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In this paper, we observe a special class of graphs known as multilayered graphs and their subclasses, namely multilayered cycles and multilayered paths. These graphs model layouts of shopping malls, city street grids, and even resemble the topology of certain famous board games. We analyze the values of all vertex spans (strong, direct, and Cartesian span) for these subclasses of graphs. Surprisingly, our results for multilayered cycles reveal that, regardless of the chosen movement rules, the span values depend solely on the length of the individual cycles, rather than the number of layers. This finding carries significant implications for the application of graph spans in maintaining safety distances. [ABSTRACT FROM AUTHOR]

The mathematical side of applied problems in multiple subject areas (biology, pattern recognition, etc.) is reduced to the problem of discrete optimization in the following mathematical method. We were provided a network and graphs in its leaves, for which we needed to find a rearrangement of graphs by non-leaf nodes, in which the given functional reached its minimum. Such a problem, even in the simplest case, is NP-hard, which means unavoidable restrictions on the network, on graphs, or on the functional. In this publication, this problem is addressed in the case of all graphs being so-called "structures", meaning directed-loaded graphs consisting of paths and cycles, and the functional as the sum (over all edges in the network) of distances between structures at the endpoints of every edge. The distance itself is equal to the minimal length of sequence from the fixed list of operations, the composition of which transforms the structure at one endpoint of the edge into the structure at its other endpoint. The list of operations (and their costs) on such a graph is fixed. Under these conditions, the given discrete optimization problem is called the reconstruction problem. This paper presents novel algorithms for solving the reconstruction problem, along with full proofs of their low error and low polynomial complexity. For example, for the network, the problem is solved with a zero error algorithm that has a linear polynomial computational complexity; and for the tree the problem is solved using an algorithm with a multiplicative error of at most two, which has a second order polynomial computational complexity. [ABSTRACT FROM AUTHOR]

A methodology is proposed for modifying computer ontologies (CO) for electronic courses (EC) in the field of information and communication technologies (ICT) for universities, schools, extracurricular institutions, as well as for the professional retraining of specialists. The methodology includes the modification of CO by representing the formal ontograph of CO in the form of a graph and using techniques for working with the graph to find optimal paths on the graph using applied software (SW). A genetic algorithm (GA) is involved in the search for the optimal CO. This will lead to the division of the ontograph into branches and the ability to calculate the best trajectory in a certain sense through the EC educational material, taking into account the syllabus. An example is considered for the ICT course syllabus in terms of a specific topic covering the design and use of databases. It is concluded that for the full implementation of this methodology, a tool is needed that automates this procedure for developing EC and/or electronic textbooks. An algorithm and a prototype of software tools are also proposed, integrating machine methods of working with CO and graphs. [ABSTRACT FROM AUTHOR]