A simplicial complex is uniquely determined by its set of discrete Morse functions

TL;DR

This paper proves that a connected simplicial complex can be uniquely reconstructed from its set of discrete Morse functions, providing a complete characterization that answers a longstanding open question.

Contribution

It establishes that the complex of discrete Morse functions uniquely determines the original simplicial complex, resolving a question posed by Chari and Joswig.

Findings

Connected simplicial complex is uniquely determined by its discrete Morse functions

In 1D, the complex of rooted forests determines the graph G

Addresses a question raised by Chari and Joswig

Abstract

We prove that a connected simplicial complex is uniquely determined by its complex of discrete Morse functions. This settles a question raised by Chari and Joswig. In the 1-dimensional case, this implies that the complex of rooted forests of a connected graph G completely determines G.