  In graph theoretical models of the spread of disease through populations, the
spread of opinion through social networks, and the spread of faults through
distributed computer networks, vertices are in two states, either black or
white, and these states are dynamically updated at discrete time steps
according to the rules of the particular conversion process used in the model.
This paper considers the irreversible k-threshold and majority conversion
processes. In an irreversible k-threshold (resp., majority) conversion process,
a vertex is permanently colored black in a certain time period if at least k
(resp., at least half) of its neighbors were black in the previous time period.
A k-conversion set (resp., dynamic monopoly) is a set of vertices which, if
initially colored black, will result in all vertices eventually being colored
black under a k-threshold (resp., majority) conversion process. We answer
several open problems by presenting bounds and some exact values of the minimum
number of vertices in k-conversion sets and dynamic monopolies of complete
multipartite graphs, as well as of Cartesian and tensor products of two graphs.
