HOMOLOGICAL PROPERTIES OF CONTRACTIBLE TRANSFORMATIONS OF GRAPHS JES ´ US F. ESPINOZA, MART ´ IN-EDUARDO FR ´ IAS-ARMENTA, AND H ´ ECTOR A. HERN ´ ANDEZ Abstract. In [1, 2], 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 the result in [2] 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. 1. Introduction In graph theory, several reductions have been studied that leave the homology invariant. In [1, 2], A. Ivashchenko shows a family of graphs constructed from K(1) by contractible transformations (as in Definition 1), and he proves that such transformations do not change the homology groups of graphs. He started the study of these transformations because these are used in the theory of molecular spaces, digital topology. Modern references are [3, 4]. In [5], ws -dismantling and the collapse of graph are studied; see Remark 3. A graph is essentially collapsible if its complete complex is collapsible; see Section 3. Neither change the homology groups of graphs ([5]). In Section 2, we introduce the graph homology as given in [1] and the contractible transformations as were presented in [2]. We conclude the section with Lemma 2, by showing the existence of an induced homomorphism between the graph homology groups of two graphs, given by the image of contractible transformations. In Section 3, we start with some terminology and notation about simplicial complexes, and we give the definition of collapsible graph in terms of the collapsibility of the clique complex. Furthermore, we prove that any contractible graph (in the sense of Ivashchenko [2]) is also a collapsible graph in Theorem 4. As a consequence, the homology groups of any contractible graphs are trivial, as was proven in [1]. We conjecture than any collapsible graph is also a contractible one. We show computational evidence that supports the conjecture through some algorithms. The scripts were written in C/C++, they are free/open and available on-line in [6]. However, we can not follow a similar arguments than used in Theorem 4. We start Section 4 with an example to show that Theorem 4 is not trivially reversible. We conclude Section 5 with an application of this work to the computation of the persistent homology of the filtered Vietoris-Rips complex. This is a topic of interest in topological data analysis. Date : March 1, 2022. 2010 Mathematics Subject Classification. 05C25, 05C85, 55U05, 55U15. Key words and phrases. Contractible transformations; collapsible graph; Vietoris-Rips complex; persistent homology. 1 arXiv:1808.07461v1 [math.CO] 22 Aug 2018