Online force-directed algorithms for visualization of dynamic graphs

Introduction to the Class of Prefractal Graphs

Fractals are already firmly rooted in modern science. Research continues on the fractal properties of objects in physics, chemistry, biology and many other scientific fields. Fractal graphs as a discrete representation are used to model and describe the structure of various objects and processes, both natural and artificial. The paper proposes an introduction to prefractal graphs. The main definitions and notation are proposed—the concept of a seed, the operations of processing a seed, the procedure for generating a prefractal graph. Canonical (typical) and non-canonical (special) types of prefractal graphs are considered separately. Important characteristics are proposed and described—the preservation of adjacency of edges for different ranks in the trajectory. The definition of subgraph-seeds of different ranks is given separately. Rules for weighting a prefractal graph by natural numbers and intervals are proposed. Separately, the definition of a fractal graph as infinite is given, and the differences between the concepts of fractal and prefractal graphs are described. At the end of the work, already published works of the authors are proposed, indicating the main backlogs, as well as a list of directions for new research. This work is the beginning of a cycle of works on the study of the properties and characteristics of fractal and prefractal graphs.

Research of NP-Complete Problems in the Class of Prefractal Graphs

NP-complete problems in graphs, such as enumeration and the selection of subgraphs with given characteristics, become especially relevant for large graphs and networks. Herein, particular statements with constraints are proposed to solve such problems, and subclasses of graphs are distinguished. We propose a class of prefractal graphs and review particular statements of NP-complete problems. As an example, algorithms for searching for spanning trees and packing bipartite graphs are proposed. The developed algorithms are polynomial and based on well-known algorithms and are used in the form of procedures. We propose to use the class of prefractal graphs as a tool for studying NP-complete problems and identifying conditions for their solvability. Using prefractal graphs for the modeling of large graphs and networks, it is possible to obtain approximate solutions, and some exact solutions, for problems on natural objects—social networks, transport networks, etc.