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.