Ascending and descending regions of a discrete Morse function

Authors:
Gregor Jerše;Neža Mramor Kosta
Affiliations:
Institute of Mathematics, Physics and Mechanics, Ljubljana, Jadranska 19, Slovenia;University of Ljubljana, Faculty of Computer and Information Science and Institute of Mathematics, Physics and Mechanics, Ljubljana, Jadranska 19, Slovenia
Venue:
Computational Geometry: Theory and Applications
Year:
2009

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Abstract

We present an algorithm which produces a decomposition of a regular cellular complex with a discrete Morse function analogous to the Morse-Smale decomposition of a smooth manifold with respect to a smooth Morse function. The advantage of our algorithm compared to similar existing results is that it works, at least theoretically, in any dimension. Practically, there are dimensional restrictions due to the size of cellular complexes of higher dimensions, though. We prove that the algorithm is correct in the sense that it always produces a decomposition into descending and ascending regions of the critical cells in a finite number of steps, and that, after a finite number of subdivisions, all the regions are topological disks. The efficiency of the algorithm is discussed and its performance on several examples is demonstrated.