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

Abstract

Modern network systems (unmanned aerial vehicles groups, social networks, network production chains, transport and logistics networks, communication networks, and cryptocurrency networks) are distinguished by their multielement nature and the dynamics of connections between their 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 lead to an additional complication of the search for solutions to discrete optimization problems. At the same time, for some subclasses of dynamic graphs that are used to model the structures of network system, conditions for the solvability of a number of NP-complete problems can be distinguished. This subclass of dynamic graphs includes prefractal graphs. This article investigates NP-complete problems on prefractal graphs: a Hamiltonian cycle, a skeleton with the maximum number of pendant vertices, a monochromatic triangle, a clique, and an independent set. The conditions under which it is possible to obtain an answer about the existence of and construct polynomial (when fixing the number of seed vertices) algorithms for finding solutions for some problems are identified.