On the Number of Zero Forcing Minimal Forts on Trees

Abstract

We solve a conjecture by Becker et al. (arXiv:2404.05963) on the topic of zero forcing regarding the number of minimal forts of a tree. They conjectured and we prove $\mathcal{F}_{T_n} \le \binom{n}{2} \mathcal{F}_{P_n}$ where $\mathcal{F}_{T_n}$ is the maximum number of minimal forts on a tree on $n$ vertices and $\mathcal{F}_{P_n}$ is the number of minimal forts of the path graph on $n$ vertices. Our solution relies on both a computational and theoretical approach. Computationally, we introduce and implement an efficient algorithm to compute the exact number of minimal forts for small trees; this is used to establish the large base case required for our strong induction. Theoretically, we provide an adaptation of the recursion relation that defines $\mathcal{F}_{P_n}$ that applies for all forests; this is used in the induction step to establish the result.