Optimal Sequential and Parallel Algorithms for Cut Vertices and Bridges on Trapezoid Graphs

Let G be a graph. A component of G is a maximal connected subgraph in G. A vertex v is a cut vertex of G if k(G-v) > k(G), where k(G) is the number of components in G. Similarly, an edge e is a bridge of G if k(G-e) > k(G). In this paper, we will propose new O(n) algorithms for finding cut vertices and bridges of a trapezoid graph, assuming the trapezoid diagram is given. Our algorithms can be easily parallelized on the EREW PRAM computational model so that cut vertices and bridges can be found in O(log n) time by using O(n / log n) processors

An Intersection Model for Multitolerance Graphs: Efficient Algorithms and Hierarchy

Tolerance graphs model interval relations in such a way that intervals can tolerate a certain degree of overlap without being in conflict. This class of graphs has attracted many research efforts, mainly due to its interesting structure and its numerous applications, especially in DNA sequence analysis and resource allocation, among others. In one of the most natural generalizations of tolerance graphs, namely multitolerance graphs, two tolerances are allowed for each interval—one from the left and one from the right side of the interval. Then, in its interior part, every interval tolerates the intersection with others by an amount that is a convex combination of its two border-tolerances. In the comparison of DNA sequences between different organisms, the natural interpretation of this model lies on the fact that, in some applications, we may want to treat several parts of the genomic sequences differently. That is, we may want to be more tolerant at some parts of the sequences than at others. These two tolerances for every interval—together with their convex hull—define an infinite number of the so called tolerance-intervals, which make the multitolerance model inconvenient to cope with. In this article we introduce the first non-trivial intersection model for multitolerance graphs, given by objects in the 3-dimensional space called trapezoepipeds. Apart from being important on its own, this new intersection model proves to be a powerful tool for designing efficient algorithms. Given a multitolerance graph with n vertices and m edges along with a multitolerance representation, we present algorithms that compute a minimum coloring and a maximum clique in optimal O(nlogn) time, and a maximum weight independent set in O(m+nlogn) time. Moreover, our results imply an optimal O(nlogn) time algorithm for the maximum weight independent set problem on tolerance graphs, thus closing the complexity gap for this problem. Additionally, by exploiting more the new 3D-intersection model, we completely classify multitolerance graphs in the hierarchy of perfect graphs. The resulting hierarchy of classes of perfect graphs is complete, i.e. all inclusions are strict

An Intersection Model for Multitolerance Graphs: Efficient Algorithms and Hierarchy

Tolerance graphs model interval relations in such a way that intervals can tolerate a certain degree of overlap without being in conflict. This class of graphs has attracted many research efforts, mainly due to its interesting structure and its numerous applications, especially in DNA sequence analysis and resource allocation, among others. In one of the most natural generalizations of tolerance graphs, namely multitolerance graphs, two tolerances are allowed for each interval—one from the left and one from the right side of the interval. Then, in its interior part, every interval tolerates the intersection with others by an amount that is a convex combination of its two border-tolerances. In the comparison of DNA sequences between different organisms, the natural interpretation of this model lies on the fact that, in some applications, we may want to treat several parts of the genomic sequences differently. That is, we may want to be more tolerant at some parts of the sequences than at others. These two tolerances for every interval—together with their convex hull—define an infinite number of the so called tolerance-intervals, which make the multitolerance model inconvenient to cope with. In this article we introduce the first non-trivial intersection model for multitolerance graphs, given by objects in the 3-dimensional space called trapezoepipeds. Apart from being important on its own, this new intersection model proves to be a powerful tool for designing efficient algorithms. Given a multitolerance graph with n vertices and m edges along with a multitolerance representation, we present algorithms that compute a minimum coloring and a maximum clique in optimal O(nlogn) time, and a maximum weight independent set in O(m+nlogn) time. Moreover, our results imply an optimal O(nlogn) time algorithm for the maximum weight independent set problem on tolerance graphs, thus closing the complexity gap for this problem. Additionally, by exploiting more the new 3D-intersection model, we completely classify multitolerance graphs in the hierarchy of perfect graphs. The resulting hierarchy of classes of perfect graphs is complete, i.e. all inclusions are strict

Grafos PI

Orientadores: Celia Picinin de Mello, Anamaria GomideDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Uma representação PI consiste em duas retas paralelas, r e s, e triângulos com um vértice em r e um lado em s. Considere R uma representação PI. O grafo interseção de R é chamado grafo P I quando cada vértice do grafo corresponde a um triângulo de R e existe aresta entre dois vértices se, e somente se, os triângulos correspondentes se intersectam. Segundo o livro Graph Classes - a Survey (1999) [3], escrito por Brandstiidt, Le e Spinrad, os problemas de reconhecer e de caracterizar a classe dos grafos PI ainda não estão resolvidos. Essa é a principal motivação para o estudo da classe PI. Nesta dissertação, apresentamos um estudo dos grafos PI baseado nas suas relações com outras classes de grafos tais como os grafos de intervalos e permutação, que são classes amplamente conhecidas de grafos interseção, e os grafos trapezóides, que possuem uma estrutura muito semelhante à dos grafos PI. Esta dissertação é uma síntese de trabalhos existentes sobre a classe PI e apresenta novas condições necessárias e/ou suficientes para que um grafo seja PIAbstract: A PI-representation consists of two parallellines, r and s, and triangles with one vertex on r and the other two on s. Let R be a PI-representation. The intersection graph of R is called PI graph when each vertex in the graph corresponds to a triangle in R and there exists an edge between two vertices if and only if their corresponding triangles intersect. According to the book Graph Classes - a Survey (1999) [3], by Brandstiidt, Le and Spinrad, the PI graph characterization and recognition problems are still open. This is the main motivation for the study of the PI graph class. In this dissertation, we present a study of PI graphs based on their relationship with other graph classes such as the interval and permutation graphs, which are well known intersection graph classes, and trapezoid graphs, which have a very similar structure to that of PI graphs. This dissertation is a survey on existing work on the PI graph class and presents new necessary andj or sufficient conditions for a graph to be PIMestradoTeoria da ComputaçãoMestre em Ciência da Computaçã