Fort Abundance in Zero Forcing

Abstract

This paper paper concerns the study of forts, the sets that obstruct zero forcing. We show that every block graph on $n$ vertices has at least $n/3$ minimal forts, extending a recent bound for trees by Cameron and and Li (arXiv:2512.12874). The number of forts, and with it the number of compatible collections, grows exponentially for every tree, every connected non-star graph of bounded degree, and every connected graph of linear minimum degree, answering a question in the negative by Hicks et al. (INFORMS Journal on Computing 2022) for these graph classes. We finish by counting the minimal forts of a tree exactly, in linear time and space.