AI 中文总结
本文研究零强制中的堡垒，证明块图至少有n/3个极小堡垒，并指出多类图堡垒数量指数增长，且能线性时间精确计数树的极小堡垒。
AI 中文摘要
本文研究堡垒，即阻碍零强制的集合。我们证明每个具有 $n$ 个顶点的块图至少有 $n/3$ 个极小堡垒，扩展了 Cameron 和 Li（arXiv:2512.12874）最近关于树的界。对于每棵树、每个有界度的连通非星图以及每个线性最小度的连通图，堡垒数量以及与之相关的相容集合数量呈指数增长，从而对 Hicks 等人（INFORMS Journal on Computing 2022）提出的问题给出了否定回答。最后，我们在线性时间和空间内精确计算一棵树的极小堡垒数量。
英文摘要
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.
