Before revision:  Our algorithm is also conceptually simpler than Fulek and Kyn\v{c : We perform a sequence of \emph{local operations} that gradually "untangles" the image \varphi(G) into an embedding \psi(G), or reports that \varphi is not a weak embedding.
After revision:  Our algorithm is also conceptually simpler than : We perform a sequence of \emph{local operations} that gradually "untangles" the image \varphi(G) into an embedding \psi(G), or reports that \varphi is not a weak embedding.