  The concept of zero forcing was introduced in the context of linear algebra,
and was further studied by both graph theorists and linear algebraists. It is
based on the process of activating vertices of a graph $G$ starting from a set
of vertices that are already active, and applying the rule that an active
vertex with exactly one non-active neighbor forces that neighbor to become
active. A set $S\subset V(G)$ is called a zero forcing set of $G$ if initially
only vertices of $S$ are active and the described process enforces all vertices
of $G$ to become active. The size of a minimum zero forcing set in $G$ is
called the zero forcing number of $G$. While a minimum zero forcing set can
only be unique in edgeless graphs, we consider the weaker uniqueness condition,
notably that for every two minimum zero forcing sets in a graph $G$ there is an
automorphism that maps one to the other. We characterize the class of trees
that enjoy this condition by using properties of minimum path covers of trees.
In addition, we investigate both variations of uniqueness for several concepts
of Grundy domination, which first appeared in the context of domination games,
yet they are also closely related to zero forcing. For each of the four
variations of Grundy domination we characterize the graphs that have only one
Grundy dominating set of the given type, and characterize those forests that
enjoy the weaker (isomorphism based) condition of uniqueness. The latter
characterizations lead to efficient algorithms for recognizing the
corresponding classes of forests.
