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

Maximizing the Strong Triadic Closure in Split Graphs and Proper Interval Graphs

In social networks the Strong Triadic Closure is an assignment of the edges with strong or weak labels such that any two vertices that have a common neighbor with a strong edge are adjacent. The problem of maximizing the number of strong edges that satisfy the strong triadic closure was recently shown to be NP-complete for general graphs. Here we initiate the study of graph classes for which the problem is solvable. We show that the problem admits a polynomial-time algorithm for two unrelated classes of graphs: proper interval graphs and trivially-perfect graphs. To complement our result, we show that the problem remains NP-complete on split graphs, and consequently also on chordal graphs. Thus we contribute to define the first border between graph classes on which the problem is polynomially solvable and on which it remains NP-complete

Systemic Modeling of Biomolecular Interaction Networks

For more than the entire past century, classical experimental methodologies have dominated biological research, providing a wealth of information about individual molecular species in cells and their functions. However, there is an increasing and strong level of evidence suggesting that an isolated biological function can only rarely be attributed to an individual biological molecule. Instead, more recently, it is argued that most biological characteristics are due to complex interactions between the cell’s numerous constituents, such as proteins, DNA and RNA. Therefore, a major challenge for the biological sciences in this century is to unravel the structure and the dynamics of these complex intracellular interactions at a systems level.   Many types of statistical and computational models have been built and applied to study cellular behavior and in this research work, we focus on two distinct instances, one from each of the two broad types of models used in computational systems biology: i) statistical inference models applied to gene regulatory interaction networks and ii) biochemical reaction models applied to protein-protein interaction networks.   For our first research problem, we focused on microRNA-mediated gene regulatory networks. MicroRNAs (miRNAs) are small non-coding ribonucleic acids (RNAs) that extensively regulate gene expression in metazoan animals, plants and protozoa. With the goal to gain a systemic understanding of miRNA-mediated interaction networks, we developed IntegraMiR, a novel integrative analysis method that can be used to infer certain types of regulatory loops of dysregulated miRNA/Transcription Factor (TF) interactions which appear at the transcriptional, post-transcriptional and signaling levels in a statistically over-represented manner. We demonstrate instances of the results in a number of distinct biological settings, which are known to play crucial roles in the contexts of prostate cancer and autism spectrum disorders.   To study the dynamics of biomolecular interaction networks, we focused on a protein-protein interaction network in living cells. Our collaborators at the School of Medicine planned to synthetically develop and characterize a biomaterial, which was produced by this protein-protein interaction network, and which would act as a molecular sieve to control the passage of biomolecules in living cells. And we wanted to computationally model the dynamic formation of this biomolecular sieve, termed a hydrogel, and characterize its properties that were relevant to the experimental work. The resulting model presented us and our experimental collaborators with a systemic and deeper understanding of the problem of gel synthesis, which guided the experimental design and provided further validation of the subsequent experimental findings and conclusions