Minimal Zero Forcing Sets

In this paper, we study minimal (with respect to inclusion) zero forcing sets. We first investigate when a graph can have polynomially or exponentially many distinct minimal zero forcing sets. We also study the maximum size of a minimal zero forcing set $\overline{\operatorname{Z}}(G)$, and relate it to the zero forcing number $\operatorname{Z}(G)$. Surprisingly, we show that the equality $\overline{\operatorname{Z}}(G)=\operatorname{Z}(G)$ is preserved by deleting a universal vertex, but not by adding a universal vertex. We also characterize graphs with extreme values of $\overline{\operatorname{Z}}(G)$ and explore the gap between $\overline{\operatorname{Z}}(G)$ and $\operatorname{Z}(G)$.