  The Reeb graph has been utilized in various applications including the
analysis of scalar fields. Recently, research has been focused on using
topological signatures such as the Reeb graph to compare multiple scalar fields
by defining distance metrics on the topological signatures themselves. Here we
survey five existing metrics that have been defined on Reeb graphs: the
bottleneck distance, the interleaving distance, functional distortion distance,
the Reeb graph edit distance, and the universal edit distance. Our goal is to
(1) provide definitions and concrete examples of these distances in order to
develop the intuition of the reader, (2) visit previously proven results of
stability, universality, and discriminativity, (3) identify and complete any
remaining properties which have only been proven (or disproven) for a subset of
these metrics, (4) expand the taxonomy of the bottleneck distance to better
distinguish between variations which have been commonly miscited, and (5)
reconcile the various definitions and requirements on the underlying spaces for
these metrics to be defined and properties to be proven.
