Classical reconstruction from (n−1)-vertex induced subgraphs (Kelly–Ulam type)

Determine whether every undirected n-vertex graph is uniquely determined, up to isomorphism, by the multiset of its induced subgraphs on n−1 vertices.

Background

As background motivation, the paper recalls the classical graph reconstruction problem from structural graph theory: reconstructing an n-vertex graph from its deck of (n−1)-vertex induced subgraphs.

Despite substantial study, whether arbitrary graphs are reconstructible from their (n−1)-vertex decks remains unresolved.

References

Determining whether this is possible for arbitrary graphs is a famous unsolved problem .

— Graph Reconstruction from Noisy Random Subgraphs  (2405.04261 - McGregor et al., 2024) in Section 1 (Introduction)

In the simplest version of the problem, we have a graph $G$, and we are permitted to look at the collection of $n$ cards, which are the subgraphs made by removing a single vertex in $G$. The goal is to reconstruct $G$ from its cards, and the famous reconstruction conjecture claims that all graphs are reconstructible from their deck.

— Degree Sequence Reconstruction from Subgraph Traces  (2609.09397 - Gandikota et al., 8 Sep 2026) in Section 1, Introduction

Let $G$ be a point determining graph such that each two point determining cards of $G$ are isomorphic. Can we reconstruct $G$ by its point determining cards?

— Recognizability of Being Point Determining  (2609.34775 - Haghighi et al., 28 Sep 2026) in Section 4, Conclusion