Indexed on: 22 Aug '18Published on: 22 Aug '18Published in: arXiv - Mathematics - Combinatorics
In a known papers, A. Ivashchenko shows the family of contractible graphs, constructed from $K(1)$ by contractible transformations, and he proves that such transformations do not change the homology groups of graphs. In this paper, we show that a contractible graph is actually a collapsible graph (in the simplicial sense), from which the invariance of the homology follows. In addition, we extend a result of A. Ivashchenko about graph homology, to a filtration of graphs, and we prove that the persistent homology is preserved with respect to contractible transformations. We apply this property as an algorithm to preprocess a data cloud and reduce the computation of the persistent homology for the filtered Vietoris-Rips complex.