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

Two graphs compared by a vertex bijection. — A structural graph diagram of the statement's mathematical objects.

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 \(\mathrm P\)?

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

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]

1Records

Record	Kind	Assessment
	Result	Supported
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 →
	Result	Supported
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 →
	Trace	Supported
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 →
	Trace	Reported
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 →
No records match these filters.

Notes and companion materialContext, examples, and computations

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 connectTyped relations and evidence flow

ProblemIs graph isomorphism solvable in polynomial time?

Claim 1Current status and exact unresolved remainderin this packetSupported
Route 1Dated source and duplicate auditchecksSupported
Claim 2Strongest checked neighboring resultinformsSupported
Route 2Work at the unresolved boundaryaddressesReported

2See also

Is VP equal to VNP?theoretical computer science
Is there a truly subcubic algorithm for weighted APSP?theoretical computer science
Strong Exponential Time Hypothesistheoretical computer science

How to cite

TheoremDB contributors, “Is graph isomorphism solvable in polynomial time?,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/graph-isomorphism-in-p

This page as plain text: graph-isomorphism-in-p.md

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

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?

1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1Lean verification

1References

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Definitions

What counts as a solution

What's known

Formal statement

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1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance