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## Decomposing 1-Sperner Hypergraphs

A hypergraph is Sperner if no hyperedge contains another one. A Sperner hypergraph is equilizable (resp., threshold) if the characteristic vectors of its hyperedges are the (minimal) binary solutions to a linear equation (resp., inequality) with positive coefficients. These combinatorial notions have many applications and are motivated by the theory of Boolean functions and integer programming. We introduce in this paper the class of 1-Sperner hypergraphs, defined by the property that for every two hyperedges the smallest of their two set differences is of size one. We characterize this class of Sperner hypergraphs by a decomposition theorem and derive several consequences from it. In particular, we obtain bounds on the size of 1-Sperner hypergraphs and their transversal hypergraphs, show that the characteristic vectors of the hyperedges are linearly independent over the reals, and prove that 1-Sperner hypergraphs are both threshold and equilizable. The study of 1-Sperner hypergraphs is motivated also by their applications in graph theory, which we present in a companion paper.

The paper is devoted to the usage of the visual analytics methods and means for systematiс exploration of the results of a multi-parameter data of social Web-based service users. These data include language characteristics of the users’ comments and posts obtained from the social services they use, as well as psychological and social characteristics obtained from their profiles and from the results of surveys they fulfilled. Suggested visual analytics tools enable to present the correlations between different users’ characteristics in an observable form and to help proposing and testing hypotheses without repeating the initial data processing experiment, using just visual analytics tools to produce new results. This in turn enables to uncover and study the regularities in the input data. Semograph linguistic analysis software is suggested as a tool to collect and preprocess the data. SciVi ontology-driven multiplatform adaptive scientific visualization system is proposed to be a visual analytics tool. Because the input data have a lot of interconnections, the described visual analytics tools are based on a graph data representation model. Circular graph with adjustable hierarchical ring scale and free structured graph are supported within SciVi to ensure advanced visual analytics. The paper presents the main interactive features and implementation details of suggested tools. In particular, different filtering mechanisms for nodes and arcs are presented, as well as the means to navigate through different input data slices. The implemented visual analytics tools are tested by solving the real world problems related to the results of psychological surveys. The survey was conducted among the users of VKontakte social network. The dependencies between their psychological characteristics and verbal behavior are discovered.

The approaches based on applying of metamodeling and domain-specific languages are widely used in software engineering. There are many different tools for creating visual domain-specific modeling languages with a possibility of determining user’s graphical notations. However, these tools possess disadvantages. The article presents an approach to the development of language workbench that allows to eliminate some restrictions of existing DSM-platforms. The MetaLanguage system is designed for creation of visual dynamic adaptable domain-specific modeling languages and for models construction with these languages. It allows executing transformations of the created models in various textual and graphical notations. Basic metalanguage constructions of this system are described. The formal description of modeling languages metamodel used in MetaLanguage is given. The architecture of MetaLanguage toolkit is presented.

In this paper we investigate the eigenvalue statistics of exponentially weighted ensembles of full binary trees and *p*-branching star graphs. We show that spectral densities of corresponding adjacency matrices demonstrate peculiar ultrametric structure inherent to sparse systems. In particular, the tails of the distribution for binary trees share the 'Lifshitz singularity' emerging in the one-dimensional localization, while the spectral statistics of *p*-branching star-like graphs is less universal, being strongly dependent on *p*. The hierarchical structure of spectra of adjacency matrices is interpreted as sets of resonance frequencies, that emerge in ensembles of fully branched tree-like systems, known as dendrimers. However, the relaxational spectrum is not determined by the cluster topology, but has rather the number-theoretic origin, reflecting the peculiarities of the rare-event statistics typical for one-dimensional systems with a quenched structural disorder. The similarity of spectral densities of an individual dendrimer and of an ensemble of linear chains with exponential distribution in lengths, demonstrates that dendrimers could be served as simple disorder-less toy models of one-dimensional systems with quenched disorder.

This book constitutes the revised selected papers of the 43rd International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2017, held in Eindhoven, The Netherlands, in June 2017.

The 31 full papers presented in this volume were carefully reviewed and selected from 71 submissions. They cover a wide range of areas, aiming at connecting theory and applications by demonstrating how graph-theoretic concepts can be applied in various areas of computer science. Another focus is on presenting recent results and on identifying and exploring promising directions of future research.

We introduce a new series Rk, k = 2, 3, 4, ..., of integer valued weight systems. The value of the weight system Rk on a chord diagram is a signed number of cycles of even length 2k in the intersection graph of the diagram.Weshow that this value depends on the intersection graph only. We check that for small orders of the diagrams, the value of the weight system Rk on a diagram of order exactly 2k coincides with the coefficient of ck in the value of the sl2-weight system on the projection of the diagram to primitive elements.

We study the following computational problem: for which values of k, the majority of n bits MAJn can be computed with a depth two formula whose each gate computes a majority function of at most k bits? The corresponding computational model is denoted by MAJk o MAJk. We observe that the minimum value of k for which there exists a MAJk o MAJk circuit that has high correlation with the majority of n bits is equal to Θ(n1/2). We then show that for a randomized MAJk o MAJk circuit computing the majority of n input bits with high probability for every input, the minimum value of k is equal to n2/3+o(1). We show a worst case lower bound: if a MAJk o MAJk circuit computes the majority of n bits correctly on all inputs, then k ≥ n13/19+o(1). This lower bound exceeds the optimal value for randomized circuits and thus is unreachable for pure randomized techniques. For depth 3 circuits we show that a circuit with k = O(n2/3) can compute MAJn correctly on all inputs.

In this article we use the modular decomposition technique for exact solving the weighted maximum clique problem. Our algorithm takes the modular decomposition tree from the paper of Tedder et. al. and finds solution recursively. Also, we propose algorithms to construct graphs with modules. We show some interesting results, comparing our solution with Ostergards algorithm on DIMACS benchmarks and on generated graphs.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.