Classification of graphs based on homotopy equivalence. Homotopy equivalent graphs. Basic graphs and complexity of homotopy equivalence classes of graphs

TL;DR

This paper introduces a novel graph classification method based on homotopy equivalence, using contractible transformations to group graphs with similar topological properties, supported by computer experiments.

Contribution

It defines homotopy equivalence classes of graphs, introduces basic graphs, and demonstrates a new classification approach grounded in topological and computational analysis.

Findings

Graphs in the same homotopy class share topological properties

Basic graphs with minimal points and edges represent classes

Computer experiments show strong connection between topological spaces and graphs

Abstract

Graph classification plays an important role is data mining, and various methods have been developed recently for classifying graphs. In this paper, we propose a novel method for graph classification that is based on homotopy equivalence of graphs. Graphs are called homotopy equivalent if one of them can be converted to the other one by contractible transformations. A basic graph and the complexity of a homotopy equivalence class are defined and investigated. It is shown all graphs belonging to a given homotopy equivalence class have similar topological properties and are represented by a basic graph with the minimal number of points and edges. Diagrams are given of basic graphs with the complexity N<7. The advantage of this classification is that it relies on computer experiments demonstrating a close connection between homotopy equivalent topological spaces and homotopy equivalent graphs.