Discrete Morse theory for the barycentric subdivision

Let $F$ be a discrete Morse function on a simplicial complex $L$. We construct a discrete Morse function $\mathrm{\Delta }(F)$ on the barycentric subdivision $\mathrm{\Delta }(L)$. The constructed function $\mathrm{\Delta }(F)$ “behaves the same way” as $F$, i. e. has the same number of critical simplexes and the same gradient path structure.