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Fort Abundance in Zero Forcing

  • Aida Abiad
  • Sina Ghasemi Nezhad
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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.

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Cite

BibTeXRIS
BibTeX
@misc{abiad2026fort,
  title = {{Fort Abundance in Zero Forcing}},
  author = {Aida Abiad and Sina Ghasemi Nezhad},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11754},
  eprint = {2609.11754},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11754},
}
RIS
TY  - UNPB
TI  - Fort Abundance in Zero Forcing
AU  - Aida Abiad
AU  - Sina Ghasemi Nezhad
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11754
UR  - https://arxiv.org/abs/2609.11754
AB  - 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.
ER  -