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 study 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

It is not known whether the GI problem is NP-complete or solvable in polynomial time; the best existing results establish that the GI problem can be solved in quasipolynomial time .

— Optimal graph joining with applications to isomorphism detection and identification  (2511.14862 - Hoàng et al., 18 Nov 2025) in Section 5, Related Work, paragraph beginning “The graph isomorphism (GI) problem is to determine whether two given graphs are isomorphic”