Star clusters in the Matching, Morse, and Generalized complex of discrete Morse functions

In this paper, we determine the homotopy type of the complex of discrete Morse functions and matching complex of multiple families of complexes by utilizing star cluster collapses and the Cluster Lemma. We compute the homotopy type of the complex of discrete Morse functions of an extended notion of a star graph, as well as the homotopy type of the matching complex of a Dutch windmill graph. Additionally, we provide alternate computations of the homotopy type of the complex of discrete Morse functions of paths, the homotopy type of the matching complex of paths, and the homotopy type of the matching complex of cycles. We then use this same method of computing homotopy types to investigate the relationship between the homotopy type of the matching complex and the generalized Morse complex. 2020 Mathematics Subject Classification. (Primary) 57Q70; (Secondary) 55P10, 55U10, 05C70.