We introduce two new approaches to compute the Persistent Homology (PH) of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the input sequence by using special types of collapses (strong and edge collapse) and to compute the PH of an induced sequence of smaller size that has the same PH as the initial one. Our first approach uses strong collapse which is introduced by J. Barmak and E.Miniam [DCG (2012)]. Strong collapse consists of removal of special vertices called dominated vertices from a simplicial complex. In the second approach, we extend the notions of dominated vertex to a simplex of any dimension. Domination of edges appears to be very powerful and we study it in the case of flag complexes in more detail. As a result and as demonstrated by numerous experiments on publicly available data sets, our approaches are extremely fast and memory efficient in practice.
