{
  "sources": [
    {
      "key": "rashidi2020petersen",
      "kind": "doi",
      "identifier": "10.13069/jacodesmath.729465",
      "title": "Computing the zero forcing number for generalized Petersen graphs",
      "authors": [
        "Rashidi, Saeedeh",
        "Poursalavati, Nosratollah Shajareh",
        "Tavakkoli, Maryam"
      ],
      "year": 2020,
      "venue": "Journal of Algebra Combinatorics Discrete Structures and Applications",
      "url": "https://doi.org/10.13069/jacodesmath.729465",
      "verified": true,
      "verifiedAt": "2026-10-30T17:22:32.307Z",
      "verificationMethod": "crossref",
      "accessLevel": "full-text",
      "notes": "Full text read (open-access PDF, CC BY 4.0, saved at sources/fulltext/rashidi2020.pdf). Thm 2.1: n=rk+s => Z<=r(s+2); Thm 2.2: Z<=2k+2; Thm 2.6: Z(P(2k+1,k))=6 for k>=5; Remark 2.4 attributes Z(P(24r,5))=12 to Alameda et al.; Cor 3.5 lower bound min{k,floor(n/k)}+1. Thm 3.6 (k=3) is known to be false at n=12.",
      "claimedTitle": "Computing the zero forcing number for generalized Petersen graphs",
      "titleSimilarity": 1
    },
    {
      "key": "aim2008zeroforcing",
      "kind": "doi",
      "identifier": "10.1016/j.laa.2007.10.009",
      "title": "Zero forcing sets and the minimum rank of graphs",
      "authors": [
        "AIM Minimum Rank – Special Graphs Work Group"
      ],
      "year": 2008,
      "venue": "Linear Algebra and its Applications",
      "url": "https://doi.org/10.1016/j.laa.2007.10.009",
      "verified": true,
      "verifiedAt": "2026-10-30T17:16:23.957Z",
      "verificationMethod": "crossref",
      "accessLevel": "metadata",
      "claimedTitle": "Zero forcing sets and the minimum rank of graphs",
      "titleSimilarity": 1
    },
    {
      "key": "arxiv2610_24059",
      "kind": "arxiv",
      "identifier": "2610.24059",
      "title": "Sharp zero forcing numbers for the generalized Petersen graphs $P(n,3)$ and $P(n,4)$",
      "authors": [
        "Arnav Krishnan",
        "Haobo Ma",
        "Wenlin Zhang"
      ],
      "year": 2026,
      "venue": "arXiv",
      "url": "https://arxiv.org/abs/2610.24059",
      "verified": true,
      "verifiedAt": "2026-10-30T17:22:39.119Z",
      "verificationMethod": "arxiv-api",
      "accessLevel": "full-text",
      "notes": "Full text read (arXiv HTML + PDF saved at sources/fulltext/arxiv_2610.24059.pdf). Z(P(n,3))=8 for n>=13, Z(P(n,4))=10 for n>=18, thresholds sharp; lower bounds by boundary of first forcing sources + column deletion + forts; Lean 4 for k=3, DRAT for k=4. Contains no statement about k=5.",
      "claimedTitle": "Sharp zero forcing numbers for the generalized Petersen graphs P(n,3) and P(n,4)",
      "titleSimilarity": 1
    },
    {
      "key": "arxiv2607_19412",
      "kind": "arxiv",
      "identifier": "2607.19412",
      "title": "A correction to the Zero Forcing Number of the Generalized Petersen Graphs $P(n,3)$",
      "authors": [
        "Arnav Krishnan"
      ],
      "year": 2026,
      "venue": "arXiv",
      "url": "https://arxiv.org/abs/2607.19412",
      "verified": true,
      "verifiedAt": "2026-10-30T17:22:39.147Z",
      "verificationMethod": "arxiv-api",
      "accessLevel": "full-text",
      "notes": "Full text read (arXiv HTML + PDF saved at sources/fulltext/arxiv_2607.19412.pdf). Z(P(12,3))=7 corrects Rashidi Thm 3.6; table 7<=n<=20; states exhaustive search confirms Z(P(11,5))=6; k>=4 stabilization thresholds left open.",
      "claimedTitle": "A correction to the Zero Forcing Number of the Generalized Petersen Graphs P(n,3)",
      "titleSimilarity": 1
    },
    {
      "key": "brimkov2019computational",
      "kind": "doi",
      "identifier": "10.1016/j.ejor.2018.09.030",
      "title": "Computational approaches for zero forcing and related problems",
      "authors": [
        "Brimkov, Boris",
        "Fast, Caleb C.",
        "Hicks, Illya V."
