Searching isomorphic graphs

TL;DR

This paper presents an algorithm for testing graph isomorphism by constructing auxiliary graphs using breadth-first search, enabling vertex positioning and a bijective mapping with a time complexity of O(n^5).

Contribution

It introduces a novel method of using auxiliary graphs and vertex positioning to determine graph isomorphism, with a specific algorithm and complexity analysis.

Findings

Algorithm constructs auxiliary graphs for isomorphism testing.

Provides a bijective vertex mapping for isomorphic graphs.

Runs in O(n^5) time complexity.

Abstract

To determine that two given undirected graphs are isomorphic, we construct for them auxiliary graphs, using the breadth-first search. This makes capability to position vertices in each digraph with respect to each other. If the given graphs are isomorphic, in each of them we can find such positionally equivalent auxiliary digraphs that have the same mutual positioning of vertices. Obviously, if the given graphs are isomorphic, then such equivalent digraphs exist. Proceeding from the arrangement of vertices in one of the digraphs, we try to determine the corresponding vertices in another digraph. As a result we develop the algorithm for constructing a bijective mapping between vertices of the given graphs if they are isomorphic. The running time of the algorithm equal to $O(n^5)$, where $n$ is the number of graph vertices.