Tight Bounds on the Minimum Size of a Dynamic Monopoly

Assume that you are given a graph $G=(V,E)$ with an initial coloring, where each node is black or white. Then, in discrete-time rounds all nodes simultaneously update their color following a predefined deterministic rule. This process is called two-way $r$-bootstrap percolation, for some integer $r$, if a node with at least $r$ black neighbors gets black and white otherwise. Similarly, in two-way $\alpha$-bootstrap percolation, for some $0<\alpha<1$, a node gets black if at least $\alpha$ fraction of its neighbors are black, and white otherwise. The two aforementioned processes are called respectively $r$-bootstrap and $\alpha$-bootstrap percolation if we require that a black node stays black forever. For each of these processes, we say a node set $D$ is a dynamic monopoly whenever the following holds: If all nodes in $D$ are black then the graph gets fully black eventually. We provide tight upper and lower bounds on the minimum size of a dynamic monopoly.