Reeb graphs of curves are stable under function perturbations |
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Authors: | B Di Fabio C Landi |
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Institution: | 1. ARCES, Università di Bologna, , I‐40125 Bologna, Italy;2. Dipartimento di Scienze e Metodi dell'Ingegneria, Università di Modena e Reggio Emilia, Via Amendola 2, Pad. Morselli, I‐42100 Reggio Emilia, Italy, ARCES, Università di Bologna, , I‐40125 Bologna, Italy |
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Abstract: | 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. |
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Keywords: | shape similarity editing distance Morse function natural stratification natural pseudo‐distance |
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