{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-minimum-vertex-cover",
  "semantic_family_id": "sf-minimum-vertex-cover",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "最小頂點覆蓋"
  },
  "summary": {
    "zh-Hant": "有限simple graph中，cardinality在全部vertex covers裡最小的候選是minimum vertex cover。"
  },
  "statement": {
    "zh-Hant": "對bounded finite simple undirected graph完整枚舉全部vertex subsets，逐一做all-edge cover audit，得到minimum cardinality與全部minimizers。給定nonminimum cover時須明示更小cover witness；inclusion-minimal只表示刪任一點即失去cover，不等於minimum。empty graph、multiple minima與minimal-but-not-minimum均保留。不得混同maximal、edge cover、independent set，也不使用König theorem或weighted/directed variants。"
  },
  "formula": {
    "tex": "V(G,C)=k",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>V</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>C</mi><mo>)</mo></mrow><mo>=</mo><mi>k</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "V",
          "glyph": "V"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "G",
            "glyph": "G"
          },
          {
            "type": "symbol",
            "name": "C",
            "glyph": "C"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "k",
        "glyph": "k"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "7b1db9cb23a81d6de257377e1a42e24f76b161726ed521685b799025ae39910c"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "vertices finite unique，edges為loop-free duplicate-free unordered pairs。"
    },
    {
      "id": "a2",
      "zh-Hant": "頂點數有界以完整枚舉全部subsets。"
    }
  ],
  "symbols": [
    {
      "token": "V",
      "role": "minimum_vertex_cover_operator",
      "role_zh": "最小頂點覆蓋算子",
      "scope": "representative_formula"
    },
    {
      "token": "G",
      "role": "finite_simple_graph",
      "role_zh": "有限簡單圖",
      "scope": "representative_formula"
    },
    {
      "token": "C",
      "role": "vertex_cover",
      "role_zh": "頂點覆蓋",
      "scope": "representative_formula"
    },
    {
      "token": "k",
      "role": "minimum_cover_cardinality",
      "role_zh": "最小覆蓋基數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-vertex-cover",
      "title_zh": "頂點覆蓋",
      "reason_zh": "cover qualification。"
    },
    {
      "id": "mko-optimization-problem",
      "title_zh": "最佳化問題",
      "reason_zh": "minimum語義。"
    },
    {
      "id": "mko-cardinality-comparison",
      "title_zh": "基數比較",
      "reason_zh": "cover size比較。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "所有subsets完整資格核對。",
      "minimum size與全部minimizers同時輸出。",
      "nonminimum與minimal-not-minimum給明示反例。"
    ],
    "common_misconception_zh": "把inclusion-minimal或maximal誤作minimum。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理小型有限simple graph的exact enumeration；不宣稱多項式時間或近似比。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入vertex-cover decision problem與approximation boundaries。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-minimum-vertex-cover.md 整理；2026-10-15 脈衝 R06（使用者手動接續 2026-11-15 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}