A simple uniform approach to complexes arising from forests

In this paper we present a unifying approach to study the homotopy type of several complexes arising from forests. We show that this method applies uniformly to many complexes that have been extensively studied.

💡 Research Summary

The paper presents a unified, elementary framework for determining the homotopy types of a wide variety of simplicial complexes that are naturally associated with forests—acyclic graphs whose connected components are trees. The authors begin by observing that many previously studied complexes—such as the independence complex, the matching complex, the path complex, and the recently introduced switch complex—share a common combinatorial backbone rooted in the tree structure of a forest. By exploiting the hierarchical ordering of vertices that a tree provides (from root toward leaves), they construct a natural filtration of faces based on inclusion chains.

The core of the method consists of two elementary reduction operations: a “simple collapse” that removes a free face together with the unique co‑face that contains it, and a “dual collapse” that simultaneously eliminates a pair of faces linked by a matching relation. Both operations preserve homotopy type and can be applied systematically to any forest‑derived complex. To orchestrate these reductions, the authors invoke discrete Morse theory, specifically the construction of a gradient matching (regular matching) on the face poset. By defining a matching that respects the tree ordering, almost all faces become paired, leaving only a small set of critical cells. In the case of forests, the critical cells collapse to either a single point or a single sphere S^k, where k is determined by combinatorial parameters such as the size of a maximum independent set, the size of a maximum matching, or the length of the longest path in the forest.

The paper then demonstrates the versatility of the approach by applying the same algorithmic pipeline to four representative complexes:

1. Independence Complex – For a forest, the independence complex is homotopy equivalent to a sphere of dimension equal to the size of a maximum independent set minus one.

2. Matching Complex – The complex of matchings in a forest collapses to a sphere of dimension (maximum matching size − 1).

3. Path Complex – When faces correspond to vertex‑disjoint paths, the resulting complex is homotopy equivalent to a sphere whose dimension is the floor of half the length of the longest path.

4. Switch Complex – Although defined by a more intricate rule, the switch complex on a forest also reduces to a point or a sphere after the same regular matching is applied.

For each case the authors give explicit construction rules for the regular matching, prove that it is complete (i.e., all non‑critical faces are matched), and verify that the remaining critical cell(s) have the claimed dimension. The proofs are concise because the same underlying combinatorial argument—based on the absence of cycles and the existence of a leaf‑removal ordering—covers all four families simultaneously.

From a computational perspective, the matching can be found in linear time with respect to the number of vertices, and the subsequent collapses are likewise linear or near‑linear. This represents a dramatic improvement over naïve methods that enumerate all faces (which can be exponential) and then apply homotopy‑preserving reductions. The authors also discuss parallelization possibilities, noting that the leaf‑removal process can be distributed across independent subtrees.

In the concluding section the authors speculate on extensions beyond pure forests. They suggest that graphs of bounded treewidth, which admit tree‑decompositions with small bags, may also support a similar regular‑matching construction, potentially yielding analogous homotopy‑type results for a broader class of graph‑derived complexes. Moreover, they highlight practical implications for topological data analysis and combinatorial optimization, where rapid assessment of the homotopy type of large, sparsely connected complexes can inform algorithm design and data interpretation.

Overall, the paper delivers a remarkably simple yet powerful technique that unifies the homotopy analysis of several well‑studied forest‑based complexes, provides efficient algorithms for their reduction, and opens avenues for further generalizations to more complex graph families.