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.

Highlights

  • Let is the set of all n-vertex undirected graphs without loops and multiple edges.Let, further, there is a graph = (, ) ∈, where = {, ... , } is the set of graph vertices and = {, ... , } is the set of graph edges

  • Effective algorithms for solving this problem were found for some narrow classes of graphs [7] - [9]

  • If graphs and are isomorphic and two auxiliary positionally equivalent digraphs ⃗( ) and ⃗ (u) are found, any bijective mapping, which convert the graph into the graph, is determined by pairs of vertices of the digraphs with equal characteristics

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Summary

Introduction

Let is the set of all n-vertex undirected graphs without loops and multiple edges. Graphs = ( , ), = ( , ) ∈ are called isomorphic if between their vertices there exists oneto-one (bijective) mapping : ↔ such that if = { , } ∈ the corresponding edge is = { ( ), ( )} ∈ , and [1, 2]. Effective (polynomial-time) algorithms for solving this problem were found for some narrow classes of graphs [7] - [9]. The purpose of this article is to propose a polynomial-time algorithm of searching isomorphic graphs. Plotnikov; Searching Isomorphic Graphs, Transactions on Networks and Communications, Volume 5 No 5, October (2017); pp: 39-54

Basic Definitions
The Basics of the Algorithm
Vertex Characteristics of the Digraphs
The Search Algorithm
Conclusion
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