Graph Isomorphism Checker

G ≅ H structural match

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About Graph Isomorphism Checker

A graph isomorphism checker testing if G≅H: exists bijection f:V(G)→V(H) preserving edges. Checks necessary conditions: same n, |E|, degree sequence, spectrum. Graph isomorphism is in quasi-polynomial time (Babai, 2015). Client-side.

Graph Isomorphism Checker Features

  • G≅H check
  • Degree sequence
  • Invariants
  • Common examples
  • Necessary conditions
Graph isomorphism: G≅H iff ∃ bijection f:V(G)→V(H) with uv∈E(G) ↔ f(u)f(v)∈E(H). Testing GI is between P and NP-complete (neither known). Babai (2015): quasi-polynomial time n^{O(log n)}. Practical: usually fast.

How to Use

Select graph pair:

  • Invariants: Quick rejection
  • Degree seq: Same?
  • Result: Isomorphic?

Graph Invariants

Necessary (not sufficient) for isomorphism: same |V|, |E|, degree sequence, girth, diameter, spectrum, chromatic number, etc. If ANY invariant differs → not isomorphic. All same → might be isomorphic (need explicit check).

Babai's Breakthrough

László Babai (2015): GI ∈ quasi-polynomial time exp(O(log n)^c). Revolutionary! Previous best: exp(√(n log n)). Still open whether GI ∈ P. GI is one of few natural problems between P and NP-complete.

Step-by-Step Instructions

  1. 1Select two graphs.
  2. 2Compare invariants.
  3. 3Check degree seq.
  4. 4Test isomorphism.
  5. 5Find bijection.

Graph Isomorphism Checker — Frequently Asked Questions

Is graph isomorphism NP-complete?+

Unknown! It's one of few natural problems not known to be P or NP-complete. Believed to be intermediate. Babai's quasi-polynomial algorithm suggests it's 'closer to P'. No known polynomial algorithm, no known NP-completeness reduction.

What invariants are checked?+

Order |V|, size |E|, degree sequence, number of triangles, girth, diameter, spectrum of adjacency/Laplacian matrices. Non-isomorphic graphs can share ALL these invariants (called co-spectral). Complete invariants exist but are expensive.

When is isomorphism easy?+

Planar graphs: linear time (Hopcroft-Wong). Trees: linear time. Bounded degree: polynomial. Bounded treewidth: polynomial. Most random graphs: distinguished by degree sequence alone. Hard cases are rare in practice.

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