Dynamic monopolies in simple graphs

This paper studies a repetitive polling game played on an n-vertex graph G. At first, each vertex is colored, Black or White. At each round, each vertex (simultaneously) recolors itself by the color of the majority of its closed neighborhood. The variants of the model differ in the choice of a particular tiebreaking rule. We assume the tie-breaking rule is Prefer-White and we study the relation between the notion of “dynamic monopoly” and “vertex cover” of G. In particular, we show that any vertex cover of G is a dynamic monopoly or reaches a 2−periodic coloring. Moreover, we compute dyn(G) for some special classes of graphs including paths, cycles and links of some graphs.