Dynamic monopolies in two-way bootstrap percolation

We study an extremal question for the (two-way) r−bootstrap percolation processes. Given a graph and an initial configuration where each vertex is active or inactive, in the r−bootstrap percolation process the following rule is 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 the two-way r−bootstrap percolation, each vertex gets active if it has at least r active neighbors, and inactive otherwise. We consider the following question on thed-dimensional torus: how many vertices must be initially active so that the whole graph becomes active? Our results settle an open problem by Balister et al. (2010) and generalize the results by Flocchini et al. (2004).