The equivalence of $athfrak{I}$-contractibility and homotopical collapsibility

Let an $athfrak{I}$-contractible graph be a graph reducible to a single vertex by $athfrak{I}$-contractible transformations. An expansion is the inverse of a collapse, and a graph is homotopically collapsible when its clique complex can be transformed to a point by a sequence of collapses and expansions. Equivalence conjecture. The family of $athfrak{I}$-contractible graphs is the same as the family of homotopically collapsible graphs. The supplied text provides no separate resolution evidence for this claim.