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”

It is in~NP, not known to be NP-complete and not known to be in~PTIME, and it has no logical characterization to be complete for.

— Descriptive Complexity in Lean: Completeness by First-Order Reductions  (2609.18261 - Senellart et al., 16 Sep 2026) in Section 4.3, “The GI degree” (subsection label sec:problems:gi)

Graph isomorphism is not known to be NP-complete, but it is a useful illustration of the association-graph principle and has a well-established continuous clique interpretation \citep{Pelillo1999}; association graphs underlie continuous methods for matching, common subgraphs, and shape correspondence \citep{pelillo-siddiqi-zucker-assoc-1999,leordeanu-hebert-spectral-2005}, and clique and constraint models are established approaches to maximum common subgraph \citep{mccreesh-clique-constraint-mcs-2016}.

— Standard Quadratic Formulations of Many NP Problems: A Simplex-Based Compilation Framework for Combinatorial Optimization  (2610.01911 - Miri et al., 1 Oct 2026) in Section 4, subsection “Graph isomorphism and association graphs”