Further References

Join trees have been harnessed for a similar purpose by Saikia, Seidel, and Weinkauf (2014). Further methods to compare Reeb graphs have been proposed by Bauer, Ge, and Wang (2014) and for the special case of functions on orientable surfaces by Di Fabio and Landi (2014). The two distances from (Bauer, Ge, and Wang 2014) and (de Silva, Munch, and Patel 2016) have been shown to be equivalent in some sense by Bauer, Munch, and Wang (2015). Moreover the work presented here borrows from (Bubenik, de Silva, and Scott 2014) on several aspects. There is much more prior art and the papers we mentioned merely represent our primary references. Most of the papers mentioned by de Silva, Munch, and Patel (2016) and Bubenik, de Silva, and Scott (2014) in their introductions are antecedents to our work as well. We presented some of the abstract ideas we use here in the form of a poster (Fluhr 2017). Little did we know that de Silva, Munch, and Stefanou (2017) were developing a very similar framework. Some differences between their approach and ours is that they work in a very general setup with an arbitrary metric space whereas we merely consider the real numbers as a base space. This made it feasible for us to treat the absolute and the relative interleaving distance in one go.