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

TL;DR

This paper introduces algebraic methods for topological graph drawing that facilitate planarity testing and visualization without geometric constructions, by representing graphs through vertex rotations and algebraic transformations.

Contribution

It develops a mathematical framework for topological graph drawing that simplifies planarity testing and related problems using algebraic methods and vertex rotations.

Findings

Reduces non-planar graph drawing to planar graph drawing with additional vertices.

Provides a unified approach for planarity testing and topological drawing.

Lays groundwork for solving graph thickness and intersection minimization problems.

Abstract

Modern methods of graph theory describe a graph up to isomorphism, which makes it difficult to create mathematical models for visualizing graph drawings on a plane. The topological drawing of the planar part of a graph allows representing the planarization process by algebraic methods, without making any geometric constructions on the plane. Constructing a rotation of graph vertices solves two most important problems of graph theory simultaneously: the problem of testing a graph for planarity and the problem of constructing a topological drawing of a planar graph. It is shown that the problem of constructing a drawing of a non-planar graph can be reduced to the problem of constructing a drawing of a planar graph, taking into account the introduction of additional vertices characterizing the intersection of edges. Naturally, the development of such a mathematical structure will make it possible to solve the following important problems of graph theory: testing the planarity of a graph, identifying the largest planar subgraph of a graph, determining the thickness of a graph, obtaining a graph with a minimum number of intersections, etc.