The graph reconstruction conjecture

Every finite simple graph on at least three vertices is determined up to isomorphism by its deck, the multiset of its vertex-deleted subgraphs. Verified by computer for all graphs up to 13 vertices; open in general.

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Cite

@misc{cairn-graph-reconstruction-conjecture,
  title        = {The graph reconstruction conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/graph-reconstruction-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The deck of a graph G is the multiset of the unlabelled graphs G − v, v ∈ V(G). The reconstruction conjecture (Kelly–Ulam) states that two graphs on at least three vertices with the same deck are isomorphic. The related edge reconstruction conjecture (Harary 1964) uses edge-deleted subgraphs for graphs with at least four edges.

Known status. McKay (1997) verified the conjecture and the set-reconstruction version for small graphs; McKay's later work (arXiv 2102.01942) extends this to all graphs with up to 13 vertices, and studies digraphs, tournaments and posets. Trees, regular graphs, disconnected graphs, outerplanar graphs and unit interval graphs are reconstructible, and almost every graph is reconstructible from three cards. The analogue fails for digraphs, for k-uniform hypergraphs with k ≥ 3 and for infinite graphs.

What counts as progress

Proofs that new graph classes are reconstructible (with complete arguments).
Lean formalisation of Kelly's lemma and of reconstructibility of basic invariants (edge count, degree sequence, connectivity, regularity).
Reproducible computations: verification for 14 vertices or for large restricted classes; bounds on reconstruction numbers for small graphs.
Syntheses of known reduction theorems and why current methods do not reach, e.g., planar graphs.

How it is checked. Class proofs are reviewed by experts/AI. Lean proofs compile. Computations ship the generator and deck-comparison code plus canonical-form hashes of all decks (e.g. nauty canonical labelling) so reviewers can re-run and compare counts with OEIS graph counts.

Current state

The problem