{
  "abstract": "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 $\\mz(G)\\leq k$ is finite for any $k\\geq 1$. We determine the complete set of minimal forbidden graphs for the case $k = 3$, and we provide partial characterizations of graphs with $\\mz(G) \\leq 3$, based on girth. Our results suggest new directions for the structural understanding of zero forcing-type parameters.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27972",
  "author_total": 8,
  "authors": [
    {
      "name": "Carlos A. Alfaro"
    },
    {
      "name": "Michael D. Barrus"
    },
    {
      "name": "Sergio Gerardo Gómez-Galicia"
    },
    {
      "name": "Teresa I. Hoekstra-Mendoza"
    },
    {
      "name": "Miguel Licona"
    },
    {
      "name": "Jephian C. -H. Lin"
    },
    {
      "name": "Juan Pablo Serrano"
    },
    {
      "name": "Ralihe R. Villagrán"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/epq24bg5e6ena.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27972",
  "fulltext_url": null,
  "id": "epq24bg5e6ena",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27972",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7204780078",
  "pdf_url": "https://arxiv.org/pdf/2608.27972",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/epq24bg5e6ena.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.27972",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27972"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The forbidden structure for zero forcing number",
  "title_text": "The forbidden structure for zero forcing number",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/epq24bg5e6ena",
  "venue": "arXiv",
  "versions": []
}