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

TL;DR

This paper introduces a topological graph drawing model based on vertex rotation and simple cycles, enabling the construction of planar representations of non-planar graphs through polynomial-time algorithms.

Contribution

It presents a novel topological model for graph visualization and a polynomial-time computational method for identifying planar subgraphs using linear algebra and structural number algebra.

Findings

Topological drawing can be based on vertex rotation and simple cycles.

A polynomial-time method for isolating planar parts of a graph.

Reduction of brute-force enumeration to a discrete optimization problem.

Abstract

A visualized graph is a powerful tool for data analysis and synthesis tasks. In this case, the task of visualization constitutes not only in displaying vertices and edges according to the graph representation, but also in ensuring that the result is visually simple and comprehensible for a human. Thus, the visualization process involves solving several problems, one of which is the problem of constructing a topological drawing of a planar part of a non-planar graph with a minimum number of removed edges. In this manuscript, we consider a mathematical model for representing the topological drawing of a graph, which is based on methods of the theory of vertex rotation with the induction of simple cycles that satisfy the Mac Lane planarity criterion. It is shown that the topological drawing of a non-planar graph can be constructed on the basis of a selected planar part of the graph. The topological model of a graph drawing allows us to reduce the brute-force enumeration problem of identifying a plane graph to a discrete optimization problem - searching for a subset of the set of isometric cycles of the graph that satisfy the zero value of the Mac Lane's functional. To isolate the planar part of the graph, a new computational method has been developed based on linear algebra and the algebra of structural numbers. The proposed method has polynomial computational complexity.