Recursive Removal-Collapsibility

• Recursive Removal-Collapsibility is a property defining the systematic reduction of discrete structures by recursively removing free faces to achieve a simpler subcomplex or a single point.
• It underpins efficient greedy collapse algorithms in simplicial and square complexes and distinguishes cases with polynomial-time solutions from NP-complete ones.
• The concept finds broad applications in computational topology, persistent homology, and constraint satisfaction in algebra and logic, emphasizing its role in structure simplification.

Recursive removal-collapsibility refers to the property of a combinatorial or algebraic structure (such as a simplicial complex, square complex, or finite relational structure) to be reduced, through a defined sequence of elementary removal operations, to a subset exhibiting some target property—typically to a single point (collapsibility) or to a subcomplex of prescribed dimension. The recursive aspect underscores a strategy in which the removal or collapse of certain “free” substructures is repeatedly and systematically applied, often using local conditions and maintaining invariants such as curvature or contractibility. This concept is pivotal in computational topology, algorithmic graph theory, algebra, and logic, where collapsibility bridges topological simplification, tractability, and model-theoretic property checking.

1. Foundational Definitions and Local Removal Strategies

In simplicial and cell complexes, recursive removal-collapsibility is formalized via the removal of “free” faces:

• A free face (simplicial or cellular) is a lower-dimensional simplex (or cell) contained in exactly one higher-dimensional cell.
• An elementary collapse consists of removing a free face together with its unique coface.
• A complex is recursively removal-collapsible if there exists a sequence of such collapses reducing the complex to a single point or a subcomplex of lower dimension, with the removal decision at each step based only on the current state.

This notion generalizes to diverse combinatorial models, including:

• Simplicial complexes, where recursive collapses correspond to the removal of facets or lower faces according to combinatorial rules.
• Square complexes, especially those with a CAT(0) metric, where removal preserves nonpositive curvature (Lazăr, 2020).
• Clique complexes of special graph families (e.g., strong Ivashchenko-contractible graphs) (Espinoza et al., 2018).

In logic and algebra, recursive removal manifests in the analysis of quantified constraint satisfaction over finite structures, where “collapsibility from a source” signifies the reduction of universally quantified variables to fixed values, witnessed by polymorphisms (“Hubie operations”) (Carvalho et al., 2015).

2. Formal Models and Variants

Simplicial and Cell Complex Models

The key structural elements are described as follows (Aronshtam et al., 2013, Paolini, 2017, Lazăr, 2020):

• Elementary Collapse: For a complex $X$ and a free face pair $(\sigma, \tau)$ (with $\tau \subset \sigma$, $\dim\sigma = \dim\tau + 1$ and $\tau$ in no other coface), remove both $\sigma$ and $\tau$.
• Sequenced Collapse: A recursive sequence $\{(K_i, (\sigma_i, \tau_i))\}$, with $K_{i+1} = K_i \setminus \{\sigma_i, \tau_i\}$, proceeding until $K_N$ is a point (or subcomplex of prescribed dimension).

Removal-Collapsibility and Its Hereditary Form

For pure simplicial complexes, the removal-collapsibility (RC) condition requires the removal of exactly $(\sigma, \tau)$0 facets (where $(\sigma, \tau)$1 is the $(\sigma, \tau)$2-th reduced Betti number) to render the complex collapsible. The hereditary RC (HRC) condition strengthens this by demanding RC for all links (Magnard et al., 2019).

• If a complex satisfies HRC, its second barycentric subdivision is vertex-decomposable, and thus shellable.

Algebraic and Logical Generalizations

In finite relational structures and digraphs, $(\sigma, \tau)$3-collapsibility from a source $(\sigma, \tau)$4 consists of reducing logical complexity by collapsing all but at most $(\sigma, \tau)$5 variables to a fixed value, testable by the existence of “Hubie operations” (multivariate polymorphisms with strong surjectivity and idempotency; (Carvalho et al., 2015)).

3. Algorithms and Complexity

Recursive removal-collapsibility underpins a family of greedy collapse algorithms, often implemented via breadth-first or depth-first identification and simultaneous removal of free faces:

• Greedy collapse algorithm (simplicial complexes):

1. Identify all free $(\sigma, \tau)$6-faces; remove with unique $(\sigma, \tau)$7-cofaces synchronously for $(\sigma, \tau)$8 rounds.
2. Add sequential round: remove free $(\sigma, \tau)$9-faces one by one in randomized or arbitrary order.
3. If all $\tau \subset \sigma$0-faces are removed, the original complex is collapsible (Aronshtam et al., 2013).

