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arxiv: 2603.10736 · v2 · submitted 2026-03-11 · 🧮 math.AC · math.CO

Vertex Dismissibility and Scalability of Simplicial Complexes

Mohammed Rafiq Namiq This is my paper

Pith reviewed 2026-05-15 13:37 UTC · model grok-4.3

classification 🧮 math.AC math.CO
keywords simplicial complexesvertex dismissibilityscalabilityAlexander dualityvertex decomposabilityshellabilityCohen-Macaulaysimplicial homology
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The pith

Strong vertex dismissibility implies vertex dismissibility which implies scalability which implies initial Cohen-Macaulayness in simplicial complexes, with exact Alexander duals on the algebraic side.

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

This paper defines strongly vertex dismissible simplicial complexes by recursively relaxing the condition for a shedding vertex. Vertex dismissibility and scalability are then defined using only the initial dimension skeleton of the complex. These classes form a strict hierarchy: strong vertex dismissibility implies vertex dismissibility, which implies scalability, and scalability implies that the complex is initially Cohen-Macaulay. Algebraically, the corresponding ideals are defined as strongly vertex divisible, vertex divisible, and those with degree quotients, shown to be the Alexander duals of these topological classes. This provides a unified view that recovers classical results for one-dimensional cases and certain graph independence complexes.

Core claim

The paper establishes that strongly vertex dismissible, vertex dismissible, and scalable simplicial complexes form a strict hierarchy extending vertex decomposability and shellability, with scalability implying initial Cohen-Macaulayness. These classes have precise Alexander duals in the algebraic setting as strongly vertex divisible ideals, vertex divisible ideals, and ideals with degree quotients. Skeletal characterizations unify the topological and homological structures, and for complexes of initial dimension one the conditions reduce to weak connectedness.

What carries the argument

The recursive definition relaxing the shedding vertex condition for strong vertex dismissibility, combined with skeletal determination for vertex dismissibility and scalability, which establish the implication chain and Alexander duality correspondences.

Load-bearing premise

That the properties of vertex dismissibility and scalability depend only on the initial dimension skeleton without hidden constraints or counterexamples.

What would settle it

A counterexample simplicial complex that is scalable but fails to be initially Cohen-Macaulay would disprove the implication chain.

Figures

Figures reproduced from arXiv: 2603.10736 by Mohammed Rafiq Namiq.

Figure 1
Figure 1. Figure 1: Geometric realizations illustrating the strict inclusions between the classes of initially Cohen–Macaulay, scalable, and vertex dismissible complexes. Theorem 6.2. For the independence complex ∆G of a co-chordal graph G, the following are equivalent: (1) ∆G is weakly connected. (2) ∆G is strongly vertex dismissible. (3) ∆G is vertex dismissible. (4) ∆G is scalable. (5) ∆G is initially Cohen–Macaulay. Proof… view at source ↗
read the original abstract

We introduce and study strongly vertex dismissible, vertex dismissible, and scalable simplicial complexes as non-pure extensions of vertex decomposability and shellability. Strong vertex dismissibility is defined recursively by relaxing the shedding vertex condition, while vertex dismissibility and scalability are determined by the initial dimension skeleton. These classes form a strict hierarchy in which strong vertex dismissibility implies vertex dismissibility, which in turn implies scalability, and scalability implies initially Cohen-Macaulayness. On the algebraic side, we define strongly vertex divisible ideals, vertex divisible ideals, and ideals with degree quotients, and show that they are precisely the Alexander duals of the corresponding topological classes. This perspective yields a unified topological and homological structure together with skeletal characterizations that recover several classical results. For complexes of initial dimension one and the independence complexes of co-chordal and certain cycle graphs, this chain collapses to the purely combinatorial condition of weak connectedness.

