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Wilfrid Laurier University

1. Gorbonos, Elizabeth. Separability and Vertex Ordering of Graphs.

Degree: 2019, Wilfrid Laurier University

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.

Subjects/Keywords: graph theory; decomposition; clique cutset; simplicial extreme; even-hole; kite; Theory and Algorithms

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Gorbonos, E. (2019). Separability and Vertex Ordering of Graphs. (Thesis). Wilfrid Laurier University. Retrieved from https://scholars.wlu.ca/etd/2148

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Gorbonos, Elizabeth. “Separability and Vertex Ordering of Graphs.” 2019. Thesis, Wilfrid Laurier University. Accessed November 28, 2020. https://scholars.wlu.ca/etd/2148.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Gorbonos, Elizabeth. “Separability and Vertex Ordering of Graphs.” 2019. Web. 28 Nov 2020.

Vancouver:

Gorbonos E. Separability and Vertex Ordering of Graphs. [Internet] [Thesis]. Wilfrid Laurier University; 2019. [cited 2020 Nov 28]. Available from: https://scholars.wlu.ca/etd/2148.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Gorbonos E. Separability and Vertex Ordering of Graphs. [Thesis]. Wilfrid Laurier University; 2019. Available from: https://scholars.wlu.ca/etd/2148

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

2. Aline Alves da Silva. DecomposiÃÃo e largura em Ãrvore de grafos planares livres de ciclos pares induzidos.

Degree: Master, 2007, Universidade Federal do Ceará

Os conceitos de DecomposiÃÃo em Ãrvore e Largura em Ãrvore foram introduzidos por Robertson e Seymour em sua sÃrie de artigos sobre menores de grafos, publicados ao longo da dÃcada de 90. Sabe-se que muitos problemas NP - difÃceis podem ser resolvidos polinomialmente para um grafo G, dada uma decomposiÃÃo em Ãrvore de G de largura limitada. Logo, limitar a largura em Ãrvore de uma classe de grafos torna-se um objeto de estudo de grande interesse. Neste contexto, a classe dos grafos planares se mostra bastante intrigante, uma vez que, apesar de possuir outras mÃtricas limitadas em valores baixos (por exemplo, nÃmero cromÃtico), nÃo possui largura em Ãrvore limitada. Desta forma, uma alternativa à restringir a classe estudada para uma subclasse dos grafos planares. Neste trabalho, nÃs investigamos a classe dos grafos planares livres de buracos pares. NÃs mostramos que se G à um grafo planar livre de buracos pares, entÃo ele nÃo contÃm uma subdivisÃo de uma grade 10  10. Portanto, se os menores grades de G sÃo obtidos de subdivisÃes G tem largura em Ãrvore no mÃximo 49. AlÃm disso, dois algoritmos nÃo exatos polinomiais para computar uma decomposiÃÃo em Ãrvore de um grafo planar livre de buracos pares sÃo apresentados, ambos baseados em caracterizaÃÃes conhecidas de tal classe de grafos. No primeiro algoritmo, uma decomposiÃÃo em Ãrvore à construÃda a partir de grafos bÃsicos pela concatenaÃÃo de decomposiÃÃes em Ãrvores de pedaÃos pequenos via os cortes clique, k-estrelas (k = 1; 2; 3) e 2-join. No segundo, uma decomposiÃÃo em Ãrvore à construÃda pela inclusÃo dos vÃrtices de G um a um, seguindo sua ordem bi-simplicial.

The definitions of tree decomposition and treewidth were introduced by Robertson and Seymour in their series of papers on graph minors, published during the nineties. It is known that many NP-hard problems can be polynomially solved if a tree decomposition of bounded treewidth is given. So, it is of interest to bound the treewidth of certain classes of graphs. In this context, the planar graphs seem to be specially challenging because, in despite of having many known bounded metrics (for example, chromatic number), they have unbounded treewidth. So, an alternative approach is to restrict ourselves to a subclass of planar graphs. In this work, we investigate the class of even-hole-free planar graphs. We show that if G is an even-hole-free planar graph, then it does not contain a subdivision of the 10Â10 grid. So, if the grid minors of G are obtained from subdivisions, then G has treewidth at most 49. Furthermore, two polynomial, non-exact algorithms to compute a tree decomposition of a even-hole-free planar graph are given, both based on known characterizations of even-hole-free graphs. In the Ârst one, a tree decomposition is built from basic graphs by concatenating the tree decomposition of small pieces via the clique, k-stars (k = 1; 2; 3) and 2-join cutsets. In the second one, a tree decomposition is built by including one by one the vertices of G, following their bi-simplicial order.

Advisors/Committee Members: Ricardo Cordeiro CorrÃa, Sulamita Klein, ClÃudia Linhares Sales.

Subjects/Keywords: CIENCIA DA COMPUTACAO; Grafos planares, grafos livres de buracos pares, largura em Ãrvore, teoria de Grafos.; Planar Graphs, even-hole-free graphs, treewidth, graph theory

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Silva, A. A. d. (2007). DecomposiÃÃo e largura em Ãrvore de grafos planares livres de ciclos pares induzidos. (Masters Thesis). Universidade Federal do Ceará. Retrieved from http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=1324 ;

Chicago Manual of Style (16th Edition):

Silva, Aline Alves da. “DecomposiÃÃo e largura em Ãrvore de grafos planares livres de ciclos pares induzidos.” 2007. Masters Thesis, Universidade Federal do Ceará. Accessed November 28, 2020. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=1324 ;.

