Merge trees in discrete Morse theory

TL;DR

This paper explores the construction and properties of merge trees induced by discrete Morse functions on trees, introducing a new equivalence notion and analyzing their combinatorial characteristics.

Contribution

It presents a method to construct induced merge trees from discrete Morse functions and defines a new equivalence concept based on these trees.

Findings

Introduces a construction method for merge trees from discrete Morse functions.

Defines a new equivalence relation for discrete Morse functions based on induced merge trees.

Counts and characterizes merge trees induced on star graphs.

Abstract

In this paper, we study merge trees induced by a discrete Morse function on a tree. Given a discrete Morse function, we provide a method to constructing an induced merge tree and define a new notion of equivalence of discrete Morse functions based on the induced merge tree. We then relate the matching number of a tree to a certain invariant of the induced merge tree. Finally, we count the number of merge trees that can be induced on a star graph and characterize the induced merge tree.