  The $P_3$-hull number of a graph is the minimum cardinality of an infecting
set of vertices that will eventually infect the entire graph under the rule
that uninfected nodes become infected if two or more neighbors are infected. In
this paper, we study the $P_3$-hull number for generalized Petersen graphs and
a number of closely related graphs that arise from surgery or more generalized
permutations. In addition, the number of components of the complement of an
infecting set of minimum cardinality is calculated for the generalized Petersen
graph and shown to always be $1$ or $2$. Moreover, infecting times for
infecting sets of minimum cardinality are studied. Bounds are provided and
complete information is given in special cases.