      ],
      "year": 2019,
      "venue": "European Journal of Operational Research",
      "url": "https://doi.org/10.1016/j.ejor.2018.09.030",
      "verified": true,
      "verifiedAt": "2026-10-30T17:18:09.229Z",
      "verificationMethod": "crossref",
      "accessLevel": "abstract",
      "claimedTitle": "Computational approaches for zero forcing and related problems",
      "titleSimilarity": 1
    },
    {
      "key": "brimkov2021improved",
      "kind": "doi",
      "identifier": "10.1287/ijoc.2020.1032",
      "title": "Improved Computational Approaches and Heuristics for Zero Forcing",
      "authors": [
        "Brimkov, Boris",
        "Mikesell, Derek",
        "Hicks, Illya V."
      ],
      "year": 2021,
      "venue": "INFORMS Journal on Computing",
      "url": "https://doi.org/10.1287/ijoc.2020.1032",
      "verified": true,
      "verifiedAt": "2026-10-30T17:18:09.297Z",
      "verificationMethod": "crossref",
      "accessLevel": "abstract",
      "claimedTitle": "Improved Computational Approaches and Heuristics for Zero Forcing",
      "titleSimilarity": 1
    },
    {
      "key": "alameda2018families",
      "kind": "doi",
      "identifier": "10.1515/spma-2018-0006",
      "title": "Families of graphs with maximum nullity equal to zero forcing number",
      "authors": [
        "Alameda, Joseph S.",
        "Curl, Emelie",
        "Grez, Armando",
        "Hogben, Leslie",
        "Kingston, O’Neill",
        "Schulte, Alex",
        "Young, Derek",
        "Young, Michael"
      ],
      "year": 2018,
      "venue": "Special Matrices",
      "url": "https://doi.org/10.1515/spma-2018-0006",
      "verified": true,
      "verifiedAt": "2026-10-30T17:18:09.354Z",
      "verificationMethod": "crossref",
      "accessLevel": "abstract",
      "claimedTitle": "Families of graphs with maximum nullity equal to zero forcing number",
      "titleSimilarity": 1
    },
    {
      "key": "watkins1969tait",
      "kind": "doi",
      "identifier": "10.1016/s0021-9800(69)80116-x",
      "title": "A theorem on tait colorings with an application to the generalized Petersen graphs",
      "authors": [
        "Watkins, Mark E."
      ],
      "year": 1969,
      "venue": "Journal of Combinatorial Theory",
      "url": "https://doi.org/10.1016/s0021-9800(69)80116-x",
      "verified": true,
      "verifiedAt": "2026-10-30T17:18:09.408Z",
      "verificationMethod": "crossref",
      "accessLevel": "metadata",
      "claimedTitle": "A theorem on tait colorings with an application to the generalized Petersen graphs",
      "titleSimilarity": 1
    },
    {
      "key": "frucht1971groups",
      "kind": "doi",
      "identifier": "10.1017/s0305004100049811",
      "title": "The groups of the generalized Petersen graphs",
      "authors": [
        "Frucht, Roberto",
        "Graver, Jack E.",
        "Watkins, Mark E."
      ],
      "year": 1971,
      "venue": "Mathematical Proceedings of the Cambridge Philosophical Society",
      "url": "https://doi.org/10.1017/s0305004100049811",
      "verified": true,
      "verifiedAt": "2026-10-30T17:18:09.464Z",
      "verificationMethod": "crossref",
      "accessLevel": "abstract",
      "claimedTitle": "The groups of the generalized Petersen graphs",
      "titleSimilarity": 1
    },
    {
      "key": "wetzler2014drattrim",
      "kind": "doi",
      "identifier": "10.1007/978-3-319-09284-3_31",
      "title": "DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs",
      "authors": [
        "Wetzler, Nathan",
        "Heule, Marijn J. H.",
        "Hunt, Warren A."
