claudio Leonardo Lucchesi / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-14T23:05:02Z (GMT). No. of bitstreams: 1 Silva_CandidaNunesda_D.pdf: 1443438 bytes, checksum: c37eb2de501a4f5b10d9c8681c3a0971 (MD5) Previous issue date: 2009 / Resumo: Esta tese trata de fluxos inteiros e colorações em grafos, problemas intimamente relacionados. Concentramos nossa atenção nas Conjeturas de Tutte sobre 5-, 4- e 3-fluxos, as quais foram propostas entre as décadas de 50 e 70 e permanecem abertas até hoje. Apresentamos três abordagens para o ataque das conjeturas, com ênfase na Conjetura dos 3-Fluxos. Na primeira abordagem propomos o estudo dos grafos fluxo-críticos, aqueles que não admitem um k-fluxo, mas que passam a admitir quando sujeitos a uma simples operação de redução. O interesse no estudo dessa classe de grafos vem da observação de que todo contra-exemplo mínimo para qualquer uma das conjeturas de Tutte é fluxo-crítico. Na segunda abordagem estudamos a conexidade cíclica do contra-exemplo mínimo para uma conjetura equivalente à Conjetura dos 3-Fluxos. Na terceira abordagem buscamos uma nova demonstração do Teorema de Grötzsch, o qual é o dual planar da Conjetura dos 3-Fluxos, que não utilize a Fórmula de Euler como a demonstração original. / Abstract: The theme of this thesis is nowhere-zero flows and colourings of graphs, two subjects that are closely related. We focus mainly on the three Conjectures of Tutte concerning 5-, 4- and 3-nowhere-zero flows. These conjectures were proposed, respectively, in the 50's, 60's and 70's; all of them remain open so far. In this thesis we present three different approaches for the study of Tutte's Conjectures, with emphasis on the 3-Flow Conjecture. In the first approach, we introduce the concept of flow-critical graph. A graph is flowcritical if it does not admit a nowhere-zero k-flow but, when a simple reduction operation is applied, the resulting graph does admit a nowhere-zero k-flow. The motivation for the study of such graphs is due to the observation that any minimum counterexample for any of Tutte's conjectures lies in this particular class. In the second approach, we study the cyclic-connectivity of a minimum counterexample for an equivalent version of the 3-Flow Conjecture. In the third approach, we give a new proof for Gr¨otzsch's Theorem that differs from the original in the fact that it does not depend on Euler's Formula. / Doutorado / Teoria dos Grafos / Doutor em Ciência da Computação

Orientadores: Pedro Jussieu de Rezende, Orlando Lee / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-23T03:33:56Z (GMT). No. of bitstreams: 1 Silva_AndreCarvalho_M.pdf: 846347 bytes, checksum: ec1253f464900a70a0ef3bb68f9af1fa (MD5) Previous issue date: 2013 / Resumo: Grafos de visibilidade entre vértices de polígonos são estruturas que resumem as informações de visibilidade de tais vértices. Existem três relevantes problemas relativos a grafos de visibilidade: caracterização, reconhecimento e reconstrução. O problema da caracterização consiste em encontrar um conjunto de condições necessárias e suficientes para a classe de grafos de visibilidade. A procura de uma forma algorítmica para se reconhecer quando um grafo é de visibilidade define o problema de reconhecimento. O problema de reconstrução trata da questão de se reconstruir um polígono a partir de um grafo de visibilidade de tal forma que este seja isomorfo ao grafo de visibilidade do polígono. Neste trabalho, abordamos estes problemas para uma subclasse destes grafos: grafos de visibilidade de leques convexos. Dois resultados principais são apresentados nesse trabalho. O primeiro deles é um conjunto de três condições necessárias que um grafo de visibilidade de um leque convexo deve satisfazer. Adicionalmente, mostramos que estas condições são necessárias e suficientes para grafos de visibilidade de pseudo-leques convexos. Um resultado colateral deste processo é a equivalência entre grafos de visibilidade entre vértices, e grafos de visibilidade vértice-aresta, para leques convexos em posição geral. O segundo resultado consiste em que podemos reduzir o problema de reconstrução de polígonos unimonótonos para o mesmo problema para leques convexos / Abstract: The (vertex) visibility graph of a polygon is a graph that gathers all the visibility information among the vertices of the polygon. Three relevant problems related to visibility graphs are: characterization, recognition and reconstruction. Characterization calls for a set of necessary and sufficient conditions that every visibility graph must satisfy. Recognition deals with the problem of determining whether a given graph is the visibility graph of some polygon. Reconstruction asks for the generation of a polygon whose visibility graph is isomorphic to a given visibility graph. In this work, we study these problems on a restricted class of polygons, namely, convex fans: polygons that contain a convex vertex in its kernel. This work is comprised of two main results. The first one presents three necessary conditions that visibility graphs of convex fans must satisfy. We also show that those conditions are necessary and sufficient for visibility graphs of convex pseudo-fans. As a byproduct, we show that we can construct the vertex-edge visibility graph of a convex fan in general position from its vertex visibility graph. In the second major result, we show that we can reduce the reconstruction problem of unimonotone polygons to the same problem for convex fans / Mestrado / Ciência da Computação / Mestre em Ciência da Computação

