A direct method for calculating cell cycles of a block map of a simple planar graph

Предлагаемый алгоритм вычисления циклов ячеек карты блока графа простого планарного является расширением классического алгоритма поиска в глубину циклов DFS-базиса. Ключевой идеей модификации алгоритма является стратегия правого обхода при прохождении графа в глубину. Начальной вершиной при правом обходе назначается вершина с минимальной координатой по оси OY. Выход из начальной вершины выполняется по ребру с минимальным полярным углом. Продолжение обхода из каждой следующей вершины осуществляется по ребру с минимальным полярным углом относительно ребра, по которому пришли в текущую вершину. Вводится двухуровневая структура вложенности циклов — основной и нулевой уровни вложенности. Все циклы базиса относятся к основному уровню. Каждый из циклов может иметь и нулевой уровень вложенности в другом основном для него цикле, если он вложен в него и не вложен ни в какой другой цикл из основного цикла. При правом обходе циклы нулевой вложенности являются смежными основному циклу и не имеют между собой общих вершин вне основного цикла. Указанные два свойства позволили в каждом цикле базиса последовательно выделить и исключить из него все циклы нулевой вложенности, применяя операцию симметрической разности. Показано, что оставшаяся часть базисного цикла является циклом ячейки карты блока. Сложность каждого шага алгоритма не превышает квадратичной относительно числа вершин планарного графа

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

A Formal Approach to Prove Compatibility in Transformation Networks

The increasing complexity of software and cyberphysical systems is handled by dividing the description of the system under construction into different models or views, each with an appropriate abstraction for the needs of specific roles. Since all such models describe the same system, they usually share an overlap of information, which can lead to inconsistencies if overlapping information is not modified uniformly in all models. A well-researched approach to make these overlaps explicit and resolve inconsistencies are incremental, bidirectional model transformations. They specify the constraints between two metamodels and the restoration of consistency between their instances. Relating more than two metamodels can be achieved by combining bidirectional transformations to a network. However, such a network may contain cycles of transformations, whose consistency constraints can be contradictory if they are not aligned with each other and thus cannot be fulfilled at the same time. Such transformations are considered incompatible. In this article, we provide a formal definition of consistency and compatibility of transformations and propose an inductive approach to prove compatibility of a given network of transformations. We prove correctness of the approach based on these formal definitions. Furthermore, we present an operationalization of the approach at the example of QVT-R. It detects contradictions between relations by transforming them into first-order logic formulae and evaluating them with an SMT solver. The approach operates conservatively, i.e., it is not able to prove compatibility in all cases, but it identifies transformations as compatible only if they actually are. We applied the approach to different evaluation networks and found that it operates conservatively and is able to properly prove compatibility in 80% of the cases, indicating its practical applicability. Its limitations especially arise from restricted capabilities of the used SMT solver, but not from conceptual shortcomings. Our approach enables multiple domain experts to define transformations independently and to check their compatibility when combining them to a network, relieving them from the necessity to align the transformations with each other a priori and to ensure compatibility manuall

On optimal and near-optimal algorithms for some computational graph problems

PhD ThesisSome computational graph problems are considered in this thesis and algorithms for solving these problems are described in detail. The problems can be divided into three main classes, namely, problems involving partially ordered sets, finding cycles in graphs, and shortest path problems. Most of the algorithms are based on recursive procedures using depth-first search. The efficiency of each algorithm is derived and it can be concluded that the majority of the proposed algorithms are either optimal and near-optimal within a constant factor. The efficiency of the algorithms is measured by the time and space requirements for their implementation.Conselho Nacional de Pesquisas,Brazil: Universidade Federal do Rio de Janeiro, Brazil