A Constructive Classification of Graphs

2022

Mathematics

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.

“…To designate a graph, the generally accepted notation G = (V, E) [34][35][36][37] is used. To define prefractal and fractal graphs, we give additional definitions.…”

Section: Operation Of Replacing Vertex By Seedmentioning

confidence: 99%

Introduction to the Class of Prefractal Graphs

2022

Mathematics

“…В настоящей работе используется общепринятое обозначение и определение графа = ( , ) [7,8]. Определение предфрактальных графов соответствует работам [5,6].…”

Section: определения и обозначенияunclassified

Identification Conditions for the Solvability of NP-Complete Problems for the Class of Prefractal Graphs

Timoshenko

Aut. Control Comp. Sci.

2022

Modern network systems (unmanned aerial vehicles groups, social networks, network production chains, transport and logistics networks, communication networks, cryptocurrency networks) are distinguished by their multi-element nature and the dynamics of connections between its elements. A number of discrete problems on the construction of optimal substructures of network systems described in the form of various classes of graphs are NP-complete problems. In this case, the variability and dynamism of the structures of network systems leads to an "additional" complication of the search for solutions to discrete optimization problems. At the same time, for some subclasses of dynamical graphs, which are used to model the structures of network systems, conditions for the solvability of a number of NP-complete problems can be distinguished. is subclass of dynamic graphs includes pre-fractal graphs.e article investigates NP-complete problems on pre-fractal graphs: a Hamiltonian cycle, a skeleton with the maximum number of pendant vertices, a monochromatic triangle, a clique, an independent set. e conditions under which for some problems it is possible to obtain an answer about the existence and to construct polynomial (when xing the number of seed vertices) algorithms for nding solutions are identi ed.

Section: определения и обозначенияunclassified

Identification Conditions for the Solvability of NP-complete Problems for the Class of Pre-fractal Graphs

Timoshenko

Model. anal. inf. sist.

2021

Modern network systems (unmanned aerial vehicles groups, social networks, network production chains, transport and logistics networks, communication networks, cryptocurrency networks) are distinguished by their multi-element nature and the dynamics of connections between its elements. A number of discrete problems on the construction of optimal substructures of network systems described in the form of various classes of graphs are NP-complete problems. In this case, the variability and dynamism of the structures of network systems leads to an "additional" complication of the search for solutions to discrete optimization problems. At the same time, for some subclasses of dynamical graphs, which are used to model the structures of network systems, conditions for the solvability of a number of NP-complete problems can be distinguished. This subclass of dynamic graphs includes pre-fractal graphs. The article investigates NP-complete problems on pre-fractal graphs: a Hamiltonian cycle, a skeleton with the maximum number of pendant vertices, a monochromatic triangle, a clique, an independent set. The conditions under which for some problems it is possible to obtain an answer about the existence and to construct polynomial (when fixing the number of seed vertices) algorithms for finding solutions are identified.