The number of excellent discrete Morse functions on graphs

AbstractIn Nicolaescu (2008) [7] the number of non-homologically equivalent excellent Morse functions defined on S2 was obtained in the differentiable setting. We carried out an analogous study in the discrete setting for some kinds of graphs, including S1, in Ayala et al. (2009) [1]. This paper completes this study, counting excellent discrete Morse functions defined on any infinite locally finite graph

Discrete Morse Functions, Vector Fields, and Homological Sequences on Trees

The goal of this project is to construct a discrete Morse function which induces both a unique gradient vector field and homological sequence on a given tree. After reviewing the basics of discrete Morse theory, we will show that the two standard notions of equivalence of discrete Morse functions, Forman and homological equivalence, are independent of one another. We then show through a constructive algorithm the existence of a discrete Morse function on a tree inducing a desired gradient vector field and homological sequence. After proving that our algorithm is correct, we give an example to illustrate its use