{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-connected-component",
  "semantic_family_id": "sf-connected-component",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "連通分量"
  },
  "summary": {
    "zh-Hant": "有限無向簡單圖中，由存在路徑所定義的頂點等價類恰為連通分量。"
  },
  "statement": {
    "zh-Hant": "在有限無向無loop簡單圖中，u與v連通若存在u到v的path；長度0 path保證反身性。此關係的等價類為互斥且覆蓋全部頂點的最大連通分量；孤立頂點形成單點分量。"
  },
  "formula": {
    "tex": "Q(G)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "Q",
        "glyph": "Q"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "G",
          "glyph": "G"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "2be16515df9a47a48483be722d7ca74dda3cce05d242d138b3352798c5102873"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "頂點標籤0到n-1與完整無向簡單邊集合明示。"
    },
    {
      "id": "a2",
      "zh-Hant": "可達性由完整BFS／DFS核對；不切換到有向圖的weak或strong connectivity。"
    }
  ],
  "symbols": [
    {
      "token": "Q",
      "role": "connected_component_partition",
      "role_zh": "連通分量分割",
      "scope": "representative_formula"
    },
    {
      "token": "G",
      "role": "simple_graph",
      "role_zh": "有限無向簡單圖",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-graph-walk-simple-path",
      "title_zh": "圖walk與simple path",
      "reason_zh": "連通性由存在path定義。"
    },
    {
      "id": "mko-equivalence-relation",
      "title_zh": "等價關係",
      "reason_zh": "可達性在無向圖上形成等價關係。"
    },
    {
      "id": "mko-graph",
      "title_zh": "有限圖",
      "reason_zh": "完整頂點與邊集合提供搜尋空間。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "每個頂點透過長度0 path與自身連通。",
      "BFS／DFS從起點得到其精確分量。",
      "所有分量互斥並覆蓋頂點；孤立點不能遺漏。"
    ],
    "common_misconception_zh": "忽略孤立頂點、把一條走訪順序當分量證書，或混用有向圖強／弱連通。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理完整有限無向簡單圖；不外推有向、無限或動態圖。",
    "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": "加入分量縮約與圖演算法複雜度。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-connected-component.md 整理；2026-10-14 脈衝 R07（使用者手動接續 2026-10-28 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}