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We study an extremal question for the (reversible) $r-$bootstrap percolationprocesses. Given a graph and an initial configuration where each vertex isactive or inactive, in the $r-$bootstrap percolation process the following ruleis applied in discrete-time rounds: each vertex gets active if it has at least$r$ active neighbors, and an active vertex stays active forever. In thereversible $r$-bootstrap percolation, each vertex gets active if it has atleast $r$ active neighbors, and inactive otherwise. We consider the followingquestion on the $d$-dimensional torus: how many vertices should be initiallyactive so that the whole graph becomes active? Our results settle an openproblem by Balister, Bollob\'as, Johnson, and Walters and generalize theresults by Flocchini, Lodi, Luccio, Pagli, and Santoro.

Dynamic Monopolies in Reversible Bootstrap Percolation