Graph Isomorphism

Graph isomorphism is a fundamental concept in graph theory that describes when two graphs are structurally identical, even if they appear visually different. Two graphs G and H are considered isomorphic if there exists a bijective mapping between their vertex sets that preserves the adjacency relationships between vertices.

Formal Definition

For two graphs G = (V₁, E₁) and H = (V₂, E₂), an isomorphism is a bijective function f: V₁ → V₂ such that:

• For any two vertices u and v in V₁, they are adjacent in G if and only if f(u) and f(v) are adjacent in H
• The mapping preserves the number of vertices and edges
• The mapping maintains the graph connectivity patterns

Properties

Isomorphic graphs share several invariant properties:

1. Same number of vertices
2. Same number of edges
3. Identical degree sequence
4. Same graph coloring
5. Equal graph cycles structures
6. Identical graph automorphism groups

Computational Complexity

The computational status of graph isomorphism testing represents a fascinating challenge in computational complexity theory:

• It is not known to be either P (complexity) or NP-complete
• It belongs to NP (complexity)
• László Babai's quasi-polynomial time algorithm (2015) represents a major breakthrough
• For certain graph classes (like planar graphs or bounded degree graphs), polynomial-time algorithms exist

Applications

Graph isomorphism has practical applications in various fields:

1. Chemistry

   • Identifying equivalent molecular structure representations
   • chemical compound database searching

2. Computer Science

   • pattern matching recognition
   • database design
   • network topology analysis

3. Social Network Analysis

   • Identifying similar social structures
   • community detection in network comparison

Testing Methods

Several approaches exist for testing graph isomorphism:

1. Canonical Labeling

   • Assigns unique labels to vertices
   • Generates canonical forms for comparison

2. Invariant Checking

   • Compares graph invariants
   • Quick preliminary screening

3. Backtracking Algorithms

   • Systematically explores possible vertex mappings
   • Implements pruning strategies for efficiency

Related Concepts

The study of graph isomorphism connects to several important theoretical concepts:

• graph automorphism (self-isomorphisms)
• subgraph isomorphism (finding structural patterns)
• graph homomorphism (relaxed structural mapping)
• graph canonization (unique representations)

Understanding graph isomorphism is crucial for anyone working in graph theory, algorithm design, or structural pattern analysis. Its unique computational complexity status makes it a continuing subject of theoretical research while its practical applications span multiple disciplines.