On Merge Trees and Discrete Morse Functions on Paths and Trees

TL;DR

This paper establishes a correspondence between merge trees and discrete Morse functions on paths and trees, providing explicit constructions and a classification framework for these functions.

Contribution

It proves that every merge tree can be represented by a discrete Morse function on a path and introduces a classification of such functions on trees.

Findings

Every merge tree corresponds to a discrete Morse function on a path.

Explicit constructions for Morse functions inducing any given merge tree.

A bijection between Morse function classes and labeled merge trees.

Abstract

In this work we answer an open question asked by Johnson--Scoville. We show that each merge tree is represented by a discrete Morse function on a path. Furthermore, we present explicit constructions for two different but related kinds of discrete Morse functions on paths that induce any given merge tree. A refinement of the used methods allows us to define notions of equivalence of discrete Morse functions on trees which give rise to a bijection between equivalence classes of discrete Morse functions and isomorphism classes of certain labeled merge trees. We also compare our results to similar ones from the literature, in particular to work by Curry.