On minimum cuts of cycles by vertices and vertex disjoint cycles

We describe a new family of di.graphs, named connectively reducible, for which we prove that the minimum cardinality of a set of vertices intersecting all cycles equals the maximum cardinality of a set of vertex disjoint cycles. In.addltion, formulate polynomial time algorithms for the problems of recognition and finding these minimum and maxium sets for digraphs of the family. Similar results hold for the currently existing families of fully reducible and cyclically reducible digraphs. Neither the fully reducible are contained nor contain the cyclically reducible. However, we show that the connectively reducible digraphs contain both of the existing families.Descrevemos uma nova família de dígrafos, denominados conexamente redutíveis, para a qual provamos que, a cardinalidade mínima de um conjunto de vértices que interceptam todos os ciclos iguala à máxima de um conjunto de ciclos disjuntos em vértices. Além disso, formulamos algoritmos polinomiais para os prpblemas de reconhecimento e determinação desses conjuntos, mínimo e máximo, para dígrafos dessa família. Resultados similares são conhecidos para os dígrafos totalmente redutíveis. Mais recentemente, uma outra família foi definida, os dígrafos ciclicamente redutíveis, que também possibilita a computação em tempo polinomial desses conjuntos mínimo e máximo. E conhecido o fato de que os dígrafos totalmente redutíveis não estão contidos nem contêm os ciclicamente redutíveis. Em contraste, provamos que os conexamente redutíveis, contêm ambas as famílias existentes

Markov-Chain-Based Heuristics for the Feedback Vertex Set Problem for Digraphs

A feedback vertex set (FVS) of an undirected or directed graph G=(V, A) is a set F such that G-F is acyclic. The minimum feedback vertex set problem asks for a FVS of G of minimum cardinality whereas the weighted minimum feedback vertex set problem consists of determining a FVS F of minimum weight w(F) given a real-valued weight function w. Both problems are NP-hard [Karp72]. Nethertheless, they have been found to have applications in many fields. So one is naturally interested in approximation algorithms. While most of the existing approximation algorithms for feedback vertex set problems rely on local properties of G only, this thesis explores strategies that use global information about G in order to determine good solutions. The pioneering work in this direction has been initiated by Speckenmeyer [Speckenmeyer89]. He demonstrated the use of Markov chains for determining low cardinality FVSs. Based on his ideas, new approximation algorithms are developed for both the unweighted and the weighted minimum feedback vertex set problem for digraphs. According to the experimental results presented in this thesis, these new algorithms outperform all other existing approximation algorithms. An additional contribution, not related to Markov chains, is the identification of a new class of digraphs G=(V, A) which permit the determination of an optimum FVS in time O(|V|^4). This class strictly encompasses the completely contractible graphs [Levy/Low88]

Fast and Processor-Efficient Parallel Algorithms for Reducible Flow Graphs

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryJoint Services Electronics Program / N00014-84-C-014

Feedback Numbers of Möbius Ladders

A subset F ⊂ V(G) is called a feedback vertex set if the subgraph G−F is acyclic. The minimum cardinality of a feedback vertex set is called the feedback number of G, which is proposed by Beineke and Vandell [1]. In this paper, we consider a particular topology graph called Möbius ladders M2n. We use f(M2n) to denote the feedback number of M2n. This paper proves that f (M2n) = [n+1/2], n≥3

Multitriangulations, pseudotriangulations and primitive sorting networks

We study the set of all pseudoline arrangements with contact points which cover a given support. We define a natural notion of flip between these arrangements and study the graph of these flips. In particular, we provide an enumeration algorithm for arrangements with a given support, based on the properties of certain greedy pseudoline arrangements and on their connection with sorting networks. Both the running time per arrangement and the working space of our algorithm are polynomial. As the motivation for this work, we provide in this paper a new interpretation of both pseudotriangulations and multitriangulations in terms of pseudoline arrangements on specific supports. This interpretation explains their common properties and leads to a natural definition of multipseudotriangulations, which generalizes both. We study elementary properties of multipseudotriangulations and compare them to iterations of pseudotriangulations.Comment: 60 pages, 40 figures; minor corrections and improvements of presentatio