The notion of a contractible transformation on a graph was introduced by Ivashchenko as a means to study molecular spaces arising from digital topology and computer image analysis, and more recently has been applied to topological data analysis. Contractible transformations involve a list of four elementary moves that can be performed on the vertices and edges of a graph, and it has been shown by Chen, Yau, and Yeh that these moves preserve the simple homotopy type of the underlying clique complex. A graph is said to be ${mathcal I}$-contractible if one can reduce it to a single isolated vertex via a sequence of contractible transformations. Inspired by the notions of collapsible and non-evasive simplicial complexes, in this paper we study certain subclasses of ${mathcal I}$-contractible graphs where one can collapse to a vertex using only a subset of these moves. Our main results involve constructions of minimal examples of graphs for which the resulting classes differ, as well as a miminal counterexample to an erroneous claim of Ivashchenko from the literature. We also relate these classes of graphs to the notion of $k$-dismantlable graphs and $k$-collapsible complexes, and discuss some open questions.
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