  Detecting on a graph the presence of the minimum number of nodes (target set)
that will be able to "activate" a prescribed number of vertices in the graph is
called the target set selection problem (TSS) proposed by Kempe, Kleinberg, and
Tardos. In TSS's settings, nodes have two possible states (active or
non-active) and the threshold triggering the activation of a node is given by
the number of its active neighbors. Dealing with fault tolerance in a majority
based system the two possible states are used to denote faulty or non-faulty
nodes, and the threshold is given by the state of the majority of neighbors.
Here, the major effort was in determining the distribution of initial faults
leading the entire system to a faulty behavior. Such an activation pattern,
also known as dynamic monopoly (or shortly dynamo), was introduced by Peleg in
1996. In this paper we extend the TSS problem's settings by representing nodes'
states with a "multicolored" set. The extended version of the problem can be
described as follows: let G be a simple connected graph where every node is
assigned a color from a finite ordered set C = {1, . . ., k} of colors. At each
local time step, each node can recolor itself, depending on the local
configurations, with the color held by the majority of its neighbors. Given G,
we study the initial distributions of colors leading the system to a k
monochromatic configuration in toroidal meshes, focusing on the minimum number
of initial k-colored nodes. We find upper and lower bounds to the size of a
dynamo, and then special classes of dynamos, outlined by means of a new
approach based on recoloring patterns, are characterized.
