{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-graph",
  "semantic_family_id": "sf-graph",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限簡單無向圖"
  },
  "summary": {
    "zh-Hant": "本輪限定有限簡單無向圖，邊是兩個相異頂點的無序集合，度數只計相鄰頂點。"
  },
  "statement": {
    "zh-Hant": "有限簡單無向圖 G=(V,E) 由有限頂點集合 V 與其二元素子集合構成的邊集合 E 組成。沒有自環或重邊；N(v)={u∈V:{u,v}∈E}，deg(v)=|N(v)|，所有頂點度數和為 2|E|。"
  },
  "formula": {
    "tex": "G(V,E)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>G</mi><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "G",
        "glyph": "G"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "V",
          "glyph": "V"
        },
        {
          "type": "symbol",
          "name": "E",
          "glyph": "E"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "9e14ce25905ccf5f99023f00c04bc85ab50f3b47719cc761422db2196eec664a"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "V 有限且頂點標籤互異；每條邊的兩端皆在 V 且相異。"
    },
    {
      "id": "a2",
      "zh-Hant": "無向邊不區分端點順序；鄰居／度數問題須指定 v∈V，空圖允許。"
    }
  ],
  "symbols": [
    {
      "token": "G",
      "role": "finite_simple_undirected_graph",
      "role_zh": "有限簡單無向圖",
      "scope": "representative_formula"
    },
    {
      "token": "V",
      "role": "vertex_set",
      "role_zh": "頂點集合",
      "scope": "representative_formula"
    },
    {
      "token": "E",
      "role": "unordered_edge_set",
      "role_zh": "無向邊集合",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-set",
      "title_zh": "集合",
      "reason_zh": "頂點和無向邊都是集合，重複表示不增加邊。"
    },
    {
      "id": "mko-relation",
      "title_zh": "二元關係與集合投影",
      "reason_zh": "相鄰關係是對稱且無自環的有限關係。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "把 {u,v} 和 {v,u} 當成同一條無向邊。集合式輸入的重複列表須正規化，不以端點排序或重複列項製造新圖。",
      "一個頂點的度數是不同鄰居數；孤立頂點度數為零。對這個簡單圖語境，一條邊在全圖度數和中貢獻兩次，故得到握手公式。",
      "圖論的圖不是函數座標圖形，也不從畫法推論相鄰性。本輪不涵蓋有向圖、多重圖、自環計數、連通性或樹的判定。"
    ],
    "common_misconception_zh": "把無向邊兩種排序算兩條，或把示意線相交誤當新增頂點。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪有限邊集與度數可完全重算；一般圖論定理與握手公式仍未附形式 Evidence。",
    "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": "綁定 SimpleGraph、有限鄰居集合、degree、空圖及握手公式的精確範圍。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-graph.md 整理；2026-10-03 脈衝 R04（依使用者要求提前完成 2026-10-04 的流程），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-03"
  }
}