Graph Structures and the Whitney Theorem

Abstract

The article considers a method for constructing the structures of an inseparable undirected graph G based on the use of the structure of its edge graph L(G). The method is grounded on the mapping of the edge graph L(G) by the subgraphs of the graph G. It is shown that the construction of the set of isometric cycles of the edge graph L(G) generates subsets of the subgraphs of the graph G. In turn, the set of generated subgraphs of the graph G allows constructing various invariants of the graph G and its topological drawing. The metric properties of the system of isometric cycles of the edge graph L(G) make it possible to determine the weights of the edges and vertices. This allows to construct the numerical characteristics of the graph structure. In turn, the numerical characteristics of the digital invariant of the edge graph make it possible to apply Whitney's theorem to solve the problem of recognizing graph isomorphism. The algorithm for constructing a digital invariant of an inseparable graph belongs to the class of polynomial problems with a computational complexity of О(n6). The digital invariant of the edge graph does not depend on the numbering of the vertices and edges of the graph. It specifies the individual characteristics of the graph, which allows determining graph isomorphism by comparing the invariants.