Dynamic Monopolies of Constant Size

The paper deals with a polling game on a graph. Initially, each vertex is colored white or black. At each round, each vertex is colored by the color shared by the majority of vertices in its neighborhood, at the previous round. (All recolorings are done simultaneously.) We say that a set W0 of vertices is a dynamic monopoly or dynamo if starting the game with the vertices of W0 colored white, the entire system is white after a finite number of rounds. D. Peleg (1998, Discrete Appl. Math.86, 262–273) asked how small a dynamic monopoly may be as a function of the number of vertices. We show that the answer is O(1).