Perfect discrete Morse functions on Stratifoldds

In this paper, we study the computation of optimal discrete Morse functions on stratifolds. In particular, we present an algorithm that efficiently computes such functions for a broad class of them. Moreover, we characterize the conditions under which these functions are perfect.

💡 Research Summary

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The paper investigates optimal and perfect discrete Morse functions (DMFs) on a special class of topological spaces known as 2‑dimensional stratifolds. A stratifold X is described as a union of compact surfaces M = {M₁,…,Mₙ} whose boundary components are glued to a collection of circles C = {c₁,…,cᵣ} via covering maps of integer degree. By fixing orientations on the surfaces and circles, the authors encode the gluing data in a weighted bipartite graph G(X): white vertices correspond to surfaces, black vertices to circles, and each edge e carries a weight w(e) equal to the degree of the attaching map (signs reflect orientation).

The central theoretical contribution consists of two theorems that give necessary and sufficient arithmetic conditions for the existence of a perfect DMF, i.e., a DMF whose number of critical i‑cells equals the i‑th Betti number βᵢ of X (over a chosen field).

• Theorem 3.1 (oriented case): If every white‑vertex weight is non‑negative (all surfaces are oriented) and there exists a prime p such that p divides the sum of the weights of all edges incident to each pair (i, j), then β₂(X; ℤₚ) equals n, the number of surface components. Consequently a DMF with exactly n critical 2‑cells exists, and because β₀ and β₁ are already minimal, the DMF is perfect.

• Theorem 3.2 (non‑oriented case): If at least one white‑vertex weight is negative (some surface is non‑orientable) and 2 divides every edge‑weight sum, then β₂(X; ℤ₂)=n, yielding a perfect DMF over ℤ₂.

If these arithmetic conditions fail, β₂ is strictly smaller than n, so a perfect DMF cannot exist; nevertheless an optimal DMF (with the fewest possible critical cells) still does.

To construct any triangulation of a finite stratifold, the authors prove that X can be represented as a disjoint union of triangulated polygons P₁,…,Pₙ (each polygon triangulates a surface Mᵢ) with identified sides according to the gluing maps. This “polygon‑with‑identified‑edges” model captures every possible simplicial decomposition of X, because each boundary circle must lie in the 1‑skeleton and each surface interior is planar.

The algorithmic core reduces the problem of finding an optimal DMF to building a gradient vector field (a Morse matching) on G(X) without closed V‑paths. The authors adapt the classic spanning‑tree construction: choose a root vertex in each white component, orient each tree edge from the farther vertex toward the root, and obtain a discrete vector field V such that −∇f = V. Since V contains no closed V‑paths, Theorem 2.1 guarantees the existence of a DMF f whose critical cells are precisely the unmatched vertices. In the oriented case, the unmatched vertices are exactly one root (a critical 0‑cell) and the n surface cells (critical 2‑cells); the number of critical 1‑cells matches β₁, making the DMF perfect when the arithmetic condition holds.

Complexity analysis shows that constructing G(X) from the CW description of X takes linear time in the number of polygons and edges, and the spanning‑tree/Morse‑matching step is also linear (or O(N log N) with standard data structures). Thus, for stratifolds the optimal DMF problem, which is MAX‑SNP‑Hard on arbitrary 2‑complexes, becomes polynomial‑time solvable.

The paper situates its contributions relative to prior work: Lewiner et al. proved optimal DMFs exist for any 2‑manifold but the construction is non‑trivial; Ayala et al. studied perfect DMFs on graphs and pseudo‑projective spaces. By focusing on stratifolds, the authors obtain a clean combinatorial criterion (divisibility by a prime) and an explicit, efficient algorithm.

Potential applications include topological data analysis where one often needs a reduced cell complex preserving homology, mesh simplification in computer graphics, and the study of spaces obtained by gluing surfaces along circles (common in 3‑manifold decompositions). The arithmetic test for perfectness provides a fast pre‑processing step to decide whether a homology‑optimal reduction is also minimal in the Morse‑theoretic sense.

In summary, the paper delivers: (1) a precise topological model of 2‑stratifolds via weighted bipartite graphs; (2) necessary and sufficient number‑theoretic conditions for the existence of perfect discrete Morse functions; (3) a linear‑time algorithm to construct optimal (and when possible, perfect) DMFs; and (4) a comprehensive bridge between combinatorial Morse theory and the geometry of stratifolds, opening avenues for both theoretical exploration and practical computation.