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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.
Given a persistence diagram with $n$ points, we give an algorithm that produces a sequence of $n$ persistence diagrams converging in bottleneck distance to the input diagram, the $i$th of which has $i$ distinct (weighted) points and is a $2$-approxim
Graphs model real-world circumstances in many applications where they may constantly change to capture the dynamic behavior of the phenomena. Topological persistence which provides a set of birth and death pairs for the topological features is one in
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
Persistence diagrams are important tools in the field of topological data analysis that describe the presence and magnitude of features in a filtered topological space. However, current approaches for comparing a persistence diagram to a set of other
In this paper, we introduce an extension of smoothing on Reeb graphs, which we call truncated smoothing; this in turn allows us to define a new family of metrics which generalize the interleaving distance for Reeb graphs. Intuitively, we chop off par