  Reeb graphs are widely used in a range of fields for the purposes of
analyzing and comparing complex spaces via a simpler combinatorial object.
Further, they are closely related to extended persistence diagrams, which
largely but not completely encode the information of the Reeb graph. In this
paper, we investigate the effect on the persistence diagram of a particular
continuous operation on Reeb graphs; namely the (truncated) smoothing
operation. This construction arises in the context of the Reeb graph
interleaving distance, but separately from that viewpoint provides a
simplification of the Reeb graph which continuously shrinks small loops. We
then use this characterization to initiate the study of inverse problems for
Reeb graphs using smoothing by showing which paths in persistence diagram space
(commonly known as vineyards) can be realized by a path in the space of Reeb
graphs via these simple operations. This allows us to solve the inverse problem
on a certain family of piecewise linear vineyards when fixing an initial Reeb
graph.

Computational Geometry