{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-graph-isomorphism",
  "semantic_family_id": "sf-graph-isomorphism",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "圖同構"
  },
  "summary": {
    "zh-Hant": "有限簡單圖同構是頂點雙射，對所有頂點對完整保持鄰接與非鄰接。"
  },
  "statement": {
    "zh-Hant": "圖同構f:V(G)到V(H)須為雙射，且對所有不同u,v，{u,v}為G的邊當且僅當{f(u),f(v)}為H的邊。只保邊而不保非邊不足。"
  },
  "formula": {
    "tex": "I(G,H)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>I</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "I",
        "glyph": "I"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "G",
          "glyph": "G"
        },
        {
          "type": "symbol",
          "name": "H",
          "glyph": "H"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "190d88e58e8012606395ef3d1cdc236f7ec8d654080780bb36e205a5cc5ed674"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫提供完整有限無向簡單圖邊集合與完整映射表。"
    },
    {
      "id": "a2",
      "zh-Hant": "度數多重集等不變量不符可否證；相同通常不足以證同構。"
    }
  ],
  "symbols": [
    {
      "token": "I",
      "role": "isomorphism_record",
      "role_zh": "圖同構紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "G",
      "role": "simple_graph",
      "role_zh": "來源圖",
      "scope": "representative_formula"
    },
    {
      "token": "H",
      "role": "simple_graph",
      "role_zh": "目標圖",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-graph",
      "title_zh": "有限圖",
      "reason_zh": "同構作用於完整有限簡單圖。"
    },
    {
      "id": "mko-bijective",
      "title_zh": "雙射",
      "reason_zh": "頂點映射必須一一且覆蓋。"
    },
    {
      "id": "mko-relation",
      "title_zh": "二元關係",
      "reason_zh": "鄰接是頂點上的對稱關係。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "先核映射是雙射。",
      "再核全部無序頂點對的adjacency iff。",
      "不變量匹配只是必要線索，不是一般充分證書。"
    ],
    "common_misconception_zh": "只比較度數序列或只核原邊映成邊就宣稱同構。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只分類指定完整有限簡單圖。",
    "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walk、path、連通分量與圖著色。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-graph-isomorphism.md 整理；2026-10-13 脈衝 R06（使用者手動接續 2026-10-26 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}