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arxiv: 2606.28610 · v1 · pith:BGV4ERIVnew · submitted 2026-06-26 · 🧮 math.CO · math.AC

Chordality, syzygies, and shellability for hypergraphic analogues of interval graphs

Anton Dochtermann , Bennet Goeckner , Marta Pavelka This is my paper

Pith reviewed 2026-06-30 00:38 UTC · model grok-4.3

classification 🧮 math.CO math.AC
keywords cointerval hypergraphsunderclosed complexeschordal hypergraphsAlexander duallinear quotientsvertex decomposableshellabilitycircuit ideals
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The pith

Cointerval hypergraphs equal underclosed complexes up to complement, and underclosed clutters are Woodroofe-chordal.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that cointerval hypergraphs and underclosed complexes are equivalent via complementation. It shows that underclosed clutters satisfy Woodroofe's chordality condition. This yields that the associated Alexander dual complexes are vertex decomposable, so the circuit ideals have linear quotients. The dual complexes also admit shellings ordered by their underclosed vertex sets.

Core claim

Underclosed clutters are chordal in the sense of Woodroofe. Their Alexander dual complexes are therefore vertex decomposable and shellable via underclosed vertex orders, which implies that the corresponding circuit ideals have linear quotients.

What carries the argument

The equivalence of cointerval hypergraphs and underclosed complexes up to complementation, together with Woodroofe's chordality applied directly to underclosed clutters.

If this is right

  • The circuit ideals of these hypergraphs have linear quotients.
  • The Alexander dual complexes are vertex decomposable.
  • The dual complexes admit shellings induced by underclosed vertex orders.
  • This resolves the question of Dochtermann and Engström on linear quotients.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The equivalence may allow results on one class to transfer immediately to the other via complement.
  • Higher-degree versions of Fröberg's theorem could be tested first on underclosed clutters.
  • Vertex decomposability might serve as a uniform test for chordality across other hypergraph analogues.

Load-bearing premise

The prior definitions of cointerval hypergraphs and underclosed complexes are compatible for equivalence up to complementation, and Woodroofe's chordality applies directly to underclosed clutters without extra restrictions.

What would settle it

An explicit underclosed clutter whose Alexander dual fails to be vertex decomposable, or a cointerval hypergraph whose complement fails to be underclosed.

read the original abstract

Interval graphs are a special class of chordal graphs, and hence have connections to commutative algebra via Fr\"oberg's theorem that characterizes linear resolutions of squarefree quadratic ideals. In recent years, several hypergraphic analogues of interval and chordal graphs have been proposed, in part as an effort to extend Fr\"oberg's theorem to ideals generated in higher degree. In this paper, we study two such classes from the literature, cointerval hypergraphs and underclosed complexes, and show that they are in fact equivalent up to complementation. We then consider their place in the broader theory of higher-dimensional chordality, proving that an underclosed clutter is chordal in the sense of Woodroofe. As a consequence, we answer a question of Dochtermann and Engstr\"om by showing that the associated Alexander dual complexes are vertex decomposable, implying that the corresponding circuit ideals have linear quotients. We furthermore show that these dual complexes have shellings induced by their underclosed vertex orders.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper shows that cointerval hypergraphs and underclosed complexes are equivalent up to complementation via explicit bijections in Section 2. It proves that every circuit in an underclosed clutter satisfies Woodroofe's chordality condition (Theorem 3.4). As a consequence, the Alexander dual complexes are vertex decomposable (answering a question of Dochtermann and Engström), the corresponding circuit ideals have linear quotients, and the duals admit shellings induced by the underclosed vertex orders.

Significance. If the results hold, the work supplies a concrete, combinatorially explicit class of hypergraphs where higher-dimensional chordality holds without extra restrictions, directly linking two proposed hypergraphic analogues of interval graphs. The explicit bijections, direct chordality verification, and induced shellings are strengths that provide verifiable tools for studying syzygies and resolutions in the associated circuit ideals.

minor comments (2)
  1. [Section 2] Section 2: the bijection statements would be easier to follow with a small concrete example of a cointerval hypergraph and its underclosed complement.
  2. [Theorem 3.4] Theorem 3.4: the statement that the verification is 'direct' could note explicitly which clauses of Woodroofe's definition are checked for each circuit type.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for highlighting its significance in linking cointerval hypergraphs and underclosed complexes, and for the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation proceeds via explicit bijections establishing equivalence of cointerval hypergraphs and underclosed complexes (Section 2), followed by direct verification that every circuit in an underclosed clutter satisfies Woodroofe's chordality condition (Theorem 3.4). Vertex decomposability of the Alexander dual then follows from the external chordal-implies-vertex-decomposable implication for clutters, with shellings constructed explicitly from the underclosed vertex orders. All load-bearing steps are internal to the stated definitions or rely on independently established external results; no self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citation chains appear.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard background theorems and prior definitions of the hypergraph classes; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)
  • standard math Fröberg's theorem characterizes linear resolutions of squarefree quadratic ideals
    Invoked as the motivating connection between chordal graphs and algebra.
  • domain assumption Definitions of cointerval hypergraphs and underclosed complexes from the literature
    The paper studies these two classes and proves their equivalence.

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discussion (0)

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Reference graph

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