Orientador: Orlando Lee / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-23T03:38:43Z (GMT). No. of bitstreams: 1 Pereira_LuizFernandodeFaria_M.pdf: 862122 bytes, checksum: 06f038348723d2201293366e75808ffd (MD5) Previous issue date: 2013 / Resumo: Uma partição em caminhos de um grafo dirigido é uma partição do conjunto de vértices deste grafo em caminhos dirigidos. Dada uma métrica sobre partições em caminhos chamada k-norma, o problema de interesse é estabelecer para um dado grafo quais das suas partições em caminhos tem a menor k-norma dentre todas as suas possíveis partições em caminhos. Chamamos estas partições de k-ótimas. Na década de 1980, Claude Berge conjecturou que para toda partição k-ótima, existe um conjunto de k conjuntos independentes disjuntos que, em certo sentido, interceptam o maior número possível de caminhos desta partição. A validade ou a falsidade desta proposição ainda não foi demonstrada, e ela é conhecida como a conjectura de Berge sobre partições em caminhos. Nesta dissertação, fizemos um estudo geral sobre a conjectura de Berge, sua história recente, e o trabalho matemático que foi desenvolvido sobre ela. Exibimos demonstrações para diversos casos particulares da conjectura que já foram resolvidos, como para grafos bipartidos, hamiltonianos, acíclicos, partições compostas somente de caminhos curtos, partições compostas somente de caminhos longos, e para valores fixos de k. Uma parte significativa do trabalho foi dedicada à reescrita da demonstração recente do caso particular onde k = 2, feita por Eli Berger e Irith Hartman, e uma análise do método usado / Abstract: A path partition of a directed graph is a partition of its vertex set into directed paths. Given a metric over path partitions called the k-norm, the problem we are interested in is to determine for a given graph which of its path partitions have the smallest k-norm among all possible path partitions. These partitions are called k-optimal. In the 1980's, Claude Berge conjectured that for every k-optimal path partition, there exists a set of k disjoint independent sets which intercepts the maximum number of paths in this partition. The validity of this proposition has not yet been demonstrated, and it is known as Berge's conjecture on path partitions. In this work, we consider Berge's conjecture, its recent history, and the related mathematical work that has been accomplished. We show proofs for many particular cases of the conjecture, including for acyclic graphs, bipartite graphs, hamiltonian graphs, partitions which include only short paths, partitions which include only long paths, and for fixed values of k. A significant part of this work was dedicated to the rewriting of a recent proof for the particular case where k = 2 by Eli Berger and Irith Hartman, and an analysis of their method / Mestrado / Ciência da Computação / Mestre em Ciência da Computação

Network theory is finding applications in the life and social sciences for ecology, epidemiology, finance and social-ecological systems. While there are methods to generate specific types of networks, the broad literature is focused on generating unweighted networks. In this paper, we present a framework for generating weighted networks that satisfy user- defined criteria. Each criterion hierarchically defines a feature of the network and, in doing so, complements existing algorithms in the literature. We use a general example of ecological species dispersal to illustrate the method and provide open- source code for academic purposes.