• Polynomial-time cases: For $\tau \subset \sigma$1-collapsibility with $\tau \subset \sigma$2 or $\tau \subset \sigma$3, greedy recursive removal yields efficient algorithms (e.g., removal of faces in decreasing dimension), with trees as the terminal structures in 1-complexes and greedy edge/triangle removal in 2-complexes (Paolini, 2017).
• NP-completeness: For $\tau \subset \sigma$4-collapsibility with $\tau \subset \sigma$5, except $\tau \subset \sigma$6, the decision problem is NP-complete. The hardness follows via dimension-raising reductions, with key base cases established via reductions from 3-SAT and other problems (Paolini, 2017).

Table: Complexity of $\tau \subset \sigma$7-Collapsibility (Paolini, 2017)

Case	Complexity	Algorithmic Principle
$\tau \subset \sigma$8 or $\tau \subset \sigma$9	Polynomial	Greedy recursive removal
$\dim\sigma = \dim\tau + 1$0, $\dim\sigma = \dim\tau + 1$1	NP-complete	Reduction from base NP-hard cases

4. Threshold Phenomena and Random Complexes

Recursive removal-collapsibility is the substrate for sharp phase transitions in the behavior of random complexes:

• In the random model $\dim\sigma = \dim\tau + 1$2 (simplicial complexes on $\dim\sigma = \dim\tau + 1$3 vertices with complete $\dim\sigma = \dim\tau + 1$4-skeleton and $\dim\sigma = \dim\tau + 1$5-faces included independently at probability $\dim\sigma = \dim\tau + 1$6), collapsibility is characterized by a threshold $\dim\sigma = \dim\tau + 1$7, with $\dim\sigma = \dim\tau + 1$8 defined by a saddle-node bifurcation in an associated fixed-point equation (Aronshtam et al., 2013).
• Below $\dim\sigma = \dim\tau + 1$9, recursive removal collapses all $\tau$0-faces a.a.s.; above $\tau$1, a “core” without free faces remains, yielding non-collapsibility.
• The analysis leverages local convergence to Poisson $\tau$2-trees, drift estimates in free face counts, and martingale concentration inequalities.

5. Structural Consequences and Decomposability

Recursive removal-collapsibility often leads to stronger combinatorial and topological properties:

• If a pure simplicial complex and all its links satisfy (hereditary) removal-collapsibility, its iterated barycentric subdivision is vertex-decomposable and hence shellable (Magnard et al., 2019). The proof introduces star-decomposability, inductively ordering faces and systematically reducing the complex via removals guided by homological invariants.
• In finite CAT(0) square 2-complexes, the CAT(0) condition is preserved throughout recursive removals, guaranteeing collapsibility in a combinatorics-free manner (Lazăr, 2020).
• For strong Ivashchenko-contractible graphs, recursive deletions of vertices with strong-contractible neighborhoods correspond exactly to simplicial collapses in the clique complex, leading to efficient algorithms for topological data analysis applications such as persistent homology computation (Espinoza et al., 2018).

6. Applications and Open Problems

Recursive removal-collapsibility is central to:

• Topological simplification in computational topology and data analysis, enabling reduction of complex size and computation in persistent homology (Espinoza et al., 2018).
• Structure theory in logic and algebra, providing a concrete connection between polymorphism clones, tractability, and the polynomially generated powers property (PGP) in constraint satisfaction problems (Carvalho et al., 2015).
• Shellability and decomposability in high-dimensional combinatorics, via removal-collapsibility conditions (Magnard et al., 2019).

Open problems include:

• Determining the complexity of collapsibility for restricted classes (flag complexes, manifolds).
• Existence of parameterized or approximation algorithms for difficult cases.
• Extension to other decomposability notions and their impact on the algebraic-topological landscape (Paolini, 2017, Magnard et al., 2019).

7. Domain-Specific Characterizations

Several domains instantiate recursive removal-collapsibility as follows:

• CAT(0) Square Complexes: Every finite CAT(0) square 2-complex is recursively removal-collapsible; at every stage, a free face exists and curvature is preserved (Lazăr, 2020).
• Simplicial Complexes (TDA): Strong Ivashchenko-contractible graphs admit deletion orders that directly correspond to a collapse of their clique complexes, leading to substantially reduced computational overhead in persistent homology (Espinoza et al., 2018).
• Finite Relational Structures: $\tau$3-collapsibility from a singleton source coincides with the existence of a Hubie operation of arity $\tau$4, allowing logical collapsibility to be checked via algebraic means (Carvalho et al., 2015).

Recursive removal-collapsibility thus unifies algorithmic, homotopical, and algebraic approaches to the simplification and analysis of complex discrete structures, with strong implications for computational complexity, topological data analysis, and combinatorial geometry.