Strong Collapses of Simplicial Complexes

Updated 3 December 2025

Strong collapses are combinatorial operations that remove dominated simplices to produce a canonical minimal core while preserving the homotopy type.
Efficient greedy algorithms and matrix pruning techniques identify and reduce dominated vertices, ensuring rapid simplification in 2D and higher-dimensional complexes.
These methods underpin advances in persistent homology and topological data analysis by yielding precise, computationally robust reductions of complex datasets.

A strong collapse of a simplicial complex is a combinatorial operation that removes dominated simplices, leading to drastic simplification while strictly preserving the homotopy type. Originating with Barmak–Minian, strong collapse plays a foundational role in homotopy-theoretic reductions, topological data analysis, and combinatorial topology. Its defining feature is the canonical nature of the resulting minimal subcomplex (the "core"), which is unique up to isomorphism for any finite complex. In two-dimensional complexes, strong collapsibility can be efficiently recognized and tightly characterized via vertex links isomorphic to stars. Strong collapse methods underpin major advances in persistent homology computations, reduction of Dowker and flag complexes, and the analysis of random simplicial complexes.

1. Definitions and Fundamental Properties

A strong collapse is defined by removing dominated simplices from a finite abstract simplicial complex $K$ . A simplex $\sigma \in K$ is dominated if its link $\operatorname{lk}_K(\sigma)$ is a simplicial cone, i.e., $\operatorname{lk}_K(\sigma) = v' * L$ for some vertex $v'$ and subcomplex $L$ (Boissonnat et al., 2023). In terms of vertices, $v$ is dominated by $v'$ iff every maximal simplex containing $v$ also contains $v'$ (Boissonnat et al., 2018, Côté, 16 Jan 2024).

An elementary strong collapse removes a dominated simplex, yielding $K \setminus \sigma$ . A sequence of such steps produces a strong collapse $K \rightsquigarrow L$ , culminating in a strongly minimal subcomplex $L$ . The unique minimal subcomplex, called the core, is denoted $K^c$ or $R_\infty(K)$ ; every finite $K$ admits a canonical core up to combinatorial isomorphism (Minian, 26 Nov 2025).

For flag complexes (clique complexes), domination is characterized graph-theoretically: a clique $\sigma$ is dominated by $v'$ if $N_G[\sigma] \subseteq N_G[v']$ , where $N_G[\sigma]$ is the common closed neighborhood in the underlying graph $G$ (Boissonnat et al., 2023).

2. Recognition Algorithms and Complexity

Strong collapsibility in two-dimensional simplicial complexes admits efficient greedy recognition (Adamaszek, 2010). At each step, any vertex whose link is a star (tree of diameter $\leq 2$ ) can be deleted; repeating this process determines whether the complex collapses to a single vertex (i.e., is strongly collapsible). For $K$ a 2-complex:

1
2
3

while there is v with lk_K(v) ≅ S_n for some n:
    K ← K \ {v}
return "strongly collapsible" if K is a single vertex else "not"

This greedy algorithm runs in linear time for stars of arbitrary size. In contrast, mixed families of trees (such as inclusion of paths of length $d\geq3$ ) yield NP-complete recognition problems.

In higher dimensions or for arbitrary complexes specified by maximal faces, core computation reduces to alternating row/column pruning in the vertex–maximal simplex matrix. Each row (vertex) is removed when its support is contained in another, followed by column pruning on the transposed matrix. Complexity bounds depend on the sparsity and dimension of the input, with typical practical behavior near-quadratic in the vertex count (Boissonnat et al., 2018, Côté, 16 Jan 2024).

3. Homotopy and Homological Implications

Each elementary strong collapse $K \rightsquigarrow K \setminus v$ admits a canonical simplicial retraction $r$ , sending $v \mapsto v'$ (the dominating vertex) and fixing all other vertices (Boissonnat et al., 2018, Minian, 26 Nov 2025). Composing such retractions over the collapse sequence produces a strong deformation retract to the core $L$ , guaranteeing homotopy equivalence $K \simeq L$ and isomorphic homology groups.

