Reeb graphs of curves are stable under function perturbations

Reeb graphs provide a method to combinatorially describe the shape of amanifold endowed with a Morse function. One question deserving attention is whetherReeb graphs are robust against function perturbations. Focusing on 1-dimensional manifolds,we define an editing distance between Reeb graphs of curves, in terms of the costnecessary to transform one graph into another through editing moves. Our main result isthat changes in Morse functions induce smaller changes in the editing distance betweenReeb graphs of curves, implying stability of Reeb graphs under function perturbations. Wealso 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.