arXiv subjects
Publications and source records attributed to Julian Brüggemann.
In this work, we introduce a combinatorial-geometric model for the space of discrete Morse functions on any CW complex $X$. We relate this version of a space of discrete Morse functions to the space of cellular filtrations of $X$ and discuss its relationship to various concepts such as smooth Morse theory, Cerf theory, complexes of discrete Morse matchings, and induced merge trees and barcodes.
In this paper, we extend the notion of a merge tree to that of a generalized merge tree, a merge tree that includes 1-dimensional cycle birth information. Given a discrete Morse function on a $1$-dimensional regular CW complex, we construct the induced generalized merge tree. We give several notions of equivalence of discrete Morse functions based on the induced generalized merge tree and how these notions relate to one another. As a consequence, we obtain a complete solution to the inverse problem between discrete Morse functions on $1$-dimensional regular CW complexes and generalized merge trees. After characterizing which generalized merge trees can be induced by a discrete Morse function on a simple graph, we give an algorithm based on the induced generalized merge tree of a discrete Morse function $f\colon X \to \mathbb{R}$ that cancels the critical simplices of $f$ and replaces it with an optimal discrete Morse function.
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.