The forbidden structure for zero forcing number

@misc{alfaro2026forbidden,
  title = {{The forbidden structure for zero forcing number}},
  author = {Carlos A. Alfaro and Michael D. Barrus and Sergio Gerardo Gómez-Galicia and Teresa I. Hoekstra-Mendoza and Miguel Licona and Jephian C. -H. Lin and Juan Pablo Serrano and Ralihe R. Villagrán},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.27972},
  eprint = {2608.27972},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27972},
}

TY  - UNPB
TI  - The forbidden structure for zero forcing number
AU  - Carlos A. Alfaro
AU  - Michael D. Barrus
AU  - Sergio Gerardo Gómez-Galicia
AU  - Teresa I. Hoekstra-Mendoza
AU  - Miguel Licona
AU  - Jephian C. -H. Lin
AU  - Juan Pablo Serrano
AU  - Ralihe R. Villagrán
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.27972
UR  - https://arxiv.org/abs/2608.27972
AB  - The {\it zero forcing number} of a graph G, Z(G), is a well-studied parameter which arises from a color changing process and has strong connections to {\it minimum rank}, {\it critical ideals} and related invariants. In this work, we consider the complementary parameter mz(G) = |V(G)| − Z(G). This parameter is monotone under taking induced subgraphs. This leads us to the study of graphs for which mz(G) is bounded, via forbidden induced subgraphs. We prove that the number of minimal forbidden graphs for graphs with mz(G)≤ k is finite for any k≥ 1. We determine the complete set of minimal forbidden graphs for the case k = 3, and we provide partial characterizations of graphs with mz(G) ≤ 3, based on girth. Our results suggest new directions for the structural understanding of zero forcing-type parameters.
ER  -