On the Number of Zero Forcing Minimal Forts on Trees

TL;DR

This paper proves a conjecture relating the maximum number of minimal forts in trees to path graphs, using computational algorithms and theoretical recursion relations.

Contribution

It establishes an upper bound on the number of minimal forts in trees, combining computational methods with theoretical recursion adaptations.

Findings

Proved the conjecture $oxed{ ext{F}_{T_n} extless= inom{n}{2} ext{F}_{P_n}}$ for trees.

Developed an efficient algorithm to compute minimal forts in small trees.

Extended recursion relations for minimal forts from paths to forests.

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.