This thesis mainly focuses on algorithmic and combinatorial questions related to some geometric problems on graphs. In the last part of this thesis, a graph coloring problem is also discussed. Boxicity and Cubicity: These are graph parameters dealing with geomet-ric representations of graphs in higher dimensions. Both these parameters are known to be NP-Hard to compute in general and are even hard to approximate within an O(n1− ) factor for any > 0, under standard complexity theoretic assumptions. We studied algorithmic questions for these problems, for certain graph classes, to yield efficient algorithms or approximations. Our results include a polynomial time constant factor approximation algorithm for computing the cubicity of trees and a polynomial time constant (≤ 2.5) factor approximation algorithm for computing the boxicity of circular arc graphs. As far as we know, there were no constant factor approximation algorithms known previously, for computing boxicity or cubicity of any well known graph class for which the respective parameter value is unbounded. We also obtained parameterized approximation algorithms for boxicity with various edit distance parameters. An o(n) factor approximation algorithm for computing the boxicity and cubicity of general graphs also evolved as an interesting corollary of one of these parameterized algorithms. This seems to be the first sub-linear factor approximation algorithm known for computing the boxicity and cubicity of general graphs. Planar grid-drawings of outerplanar graphs: A graph is outerplanar, if it has a planar embedding with all its vertices lying on the outer face. We give an efficient algorithm to 2-vertex-connect any connected outerplanar graph G by adding more edges to it, in order to obtain a supergraph of G such that the resultant graph is still outerplanar and its pathwidth is within a constant times the pathwidth of G. This algorithm leads to a constant factor approximation algorithm for computing minimum height planar straight line grid-drawings of outerplanar graphs, extending the existing algorithm known for 2-vertex connected outerplanar graphs. n−1 3 Maximum matchings in triangle distance Delaunay graphs: Delau-nay graphs of point sets are well studied in Computational Geometry. Instead of the Euclidean metric, if the Delaunay graph is defined with respect to the convex distance function defined by an equilateral triangle, it is called a Trian-gle Distance Delaunay graph. TD-Delaunay graphs are known to be equivalent to geometric spanners called half-Θ6 graphs. It is known that classical Delaunay graphs of point sets always contain a near perfect matching, for non-degenerate point sets. We show that Triangle Distance Delaunay graphs of a set of n points in general position will always l m contain a matching of size and this bound is tight. We also show that Θ6 graphs, a class of supergraphs of half-Θ6 graphs, can have at most 5n − 11 edges, for point sets in general position. Heterochromatic Paths in Edge Colored Graphs: Conditions on the coloring to guarantee the existence of long heterochromatic paths in edge col-ored graphs is a well explored problem in literature. The objective here is to obtain a good lower bound for λ(G) - the length of a maximum heterochro-matic path in an edge-colored graph G, in terms of ϑ(G) - the minimum color degree of G under the given coloring. There are graph families for which λ(G) = ϑ(G) − 1 under certain colorings, and it is conjectured that ϑ(G) − 1 is a tight lower bound for λ(G). We show that if G has girth is at least 4 log2(ϑ(G))+2, then λ(G) ≥ ϑ(G)− 2. It is also proved that a weaker requirement that G just does not contain four-cycles is enough to guarantee that λ(G) is at least ϑ(G) −o(ϑ(G)). Other special cases considered include lower bounds for λ(G) in edge colored bipartite graphs, triangle-free graphs and graphs without heterochromatic triangles.

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A (k,l)-colouring of a graph is a partition of its vertex set into k independent sets and l cliques. The bichromatic number chi^b of a graph is the minimum r$ such that the graph is (k,l)-colourable for all k+l=r. The bichromatic number is related to the cochromatic number, which can also be defined in terms of (k,l)-colourings. The bichromatic number is a fairly recent graph parameter that arises in the study of extremal graphs related to a classical result of Erd\H{o}s, Stone and Simonovits, and in the study of the edit distance of graphs from hereditary graph classes. While the cochromatic number has been well studied in the literature, there are only few known structural results for the bichromatic number. A main focus of this thesis is to establish a foundation of knowledge about the bichromatic number. The secondary focus is on $(k,l)$-colourings of certain interesting graph classes. Two known bounds for the bichromatic number are $\chi^b \leq \chi + \theta - 1$, where $\chi$ is the chromatic number and $\theta$ the clique covering number of the graph, and $\chi^b \geq \sqrt{n}$, where $n$ the number of vertices of the graph. We give a complete characterization of all graphs for which equality holds in the first bound, and show that the second bound is best possible by constructing graphs for square numbers $n$ such that equality holds in the bound. We investigate graphs for which the bichromatic number equals the cochromatic number and prove a Brooks-type theorem for the bichromatic number. Regarding $(k,l)$-colourings, we find a new algorithm for calculating the $(k,l)$-colourability of cographs and show that cographs have a particularly nice representation with regard to $(k,l)$-colourings. For proper circular arc graphs, we provide a method for $(k,l)$-colouring if $l \geq 1$, and establish an algebraic characterization for all maximally $(k,0)$-colourable proper circular arc graphs. Finally, we investigate the bichromatic number and cochromatic with respect to lexicographic products and show several nice bounds. / Graduate

Ph.D. / In this thesis we undertake a study of formal three-dimensional representational and acceptor methods. In lieu hereof then, we give a short overview of such strategies existing in the literature. Graph and graph grammar theory present us with a powerful two dimensional representational method, and we propose to extend these concepts to the three-dimensional case. We therefore give a short discussion on the theory of graphs and graph grammars. As point of departure, we review the concepts of a structure parameter and structure graph (SG) introduced by us in [BEH,93] and show that these concepts enable us to describe objects in three-dimensional space. We propose various modified graph grammar extensions that generates structure graphs, referred to as Structure Graph Grammar extensions (SGG's) by combining context provisions with the rewriting rules of the various grammar systems. This proposed methodology of ours culminates in the combination of production rule bounded contexts and globally specified contexts, thus defining Structure Graph Grammar extension 7 (SGG-7). We show the applicative value of the three dimensional generative abilities of SGG's by considering the generation of various chemical structural formulae. Brandenburg and Skodinis mentions in [BS,95] that there is a shortcoming in the theory of graph grammars in the sense that in general, there exists no accepting device for graph grammar systems. The following quote from [BS,95,p.336] illustrates this point: "There are no graph automata, which fit to the major classes of graph languages. This is a gap in the theory of graph languages." Regarding the class of languages generated by SGG-7, we propose to fill this gap by introducing an Structure Graph Automaton (SGA) to accept this class of languages.