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The Edit Distance for Reeb Graphs of Surfaces

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Abstract

Reeb graphs are structural descriptors that capture shape properties of a topological space from the perspective of a chosen function. In this work, we define a combinatorial distance for Reeb graphs of orientable surfaces in terms of the cost necessary to transform one graph into another by edit operations. The main contributions of this paper are the stability property and the optimality of this edit distance. More precisely, the stability result states that changes in the Reeb graphs, measured by the edit distance, are as small as changes in the functions, measured by the maximum norm. The optimality result states that the edit distance discriminates Reeb graphs better than any other distance for Reeb graphs of surfaces satisfying the stability property.

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Acknowledgments

The authors wish to thank Professor V. V. Sharko for his clarifications on the uniqueness property of Reeb graphs of surfaces and for indicating the reference [18]. The research described in this article has been partially supported by GNSAGA-INdAM (Italy).

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Authors and Affiliations

  1. Dipartimento di Scienze e Metodi dell’Ingegneria, Università di Modena e Reggio Emilia, Via Amendola 2, Pad, 42122, Morselli, Reggio Emilia, Italy

    Barbara Di Fabio & Claudia Landi

  2. ARCES, Università di Bologna, Via Toffano 2/2, 40125, Bologna, Italy

    Barbara Di Fabio & Claudia Landi

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  1. Barbara Di Fabio
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  2. Claudia Landi
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Correspondence to Claudia Landi.

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Di Fabio, B., Landi, C. The Edit Distance for Reeb Graphs of Surfaces. Discrete Comput Geom 55, 423–461 (2016). https://doi.org/10.1007/s00454-016-9758-6

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