Realizable piecewise linear paths of persistence diagrams with Reeb graphs

TL;DR

This paper explores how smoothing operations on Reeb graphs affect their persistence diagrams and demonstrates how certain paths in persistence diagram space can be realized by continuous Reeb graph transformations, aiding inverse problem solutions.

Contribution

It introduces a method to realize paths in persistence diagram space through Reeb graph smoothing, advancing inverse problem analysis for Reeb graphs.

Findings

Smoothing operations simplify Reeb graphs by shrinking small loops.

Certain vineyards in persistence diagram space can be realized by Reeb graph paths.

The approach aids in solving inverse problems for Reeb graphs.

Abstract

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.