Persistent pairs and connectedness in discrete Morse functions on simplicial complex I

Abstract

In this paper, we study some useful properties of persistent pairs in a discrete Morse function on a simplicial complex K. In case of dim⁡K=1 (i.e., a graph), by using the properties, we characterize strongly connectedness of critical simplices between two distinct discrete Morse functions, and relate the number of such pairs to the Euler characteristic of K.