{
  "abstract": "We determine the zero-forcing number of maximal biconnected outerplanar graphs. When the weak dual is a path, the zero-forcing number and the induced path-covering number are both two. Otherwise, the branching faces determine a canonical core and a family of terminal strips. We prove that an optimal forcing process can be chosen in a normal form governed by explicit vertex and face capacities on the core. After fixing forests on the terminal strips, the zero-forcing number is expressed through an optimization over core-edge selections and the minimum number of root connections that must be removed to obtain an executable orientation. The path-covering number is obtained from the same capacity optimization without the repair cost. We characterize executable orientations by blocking-circuit equations and identify the two-path obstruction with a first cohomology class of a signed contact multigraph. Local replacements and a finite comparison of boundary recurrences establish the normal form for graphs of arbitrary size. A fifty-six-state recurrence constructs a minimum zero-forcing set and a complete forcing chronology in linear time from indexed triangle adjacency. We also obtain exact replacement results for rooted blocks and construct infinite families in which the zero-forcing number exceeds the path-covering number.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31138",
  "author_total": 2,
  "authors": [
    {
      "name": "Zongjian Han"
    },
    {
      "name": "Yiran Liang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fc310vwb4phcp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31138",
  "fulltext_url": null,
  "id": "fc310vwb4phcp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31138",
  "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": "W7223401672",
  "pdf_url": "https://arxiv.org/pdf/2610.31138",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fc310vwb4phcp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31138",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31138"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Zero Forcing on Maximal Outerplanar Graphs: Capacities and Phase Repair",
  "title_text": "Zero Forcing on Maximal Outerplanar Graphs: Capacities and Phase Repair",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/fc310vwb4phcp",
  "venue": "arXiv",
  "versions": []
}