  This paper begins the study of reconfiguration of zero forcing sets, and more
specifically, the zero forcing graph. Given a base graph $G$, its zero forcing
graph, $\mathscr{Z}(G)$, is the graph whose vertices are the minimum zero
forcing sets of $G$ with an edge between vertices $B$ and $B'$ of
$\mathscr{Z}(G)$ if and only if $B$ can be obtained from $B'$ by changing a
single vertex of $G$. It is shown that the zero forcing graph of a forest is
connected, but that many zero forcing graphs are disconnected. We characterize
the base graphs whose zero forcing graphs are either a path or the complete
graph, and show that the star cannot be a zero forcing graph. We show that
computing $\mathscr{Z}(G)$ takes $2^{\Theta(n)}$ operations in the worst case
for a graph $G$ of order $n$.
