Abstract: The purpose of this talk is to present a version of Forman’s discrete Morse theory for simplicial complexes, based on internal strong collapses. Classical discrete Morse theory can be viewed as a generalization of Whitehead’s collapses, where each Morse function on a simplicial complex K defines a sequence of elementary internal collapses. This reduction guarantees the existence of a CW-complex that is homotopy equivalent to K, with cells corresponding to the critical simplices of the Morse function. However, this approach lacks an explicit combinatorial description of the attaching maps, which limits the reconstruction of the homotopy type of K. By restricting discrete Morse functions to those induced by total orders on the vertices, I will present a strong discrete Morse theory, generalizing the strong collapses introduced by Barmak and Minian. I will show that, in this setting, the resulting reduced CW-complex is regular, enabling to recover its homotopy type combinatorially. I will also provide an algorithm to compute this reduction and apply it to obtain efficient structures for simplicial complexes in the library of triangulations by Benedetti and Lutz.