Strong collapses strictly preserve the homotopy type, unlike general simplicial collapses, which preserve only simple-homotopy equivalence. Flag property is also preserved: strong collapse reductions in flag complexes yield subcomplexes that remain flag (Boissonnat et al., 2023). This stability under strong collapse underpins reductions in persistent homology, since barcodes computed on the sequence of cores are identical to those for the full filtration, as proven for zigzag modules, towers, and filtrations (Boissonnat et al., 2018).

4. Structures, Examples, and Counterexamples

Arc complexes serve as canonical examples. The arc complex $A(\holed n)$ of a polygon with interior marked point is a combinatorial ball and strongly collapsible: boundary-arcs are peeled off via domination by core-arcs, yielding a simplex (Panda, 16 Feb 2024). In contrast, $A(\mob[n])$, the arc complex of a Möbius strip with $n\geq4$ marked points, is not strongly collapsible. The absence of a conical link for core-arcs and stall of the collapse sequence led by boundary-arcs implies that its strong-collapse core is nontrivial.

For random clique complexes $X(n,c/n)$ , strong collapse admits asymptotic analysis (Boissonnat et al., 2023). After maximal strong collapses, the 0-core has size $(1-\gamma)(1-c\gamma) n + o(n)$ with $\gamma = e^{-c(1-\gamma)}$ . This result, derived from neighborhood pruning algorithms and fixed-point tree approximations, offers the first precise reduction laws for strong collapse on random or nonrandom complexes.

Not all combinatorial invariants are preserved. The geometric simplicial Lusternik–Schnirelmann (LS) category can strictly increase under a strong collapse: explicit examples have $gscat\,K<gscat\,M$ for $M \rightsquigarrow K$ , refuting earlier conjectures of strong homotopy invariance (Askitis, 2017).

5. Algorithmic Reductions via Relations and Dowker Complexes

Strong collapse generalizes naturally to relation-based and Dowker complexes through binary matrix representations (Côté, 16 Jan 2024). The reduction algorithm involves:

Identifying candidate pairs of vertices whose union of closed stars ( $L$ ) is contractible.
Adding a dummy vertex $z$ and coning off $L$ .
Strong collapsing each vertex in $L$ onto $z$ .

If $L$ is contractible, the gluing theorem guarantees homotopy equivalence $|K| \simeq |K'|$ . The reduction is completed in $O(n^2)$ pair tests and algebraic checks for contractibility. The method is adaptable to arbitrary relational structures—including convex polytopes and nerves of covers—where contractibility and cone-addition facilitate topological reduction.

6. Strong Collapse in Topological Data Analysis

Strong collapses are widely employed to accelerate persistent homology calculations, both for filtration-based and sequence-based approaches (Boissonnat et al., 2018, Minian, 26 Nov 2025). In Vietoris–Rips complexes arising from point clouds, strong collapse of dominated vertices yields a minimal core subsample ("δ-core" subsampling), preserving all global and local homology up to scale $\delta$ .

The $\delta$ -core subsampling algorithm iteratively removes $\delta$ -dominated points, dramatically reducing the size of complexes for subsequent homology computation, with observed near-linear complexities for geometric data. For fixed scale $\delta$ , VR $(X,\delta)$ strongly collapses to VR $(X',\delta)$ , and persistent homology barcodes are closely approximated (empirically within bottleneck distances $O(10^{-1})$ to $O(10^{0})$ ) (Minian, 26 Nov 2025). This approach surpasses farthest-point sampling and witness complex-based reductions, yielding lower matching errors and computational cost for a range of synthetic and real datasets.

7. Open Questions and Research Directions

Extensions to higher-dimensional and multiparameter random complexes, Čech and witness filtrations, and dynamically evolving data are active areas of investigation (Boissonnat et al., 2023, Minian, 26 Nov 2025). Unresolved questions include formal stability theorems for core filtrations, precise error bounds for persistence diagrams post-collapse, and combinatorial characterization of deterministic complexes admitting substantial reduction. Furthermore, the interplay between geometric LS-category and strong collapse, as well as optimal strategies for adaptive subsampling, remain open topics.

In summary, strong collapse offers an algorithmically tractable, homotopy-theoretically sound mechanism for simplifying simplicial complexes, with rigorous guarantees and broad applicability ranging from theoretical topology to computational data analysis.