  Zero forcing (also called graph infection) on a simple, undirected graph $G$
is based on the color-change rule: If each vertex of $G$ is colored either
white or black, and vertex $v$ is a black vertex with only one white neighbor
$w$, then change the color of $w$ to black. A minimum zero forcing set is a set
of black vertices of minimum cardinality that can color the entire graph black
using the color change rule. The propagation time of a zero forcing set $B$ of
graph $G$ is the minimum number of steps that it takes to force all the
vertices of $G$ black, starting with the vertices in $B$ black and performing
independent forces simultaneously. The minimum and maximum propagation times of
a graph are taken over all minimum zero forcing sets of the graph.
  It is shown that a connected graph of order at least two has more than one
minimum zero forcing set realizing minimum propagation time. Graphs $G$ having
extreme minimum propagation times $|G| - 1$, $|G| - 2$, and $0$ are
characterized, and results regarding graphs having minimum propagation time $1$
are established. It is shown that the diameter is an upper bound for maximum
propagation time for a tree, but in general propagation time and diameter of a
graph are not comparable.