MLA Handbook (7th Edition):

Silva, Aline Alves da. “DecomposiÃÃo e largura em Ãrvore de grafos planares livres de ciclos pares induzidos.” 2007. Web. 28 Nov 2020.

Vancouver:

Silva AAd. DecomposiÃÃo e largura em Ãrvore de grafos planares livres de ciclos pares induzidos. [Internet] [Masters thesis]. Universidade Federal do Ceará 2007. [cited 2020 Nov 28]. Available from: http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=1324 ;.

Council of Science Editors:

Silva AAd. DecomposiÃÃo e largura em Ãrvore de grafos planares livres de ciclos pares induzidos. [Masters Thesis]. Universidade Federal do Ceará 2007. Available from: http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=1324 ;

3. Le, Ngoc Khang. Detecting and Coloring some Graph Classes : Détection et coloration de certaines classes de graphes.

Degree: Docteur es, Informatique, 2018, Lyon

Les graphes sont des structures mathématiques utilisées pour modéliser les relations par paires entre objets. Malgré leur structure simple, les graphes ont des applications dans divers domaines tels que l'informatique, la physique, la biologie et la sociologie. L'objectif principal de ce travail est de continuer l'étude des problèmes de coloration et de détection dans le cadre de classes de graphes fermées par sous-graphes induits (que nous appelons classes de graphes héréditaires).La première classe que nous considérons est graphes sans ISK4 - les graphes qui ne contiennent aucune subdivision de en tant que sous-graphe induit. Nous montrons que le nombre chromatique de cette classe est limité à 24, une amélioration considérable par rapport à la borne existant précédemment. Nous donnons également une bien meilleure limite dans le cas sans triangle. De plus, nous prouvons qu'il existe un algorithme de complexité pour détecter cette classe, ce qui répond à une question de Chudnovsky et al. et Lévêque et al.La deuxième classe que nous étudions est celle des graphes sans trou pair et sans étoile d’articulation. Cela est motivé par l'utilisation de la technique de décomposition pour résoudre certains problèmes d'optimisation. Nous garantissons la fonction χ-bounding optimale pour cette classe. Nous montrons que la classe a rank-width bornée, ce qui implique l'existence d'un algorithme de coloration en temps polynomial. Enfin, la coloration gloutonne connexe dans les graphes sans griffes est considérée. Une façon naturelle de colorier un graphe est d'avoir un ordre de ses sommets et d'affecter pour chaque sommet la première couleur disponible. Beaucoup de recherches ont été faites pour des ordres généraux. Cependant, nous connaissons très peu de choses sur la caractérisation des bons graphes par rapport aux ordres connexes. Un graphe est bon si pour chaque sous-graphe induit connexe de , chaque ordre connexe donne à une coloration optimale. Nous donnons la caractérisation complète de bons graphes sans griffes en termes de sous-graphes induits minimaux interdits.

Graphs are mathematical structures used to model pairwise relations between objects. Despite their simple structures, graphs have applications in various areas like computer science, physics, biology and sociology. The main focus of this work is to continue the study of the coloring and detecting problems in the setting of graph classes closed under taking induced subgraphs (which we call hereditary graph classes). The first class we consider is ISK4-free graphs - the graphs that do not contain any subdivision of K4 as an induced subgraph. We prove that the chromatic number of this class is bounded by 24, a huge improvement compared to the best-known bound. We also give a much better bound in the triangle-free case. Furthermore, we prove that there exists an O(n 9) algorithm for detecting this class, which answers a question by Chudnovsky et al. and Lévêque et al. The second class we study is even-hole-free graphs with no star cutset. This was motivated by the…

Advisors/Committee Members: Trotignon, Nicolas (thesis director).

Subjects/Keywords: Coloration de graphe; Reconnaissance de graphe; Sous-graphes induits; Graphes sans ISK4; Graphes sans trou pair; Coloration gloutonne connexe; Graph coloring; Graph recognition; Induced subgraphs; ISK4-free graphs; Even-hole-free graphs; Connected greedy coloring

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Le, N. K. (2018). Detecting and Coloring some Graph Classes : Détection et coloration de certaines classes de graphes. (Doctoral Dissertation). Lyon. Retrieved from http://www.theses.fr/2018LYSEN021

Chicago Manual of Style (16th Edition):

Le, Ngoc Khang. “Detecting and Coloring some Graph Classes : Détection et coloration de certaines classes de graphes.” 2018. Doctoral Dissertation, Lyon. Accessed November 28, 2020. http://www.theses.fr/2018LYSEN021.

MLA Handbook (7th Edition):

Le, Ngoc Khang. “Detecting and Coloring some Graph Classes : Détection et coloration de certaines classes de graphes.” 2018. Web. 28 Nov 2020.

Vancouver:

Le NK. Detecting and Coloring some Graph Classes : Détection et coloration de certaines classes de graphes. [Internet] [Doctoral dissertation]. Lyon; 2018. [cited 2020 Nov 28]. Available from: http://www.theses.fr/2018LYSEN021.

Council of Science Editors:

Le NK. Detecting and Coloring some Graph Classes : Détection et coloration de certaines classes de graphes. [Doctoral Dissertation]. Lyon; 2018. Available from: http://www.theses.fr/2018LYSEN021

.