      ],
      "year": 2014,
      "venue": "Lecture Notes in Computer Science",
      "url": "https://doi.org/10.1007/978-3-319-09284-3_31",
      "verified": true,
      "verifiedAt": "2026-10-30T17:18:10.782Z",
      "verificationMethod": "crossref",
      "accessLevel": "metadata",
      "claimedTitle": "DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs",
      "titleSimilarity": 1
    },
    {
      "key": "cameron2023forts",
      "kind": "arxiv",
      "identifier": "2310.17904",
      "title": "Forts, (fractional) zero forcing, and Cartesian products of graphs",
      "authors": [
        "Thomas R. Cameron",
        "Leslie Hogben",
        "Franklin H. J. Kenter",
        "Seyed Ahmad Mojallal",
        "Houston Schuerger"
      ],
      "year": 2023,
      "venue": "arXiv",
      "url": "https://arxiv.org/abs/2310.17904",
      "verified": true,
      "verifiedAt": "2026-10-30T17:18:14.036Z",
      "verificationMethod": "arxiv-api",
      "accessLevel": "abstract",
      "claimedTitle": "Forts, (fractional) zero forcing, and Cartesian products of graphs",
      "titleSimilarity": 1
    },
    {
      "key": "davila2015girth",
      "kind": "doi",
      "identifier": "10.20429/tag.2015.020201",
      "title": "Bounds for the Zero Forcing Number of Graphs with Large Girth",
      "authors": [
        "Davila, Randy",
        "Kenter, Franklin"
      ],
      "year": 2015,
      "venue": "Theory and Applications of Graphs",
      "url": "https://doi.org/10.20429/tag.2015.020201",
      "verified": true,
      "verifiedAt": "2026-10-30T17:18:14.281Z",
      "verificationMethod": "crossref",
      "accessLevel": "abstract",
      "claimedTitle": "Bounds for the Zero Forcing Number of Graphs with Large Girth",
      "titleSimilarity": 1
    },
    {
      "key": "hogben2012propagation",
      "kind": "doi",
      "identifier": "10.1016/j.dam.2012.04.003",
      "title": "Propagation time for zero forcing on a graph",
      "authors": [
        "Hogben, Leslie",
        "Huynh, My",
        "Kingsley, Nicole",
        "Meyer, Sarah",
        "Walker, Shanise",
        "Young, Michael"
      ],
      "year": 2012,
      "venue": "Discrete Applied Mathematics",
      "url": "https://doi.org/10.1016/j.dam.2012.04.003",
      "verified": true,
      "verifiedAt": "2026-10-30T17:18:15.636Z",
      "verificationMethod": "crossref",
      "accessLevel": "metadata",
      "claimedTitle": "Propagation time for zero forcing on a graph",
      "titleSimilarity": 1
    },
    {
      "key": "bjorkman2025leaky",
      "kind": "arxiv",
      "identifier": "2508.02564",
      "title": "Leaky Forcing: Extending Zero Forcing Results to a Fault-Tolerant Setting",
      "authors": [
        "Beth Bjorkman",
        "Lei Cao",
        "Franklin Kenter",
        "Ryan Moruzzi",
        "Carolyn Reinhart",
        "Violeta Vasilevska"
      ],
      "year": 2025,
      "venue": "arXiv",
      "url": "https://arxiv.org/abs/2508.02564",
      "verified": true,
      "verifiedAt": "2026-10-30T17:20:41.465Z",
      "verificationMethod": "arxiv-api",
      "accessLevel": "abstract",
      "claimedTitle": "Leaky Forcing: Extending Zero Forcing Results to a Fault-Tolerant Setting",
      "titleSimilarity": 1
    },
    {
      "key": "fast2018degrees",
      "kind": "doi",
      "identifier": "10.1016/j.dam.2018.05.002",
      "title": "Effects of vertex degrees on the zero-forcing number and propagation time of a graph",
      "authors": [
        "Fast, Caleb C.",
        "Hicks, Illya V."