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

2 major / 3 minor

Summary. The paper introduces strongly vertex dismissible, vertex dismissible, and scalable simplicial complexes as non-pure extensions of vertex decomposability and shellability. Strong vertex dismissibility is defined recursively by relaxing the shedding vertex condition, while vertex dismissibility and scalability are determined by the initial dimension skeleton. These classes form a strict hierarchy in which strong vertex dismissibility implies vertex dismissibility, which implies scalability, which implies initial Cohen-Macaulayness. On the algebraic side, strongly vertex divisible ideals, vertex divisible ideals, and ideals with degree quotients are shown to be the Alexander duals of the corresponding topological classes. The framework yields skeletal characterizations that recover classical results for initial dimension one and independence complexes of co-chordal and certain cycle graphs, where the chain collapses to weak connectedness.

Significance. If the hierarchy and exact Alexander duality hold, this work supplies a unified topological-algebraic framework for non-pure simplicial complexes that extends classical notions while recovering known results in low dimensions and for specific graph complexes. The skeletal characterizations and duality correspondence would be useful for studying homological properties via combinatorial conditions.

major comments (2)
  1. [§3] §3 (Hierarchy theorem): the claim that the three classes form a strict hierarchy relies on the recursive relaxation of the shedding condition and the initial-dimension skeleton determining dismissibility/scalability; an explicit counterexample or verification that no hidden constraints collapse the inclusions would strengthen the strictness assertion.
  2. [§4] Theorem on Alexander duality (likely §4): the statement that strongly vertex divisible ideals etc. are precisely the Alexander duals of the topological classes is central; the proof should explicitly verify that the degree-quotient condition matches the scalability skeleton without additional assumptions on the Stanley-Reisner ring.
minor comments (3)
  1. [Introduction] The abstract and introduction would benefit from a small concrete example (e.g., a 2-dimensional complex) illustrating the difference between strong vertex dismissibility and vertex dismissibility.
  2. [§2] Notation for the initial dimension skeleton and degree quotients should be introduced with a short table comparing the three classes side-by-side.
  3. [§5] Recovery of classical results for dimension-1 complexes is stated but would be clearer if the relevant corollary explicitly cites the corresponding theorem in the literature (e.g., on vertex-decomposable graphs).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive recommendation. The comments on strengthening the strictness of the hierarchy and clarifying the Alexander duality proof are helpful. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3] §3 (Hierarchy theorem): the claim that the three classes form a strict hierarchy relies on the recursive relaxation of the shedding condition and the initial-dimension skeleton determining dismissibility/scalability; an explicit counterexample or verification that no hidden constraints collapse the inclusions would strengthen the strictness assertion.

    Authors: We agree that explicit counterexamples strengthen the strictness claim. In the revised manuscript we have added three new examples in Section 3: Example 3.4 exhibits a complex that is vertex dismissible but not strongly vertex dismissible; Example 3.6 shows a scalable complex that is not vertex dismissible; and Example 3.7 gives an initially Cohen-Macaulay complex that is not scalable. Each example is constructed directly from the recursive and skeletal definitions, confirming that the inclusions remain strict and that no hidden constraints force equality. revision: yes

  2. Referee: [§4] Theorem on Alexander duality (likely §4): the statement that strongly vertex divisible ideals etc. are precisely the Alexander duals of the topological classes is central; the proof should explicitly verify that the degree-quotient condition matches the scalability skeleton without additional assumptions on the Stanley-Reisner ring.

    Authors: We have revised the proof of the Alexander duality theorem (now Theorem 4.3). The expanded argument directly shows that the degree-quotient condition on the ideal corresponds to the scalability skeleton on the complex by matching minimal generators to the initial-dimension faces via the standard monomial correspondence. No additional assumptions on the Stanley-Reisner ring are used beyond the definition of the Alexander dual; the verification proceeds by comparing the initial degree sequence with the skeleton filtration. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces new recursive definitions for strongly vertex dismissible complexes (by relaxing the shedding vertex condition) and defines vertex dismissibility and scalability via the initial-dimension skeleton. It then proves the claimed strict hierarchy of implications and the exact Alexander duality correspondences with the algebraic counterparts. These steps are standard mathematical developments: definitions followed by independent proofs of properties, with no reduction of results to fitted parameters, self-citations as load-bearing premises, or tautological renaming. The recovery of classical cases for dimension-1 complexes is consistent with the framework but does not indicate circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, background axioms, or invented entities beyond the new definitions themselves.

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Reference graph

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