Geodesic
Discrete Morse theory for the barycentric subdivision
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVIII, Tome 462 (2017), pp. 52-64

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Let $F$ be a discrete Morse function on a simplicial complex $L$. We construct a discrete Morse function $\Delta(F)$ on the barycentric subdivision $\Delta(L)$. The constructed function $\Delta(F)$ “behaves the same way” as $F$, i.e., has the same number of critical simplices and the same gradient path structure. 
@article{ZNSL_2017_462_a2,
     author = {A. Zhukova},
     title = {Discrete {Morse} theory for the barycentric subdivision},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {52--64},
     publisher = {mathdoc},
     volume = {462},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_462_a2/}
}
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A. Zhukova. Discrete Morse theory for the barycentric subdivision. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVIII, Tome 462 (2017), pp. 52-64. http://geodesic.mathdoc.fr/item/ZNSL_2017_462_a2/