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A simplicial complex $X$ is flag if it is isomorphic to the clique complex ofsome graph. Boulet, Fieux, and Jouve have shown that simple homotopyequivalence classes of such complexes are determined by the notion of$s$-homotopy defined on its $1$-skeleton. In his work on molecular spaces,Ivaschenko introduced the notion of $\mathfrak{I}$-contractible transformationson finite graphs, and we say that two graphs are $\mathfrak{I}$-homotopyequivalent if there is a sequence of $\mathfrak{I}$-contractibletransformations that takes one to the other. We show that the $s$-homotopy typeand $\mathfrak{I}$-homotopy type of a finite graph coincide, answering aquestion posed by Boulet, Fieux, and Jouve. As a result, two graphs $G$ and $H$are $\mathfrak{I}$-homotopy equivalent if and only if their clique complexes$C(G)$ and $C(H)$ are simple homotopy equivalent. We also show that a finitegraph $G$ is $\mathfrak{I}$-contractible if and only if $C(G)$ is contractible,which answers a question posed by the first author, Espinoza, Fr\'ias-Armenta,and Hern\'andez.

Simple homotopy of flag simplicial complexes and contractible contractions of graphs