The realization problem for discrete Morse functions on trees

TL;DR

This paper introduces persistence equivalence for discrete Morse functions on graphs, especially trees, providing bounds on their count and addressing the realization problem of the persistence map.

Contribution

It defines a new equivalence notion for discrete Morse functions based on persistence diagrams and solves the realization problem for trees.

Findings

Upper bound for the number of persistence equivalent functions on a fixed graph

The upper bound is sharp for trees

Illustrative example of the construction

Abstract

We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence. Two functions are considered persistence equivalent if and only if they induce the same persistence diagram. We compare this notion of equivalence to other notions of equivalent discrete Morse functions. We then compute an upper bound for the number of persistence equivalent discrete Morse functions on a fixed graph and show that this upper bound is sharp in the case where our graph is a tree. This is a version of the "realization problem" of the persistence map. We conclude with an example illustrating our construction.