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Reeb graphs of curves are stable under function perturbations
Author(s) -
Di Fabio B.,
Landi C.
Publication year - 2012
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.2533
Subject(s) - mathematics , morse code , morse theory , bottleneck , bounded function , graph , function (biology) , combinatorics , discrete mathematics , pure mathematics , mathematical analysis , computer science , telecommunications , evolutionary biology , biology , embedded system
Reeb graphs provide a method to combinatorially describe the shape of a manifold endowed with a Morse function. One question deserving attention is whether Reeb graphs are robust against function perturbations. Focusing on one‐dimensional manifolds, we define an editing distance between Reeb graphs of curves, in terms of the cost necessary to transform one graph into another through editing moves. Our main result is that changes in Morse functions induce smaller changes in the editing distance between Reeb graphs of curves, implying stability of Reeb graphs under function perturbations. We also prove that our editing distance is equal to the natural pseudo‐distance and, moreover, that it is lower bounded by the bottleneck distance of persistent homology. Copyright © 2012 John Wiley & Sons, Ltd.