The edit distance for Reeb graphs of surfaces

TL;DR

This paper introduces a new combinatorial edit distance for Reeb graphs of surfaces, proving its stability under function perturbations and its optimal discriminative power compared to other metrics.

Contribution

It defines a novel edit distance for Reeb graphs of surfaces and establishes its stability and optimality properties.

Findings

The edit distance is stable under small changes in the input functions.

The proposed metric discriminates Reeb graphs more effectively than existing metrics.

The work provides theoretical guarantees for the metric's performance.

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 metric 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 functions, measured by the maximum norm, imply not greater changes in the corresponding Reeb graphs, measured by the edit distance. The optimality result states that our edit distance discriminates Reeb graphs better than any other metric for Reeb graphs of surfaces satisfying the stability property.