      ],
      "year": 2018,
      "venue": "Discrete Applied Mathematics",
      "url": "https://doi.org/10.1016/j.dam.2018.05.002",
      "verified": true,
      "verifiedAt": "2026-10-30T17:20:57.107Z",
      "verificationMethod": "crossref",
      "accessLevel": "metadata",
      "claimedTitle": "Effects of vertex degrees on the zero-forcing number and propagation time of a graph",
      "titleSimilarity": 1
    },
    {
      "key": "barioli2010parameters",
      "kind": "doi",
      "identifier": "10.1016/j.laa.2010.03.008",
      "title": "Zero forcing parameters and minimum rank problems",
      "authors": [
        "Barioli, Francesco",
        "Barrett, Wayne",
        "Fallat, Shaun M.",
        "Hall, H. Tracy",
        "Hogben, Leslie",
        "Shader, Bryan",
        "van den Driessche, P.",
        "van der Holst, Hein"
      ],
      "year": 2010,
      "venue": "Linear Algebra and its Applications",
      "url": "https://doi.org/10.1016/j.laa.2010.03.008",
      "verified": true,
      "verifiedAt": "2026-10-30T17:20:57.549Z",
      "verificationMethod": "crossref",
      "accessLevel": "metadata",
      "claimedTitle": "Zero forcing parameters and minimum rank problems",
      "titleSimilarity": 1
    },
    {
      "key": "ignatiev2018pysat",
      "kind": "doi",
      "identifier": "10.1007/978-3-319-94144-8_26",
      "title": "PySAT: A Python Toolkit for Prototyping with SAT Oracles",
      "authors": [
        "Ignatiev, Alexey",
        "Morgado, Antonio",
        "Marques-Silva, Joao"
      ],
      "year": 2018,
      "venue": "Lecture Notes in Computer Science",
      "url": "https://doi.org/10.1007/978-3-319-94144-8_26",
      "verified": true,
      "verifiedAt": "2026-10-30T17:20:59.170Z",
      "verificationMethod": "crossref",
      "accessLevel": "metadata",
      "claimedTitle": "PySAT: A Python Toolkit for Prototyping with SAT Oracles",
      "titleSimilarity": 1
    },
    {
      "key": "demoura2021lean4",
      "kind": "doi",
      "identifier": "10.1007/978-3-030-79876-5_37",
      "title": "The Lean 4 Theorem Prover and Programming Language",
      "authors": [
        "Moura, Leonardo de",
        "Ullrich, Sebastian"
      ],
      "year": 2021,
      "venue": "Lecture Notes in Computer Science",
      "url": "https://doi.org/10.1007/978-3-030-79876-5_37",
      "verified": true,
      "verifiedAt": "2026-10-30T17:21:00.796Z",
      "verificationMethod": "crossref",
      "accessLevel": "abstract",
      "claimedTitle": "The Lean 4 Theorem Prover and Programming Language",
      "titleSimilarity": 1
    },
    {
      "key": "sinz2005cardinality",
      "kind": "doi",
      "identifier": "10.1007/11564751_73",
      "title": "Towards an Optimal CNF Encoding of Boolean Cardinality Constraints",
      "authors": [
        "Sinz, Carsten"
      ],
      "year": 2005,
      "venue": "Lecture Notes in Computer Science",
      "url": "https://doi.org/10.1007/11564751_73",
      "verified": true,
      "verifiedAt": "2026-10-30T17:21:02.478Z",
      "verificationMethod": "crossref",
      "accessLevel": "metadata",
      "claimedTitle": "Towards an Optimal CNF Encoding of Boolean Cardinality Constraints",
      "titleSimilarity": 1
    },
    {
      "key": "hogben2022book",
      "kind": "doi",
      "identifier": "10.1090/surv/270",
      "title": "Inverse Problems and Zero Forcing for Graphs",
      "authors": [
        "Hogben, Leslie",
        "Lin, Jephian",
        "Shader, Bryan"
      ],
      "year": 2022,
      "venue": "Mathematical Surveys and Monographs",
      "url": "https://doi.org/10.1090/surv/270",
      "verified": true,
      "verifiedAt": "2026-10-30T17:21:03.776Z",
      "verificationMethod": "crossref",
      "accessLevel": "metadata",
      "claimedTitle": "Inverse Problems and Zero Forcing for Graphs",
      "titleSimilarity": 1
    }
  ]
}