  In 2018, a fort of a graph was introduced as a non-empty subset of vertices
in which no vertex outside of the set has exactly one neighbor in the set.
Since then, forts have been used to characterize zero forcing sets, model the
zero forcing number as an integer program, and generate lower bounds on the
zero forcing number of a Cartesian product. In this article, we investigate the
number of minimal forts of a graph, where a fort is minimal if every proper
subset is not a fort. In particular, we show that the number of minimal forts
of a graph of order at least six is strictly less than Sperner's bound, a
famous bound due to Emanuel Sperner (1928) on the size of a collection of
subsets where no subset contains another. Then, we derive an explicit formula
for the number of minimal forts for several families of graphs, including the
path, cycle, and spider graphs. Moreover, we show that the asymptotic growth
rate of the number of minimal forts of the spider graph is bounded above by
that of the path graph. We conjecture that the asymptotic growth rate of the
path graph is extremal over all trees. Finally, we develop methods for
constructing minimal forts of graph products using the minimal forts of the
original graphs. In the process, we derive explicit formulas and lower bounds
on the number of minimal forts for additional families of graphs, such as the
wheel, sunlet, and windmill graphs. Most notably, we show that the family of
windmill graphs has an exponential number of minimal forts with a maximum
asymptotic growth rate of cube root of three, which is the largest asymptotic
growth rate we have observed. We conjecture that there exist families of graphs
with a larger asymptotic growth rate.
