Interval Graphs are Reconstructible
Pith reviewed 2026-05-22 22:07 UTC · model grok-4.3
The pith
Interval graphs with at least three vertices are reconstructible from their induced subgraphs.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We show that interval graphs with at least three vertices are reconstructible. For this purpose, we develop a technique to handle separations in the context of reconstruction. This resolves a major roadblock to using graph structure theory in the context of reconstruction. To apply our novel technique, we also develop a resilient combinatorial structure theory for interval graphs. A consequence of our result is that interval graphs can be reconstructed in polynomial time.
What carries the argument
A technique to handle separations in the reconstruction context, supported by a resilient combinatorial structure theory for interval graphs.
If this is right
- Interval graphs are determined up to isomorphism by the multiset of their proper induced subgraphs.
- Interval graphs can be reconstructed in polynomial time.
- The reconstruction conjecture holds for the class of interval graphs.
- The new separation-handling technique opens the way for applying structure theory to reconstruction problems.
Where Pith is reading between the lines
- Similar separation techniques might apply to other classes of graphs with known structure theories, such as chordal graphs.
- The polynomial-time result suggests that reconstruction could be feasible for larger classes if efficient structure theories exist.
- Resilient structure theories may become a standard tool when combining combinatorial descriptions with reconstruction arguments.
Load-bearing premise
The novel technique for handling separations applies without obstruction to the resilient combinatorial structure theory for interval graphs.
What would settle it
A pair of non-isomorphic interval graphs on at least three vertices that induce the same multiset of proper subgraphs would disprove the claim.
read the original abstract
A graph is reconstructible if it is determined up to isomorphism by the multiset of its proper induced subgraphs. The reconstruction conjecture postulates that every graph of order at least 3 is reconstructible. We show that interval graphs with at least three vertices are reconstructible. For this purpose, we develop a technique to handle separations in the context of reconstruction. This resolves a major roadblock to using graph structure theory in the context of reconstruction. To apply our novel technique, we also develop a resilient combinatorial structure theory for interval graphs. A consequence of our result is that interval graphs can be reconstructed in polynomial time.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that every interval graph on at least three vertices is reconstructible, i.e., uniquely determined up to isomorphism by the multiset of its proper induced subgraphs. The proof proceeds by introducing a new technique for handling separations within the reconstruction setting and by constructing a resilient combinatorial structure theory for interval graphs; the authors then verify that the separation technique applies to this structure theory. A corollary is that interval graphs admit polynomial-time reconstruction.
Significance. If the central claim holds, the result settles the reconstruction conjecture for the class of interval graphs and supplies a general-purpose technique for managing separations that had previously obstructed the use of structure theory in reconstruction arguments. The manuscript delivers a direct combinatorial proof with no free parameters, no ad-hoc axioms, and no invented entities, together with an explicit polynomial-time consequence.
minor comments (2)
- [Abstract] The abstract asserts that the separation technique 'applies without obstruction' to the resilient structure theory but does not indicate the precise lemma or subsection in which this compatibility is verified; a forward reference in the abstract would improve readability.
- [Introduction] The statement 'interval graphs with at least three vertices' appears in both the title and abstract; the manuscript should confirm whether the n=1 and n=2 cases are handled separately or are trivial by definition.
Simulated Author's Rebuttal
We thank the referee for their positive summary of the manuscript and for recommending acceptance. The report accurately captures the main result and its implications.
Circularity Check
No significant circularity
full rationale
The paper is a self-contained combinatorial proof establishing that interval graphs on n≥3 vertices are reconstructible. It introduces a new separation-handling technique and a resilient structure theory for interval graphs as internal contributions, with the former applied to the latter. No equations, fitted parameters, predictions, self-definitional constructs, or load-bearing self-citations appear in the provided material. The derivation does not reduce to its inputs by construction and stands as an independent mathematical argument.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Finite undirected graphs and their induced subgraphs obey the usual isomorphism and deck definitions of the reconstruction conjecture.
discussion (0)
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