  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. We also relate these classes of graphs to the notion of
$k$-dismantlable graphs and $k$-collapsible complexes, which also leads to a
minimal counterexample to an erroneous claim of Ivashchenko from the
literature. We end with some open questions.
