Algorithmic methods of finite discrete structures. Topological graph drawing (part III)

TL;DR

This paper introduces new algorithmic methods for topological graph drawing, utilizing cycle basis selection, Monte Carlo, and steepest descent techniques to improve graph visualization and Hamiltonian cycle construction.

Contribution

It presents novel algorithms combining cycle basis selection, Monte Carlo, and steepest descent methods for topological graph drawing and Hamiltonian cycle construction.

Findings

Algorithm for topological graph drawing using cycle basis and Monte Carlo.

Steepest descent method for flat subgraph topological drawing.

New approach for constructing Hamiltonian cycles via cycle graphs.

Abstract

The manuscript considers mathematical models for creating a topological drawing of a graph based on the methods of G. Ringel's vertex rotation theory. An algorithm is presented for generating a topological drawing of a flat part of a graph based on the selection of a basis for the cycle subspace C(G) using the Monte Carlo method. A steepest descent method for constructing a topological drawing of a flat subgraph is described in the manuscript. The topological drawing of a graph is constructed using a combination of the methods of vector intersection algebra developed by L. I. Rapport. Three stages of constructing a flat subgraph of a non-separable graph are described. The issues of constructing a Hamiltonian cycle based on constructing a flat subgraph are considered. A new method for constructing a Hamiltonian cycle of a graph based on the cycle graph of a flat subgraph is described.