Complexity classification of the graph isomorphism problem

Determine whether the graph isomorphism decision problem is NP-complete or solvable in polynomial time.

Background

Graph isomorphism sits at the core of avoiding duplication in generation and underlies canonical labeling and automorphism computations. Despite extensive paper and major algorithmic advances, the precise complexity-theoretic status of GI—NP-complete versus polynomial-time solvable—remains unresolved.

References

The first problem is the graph isomorphism problem, which asks one to decide for two given graphs $G$ and $H$ whether they are isomorphic. This problem is a classical example of a problem for which it is not known whether it is NP-complete or solvable in polynomial time.

— Computer-assisted graph theory: a survey (2508.20825 - Jooken, 28 Aug 2025) in Section 2.2 (Underlying principles), first bullet on the graph isomorphism problem