A graph-theoretical approach to cancelling critical elements R. Ayala 1,2 D.Fern´andez-Ternero 2 J.A. Vilches 1,2 Departamento de Geometr´ ıa y Topolog´ ıa Universidad de Sevilla Sevilla, Spain Abstract This work is focused on the links between Forman’s discrete Morse theory and graph theory. More precisely, we are interested on putting the optimization of a discrete Morse function in terms of matching theory. It can be done by describing the process of cancellation of pairs of critical simplices by means of obtaining Morse matchings on the corresponding Hasse diagram with a greater number of edges using the combinatorial notion of transference. Keywords: Optimal discrete Morse function, Hasse diagram, maximum matching, simplicial complex. 1 Introduction R. Forman [3] introduced the notion of discrete Morse function defined on a finite cw-complex and, in this combinatorial context, he developed a discrete 1 The authors are partially supported by Plan Nacional de Investigaci´ on 2.010, Project MTM2010-20445, Espa˜ na, 2010. 2 Emails: ayala@us.es, desamfer@us.es, vilches@us.es Electronic Notes in Discrete Mathematics 37 (2011) 285–290 1571-0653/$ – see front matter © 2011 Elsevier B.V. All rights reserved. www.elsevier.com/locate/endm doi:10.1016/j.endm.2011.05.049