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Reference documentation for older polymake versions: release 3.4, release 3.3, release 3.2

# BigObject MorseMatching

from application topaz

A Morse matching is a reorientation of the arcs in the Hasse diagram of a simplicial complex such that at most one arc incident to each face is reoriented (matching condition) and the resulting orientation is acyclic (acyclicity condition). Morse matchings capture the main structure of discrete Morse functions, see

Robin Forman: Morse Theory for Cell-Complexes,
Advances in Math., 134 (1998), pp. 90-145.

This property is computed by one of two heuristics. The default heuristic is a simple greedy algorithm (greedy). The alternative is to use a canceling algorithm due to Forman (cancel) or both (both) together. Note that the computation of a Morse matching of largest size is NP-hard. See

Michael Joswig, Marc E. Pfetsch: Computing Optimal Morse Matchings
SIAM J. Discrete Math., 2006, to appear

### no category

CRITICAL_FACES

The critical faces of the computed Morse matching, i.e., the faces not incident to any reoriented arc (not matched).

Type:
Array<Set<Int>>

CRITICAL_FACE_VECTOR

The vector of critical faces in each dimension.

Type:
Array<Int>

N_CRITICAL_FACES

Number of critical faces of the computed Morse matching.

Type:
Int

SIZE

Size of the computed Morse matching.

Type:
Int

• documentation/latest/topaz/morsematching.txt