The number of excellent discrete Morse functions on graphs

In Nicolaescu (2008) [7] the number of non-homologically equivalent excellent Morse functions defined on S2S2 was obtained in the differentiable setting. We carried out an analogous study in the discrete setting for some kinds of graphs, including S1S1, in Ayala et al. (2009) [1]. This paper completes this study, counting excellent discrete Morse functions defined on any infinite locally finite graph.