Persistent Pairs And Connectedness In Discrete Morse Functions On Simplicial Complex I

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 $K$.