[#P3112] Is graph isomorphism solvable in polynomial time?

Problem. Is there a deterministic algorithm that, given two finite simple graphs on \(n\) vertices, decides whether they are isomorphic in time \(n^{O(1)}\), equivalently is graph isomorphism in \(\mathrm P\)?

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Definitions and notation

1Context

Known frontier: Babai's deterministic quasipolynomial-time algorithm is the best general worst-case bound located. Open boundary: A polynomial-time bound remains open.

2Problem setup

Definition 1 (graph isomorphism). A vertex bijection preserving adjacency in both directions.

Definition 2 (quasipolynomial time). Time exp((log n)^{O(1)}).

Remark 1. Graph isomorphism has strong practical solvers and a quasipolynomial worst-case algorithm. It is neither known NP-complete nor known to lie in P.

3What counts as a solution

Give and prove a deterministic polynomial-time isomorphism algorithm.
Or prove no such algorithm exists in a stated standard model, which would separate complexity classes.

1Status

What counts as a solution

Current status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: Babai's deterministic quasipolynomial-time algorithm is the best general worst-case bound located. Exact unresolved remainder: A polynomial-time bound remains open.^[1][2]

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Record	Kind	Assessment
OPEN as checked on 2026-08-01. Strongest checked neighboring result: Babai's deterministic quasipolynomial-time algorithm is the best general worst-case bound located. Exact unresolved remainder: A polynomial-time bound remains open.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as checked on 2026-08-01. Strongest checked neighboring result: Babai's deterministic quasipolynomial-time algorithm is the best general worst-case bound located. Exact unresolved remainder: A polynomial-time bound remains open.^[1][2] Relevance to this problem This is the dated publication status for the canonical target Is graph isomorphism solvable in polynomial time?. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The problem was checked as open on 2026-08-01. The strongest neighboring result found in the cited sources is: Babai's deterministic quasipolynomial-time algorithm is the best general worst-case bound located. The exact unresolved remainder is: A polynomial-time bound remains open. A complete resolution must meet the following acceptance conditions: - Give and prove a deterministic polynomial-time isomorphism algorithm. - Or prove no such algorithm exists in a stated standard model, which would separate complexity classes. Open full record →
Babai's deterministic quasipolynomial-time algorithm is the best general worst-case bound located.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 2 Babai's deterministic quasipolynomial-time algorithm is the best general worst-case bound located.^[1][2] Relevance to this problem Locates the present research frontier immediately below Is graph isomorphism solvable in polynomial time?. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details Babai's deterministic quasipolynomial-time algorithm is the best general worst-case bound located. This leaves the following boundary unresolved: A polynomial-time bound remains open. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target. Related: Claim 1 Open full record →
The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Babai's deterministic quasipolynomial-time algorithm is the best general worst-case bound located. Unresolved remainder: A polynomial-time bound remains open.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Babai's deterministic quasipolynomial-time algorithm is the best general worst-case bound located. Unresolved remainder: A polynomial-time bound remains open.^[1][2] Relevance to this problem Documents why Is graph isomorphism solvable in polynomial time? was treated as a distinct open target on 2026-08-01. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2 What happened The audit ran on 2026-08-01 across the cited primary literature and problem collections, Crossref, arXiv, and stable publisher records, TheoremDB published, prospecting, packet, formal, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus. Queries included: - graph isomorphism polynomial time open 2026 - best general graph isomorphism algorithm quasipolynomial The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Babai's deterministic quasipolynomial-time algorithm is the best general worst-case bound located. Unresolved remainder: A polynomial-time bound remains open. Related: Claim 1 Open full record →
A polynomial-time bound remains open.	Trace	Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 2 A polynomial-time bound remains open. Relevance to this problem Turns the remaining uncertainty in Is graph isomorphism solvable in polynomial time? into a checkable research target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded What happened Research should address this boundary directly: A polynomial-time bound remains open. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - Give and prove a deterministic polynomial-time isomorphism algorithm. - Or prove no such algorithm exists in a stated standard model, which would separate complexity classes. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: Babai's deterministic quasipolynomial-time algorithm is the best general worst-case bound located. Exact unresolved remainder: A polynomial-time bound remains open.

Equivalent-formulation queries: graph isomorphism polynomial time open 2026; best general graph isomorphism algorithm quasipolynomial
Strongest checked neighboring result: Babai's deterministic quasipolynomial-time algorithm is the best general worst-case bound located.
Exact unresolved remainder: A polynomial-time bound remains open.

How the 4 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemIs graph isomorphism solvable in polynomial time?

All 3 recorded relations between these records and the problem

2See also

Multiplicative complexity of the six-bit threshold-at-least-three functiontheoretical computer science
Polynomial determinization of two-way finite automatatheoretical computer science
Logarithmic DFA separation of binary wordstheoretical computer science

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“Is graph isomorphism solvable in polynomial time?.” TheoremDB. P3112. Problem statement; statement text SHA-256 95b034a694f7dcd7e87d18fa89a41cc39763e3220b39d64769645cc713b0dae7. https://theoremdb.org/statement/?ref=P3112

BibTeX

@misc{theoremdb-problem-95b034a694f7dcd7e87d18fa89a41cc39763e3220b39d64769645cc713b0dae7,
  title = {{Is graph isomorphism solvable in polynomial time?}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 95b034a694f7dcd7e87d18fa89a41cc39763e3220b39d64769645cc713b0dae7},
  url = {https://theoremdb.org/statement/?ref=P3112}
}

Plain text: Built Markdown snapshot

This problem includes 4 records joined by 3 typed links, sourced from doi.org^[1], current as of August 1, 2026.

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1References

Packet source. László Babai, “Graph isomorphism in quasipolynomial time [extended abstract]”. Proceedings of the forty-eighth annual ACM symposium on Theory of Computing (2016), 684-697. DOI 10.1145/2897518.2897542. main theorem. ↗preprint · primary source · arXiv:1512.03547, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Gives the exp(polylog n) general algorithm.Also cited at abstract and theorem.Also cited at L. Babai, Graph isomorphism in quasipolynomial time, STOC 2016. main theorem.Durable proceedings record for the current general upper bound.Source used to assess the problem's recorded status.For Is graph isomorphism solvable in polynomial time?: This is the dated publication status for the canonical target Is graph isomorphism solvable in polynomial time?.Source named by the research packet.
Martin Grohe and Daniel Neuen, “Recent Advances on the Graph Isomorphism Problem”. arXiv:2011.01366 (2021). Introduction, pp. 1-2, and concluding open questions. ↗preprint · secondary source · arXiv:2011.01366v2 · checked 2026-08-01Source use: original summary.Surveys the quasipolynomial frontier and explicitly records polynomial-time graph isomorphism as the main open question.

Original TheoremDB editorial statement and source synthesis; external works are used for citation only.

[#P3112] Is graph isomorphism solvable in polynomial time?

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