The family of contractible graphs, introduced by A. Ivashchenko, consists of the collection $mathfrak{I}$ of graphs constructed recursively from $K_1$ by contractible transformations. In this paper we show that every graph in a subfamily of $mathfrak{I}$ (the strongly contractible ones) is a collapsible graph (in the simplicial sense), by providing a sequence of elementary collapses induced by removing contractible vertices or edges. In addition, we introduce an algorithm to identify the contractible vertices in any graph and show that there is a natural homomorphism, induced by the inclusion map of graphs, between the homology groups of the clique complex of graphs with the contractible vertices removed. Finally, we show an application of this result to the computation of the persistent homology for the Vietoris-Rips filtration.
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