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 $n$.

中文翻译：

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迫零集的重配置图

本文开始研究迫零集的重构，更具体地说，是迫零图。给定一个基图 $G$，它的零强迫图 $\mathscr{Z}(G)$ 是顶点是 $G$ 的最小零强迫集的图，在顶点 $B$ 和 $B 之间有一条边'$ of $\mathscr{Z}(G)$ 当且仅当可以通过改变 $G$ 的单个顶点从 $B'$ 获得 $B$。结果表明，森林的迫零图是连通的，但许多迫零图是不连通的。我们刻画了零强迫图是路径或完整图的基图，并表明星形不能是零强迫图。我们表明，在最坏的情况下，计算 $\mathscr{Z}(G)$ 需要 $2^{\Theta(n)}$ 操作，对于阶 $n$ 的图 $G$。