#  Copyright (c) 1997-2021
#  Ewgenij Gawrilow, Michael Joswig, and the polymake team
#  Technische Universität Berlin, Germany
#  https://polymake.org
#
#  This program is free software; you can redistribute it and/or modify it
#  under the terms of the GNU General Public License as published by the
#  Free Software Foundation; either version 2, or (at your option) any
#  later version: http://www.gnu.org/licenses/gpl.txt.
#
#  This program is distributed in the hope that it will be useful,
#  but WITHOUT ANY WARRANTY; without even the implied warranty of
#  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
#  GNU General Public License for more details.
#-------------------------------------------------------------------------------


# @category Topology
# 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 ([[topaz::SimplicialComplex|greedy]]). The alternative is to use a canceling algorithm
# due to Forman ([[topaz::SimplicialComplex|cancel]]) or both ([[topaz::SimplicialComplex|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
#
# @relates objects/SimplicialComplex
declare object MorseMatching {


# Size of the computed Morse matching.
property SIZE : Int;


# The critical faces of the computed Morse matching, i.e., the faces not
# incident to any reoriented arc (not matched).
property CRITICAL_FACES : Array<Set>;


# The vector of critical faces in each dimension.
property CRITICAL_FACE_VECTOR : Array<Int>;


# Number of critical faces of the computed Morse matching.
property N_CRITICAL_FACES : Int;

}

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