{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-vertex-cover",
  "semantic_family_id": "sf-vertex-cover",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "頂點覆蓋"
  },
  "summary": {
    "zh-Hant": "有限simple undirected graph的頂點子集C若與每條edge至少一端相交，則C為vertex cover。"
  },
  "statement": {
    "zh-Hant": "graph須明示有限unique vertices與loop-free、duplicate-free unordered edges。cover audit遍歷全部edges；一條兩端皆不在C的edge即為反例。empty-edge graph的empty set是cover；isolated vertices不必加入；任何cover的superset仍為cover。inclusion-minimal表示刪任何一點都不再cover，minimum表示全體covers中cardinality最小，兩者不可混同。本節點不混用edge cover、independent set或directed cover，也不自動求minimum size。"
  },
  "formula": {
    "tex": "V(G,C)=1",
    "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><mn>1</mn></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": "number",
        "value": 1
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "7b7fd114bd48a6e07be0191f20ee4d09cb9e598bce05c7466c8688503e1c4f93"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "edges canonical成排序unordered pairs；candidate subset unique且屬vertex set。"
    },
    {
      "id": "a2",
      "zh-Hant": "minimum boundary只在小型完整finite graph exhaustive枚舉全部subsets。"
    }
  ],
  "symbols": [
    {
      "token": "V",
      "role": "vertex_cover_predicate",
      "role_zh": "頂點覆蓋述詞",
      "scope": "representative_formula"
    },
    {
      "token": "G",
      "role": "finite_simple_graph",
      "role_zh": "有限簡單圖",
      "scope": "representative_formula"
    },
    {
      "token": "C",
      "role": "candidate_vertex_subset",
      "role_zh": "候選頂點子集",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-graph",
      "title_zh": "圖",
      "reason_zh": "finite simple graph。"
    },
    {
      "id": "mko-subset",
      "title_zh": "子集合",
      "reason_zh": "candidate vertex subset。"
    },
    {
      "id": "mko-quantifier",
      "title_zh": "量詞",
      "reason_zh": "all-edge責任。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "全部edges逐一核至少一端被選。",
      "uncovered edge是完整反例witness。",
      "empty/isolated/superset與minimal/minimum boundaries分開。"
    ],
    "common_misconception_zh": "只覆蓋大多數edges，或把inclusion-minimal當minimum。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪只判定vertex covers與小型boundary；minimum vertex cover留待後續節點。",
    "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": "加入最小頂點覆蓋。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-vertex-cover.md 整理；2026-10-15 脈衝 R07（使用者手動接續 2026-11-14 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"
  }
}