Abstract
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 the
d-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).
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 the
d-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).
| Original language | Undefined/Unknown |
|---|---|
| Pages (from-to) | 116-126 |
| Journal | Discret. Appl. Math. |
| Volume | 262 |
| DOIs | |
| Publication status | Published - 2019 |
| Externally published | Yes |