{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-vertex-cover-01",
  "mko_id": "mko-vertex-cover",
  "title": {
    "zh-Hant": "星形圖的cover與刪點證書",
    "en": "Vertex covers and deletion certificates on a star"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-vertex-cover-01",
    "source": {
      "mko_id": "mko-vertex-cover",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "4e6ca5a6921189d4aac44783f8135dd0f613679ec348ac53ddf8b3fd648ff22b",
      "english_sha256": "9b80569183079a1991326108c486827298f81d22fc1f370d725462ef96550f0a"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "逐邊核cover並區分minimal與minimum。",
      "en": "Audit covers edgewise and distinguish minimal from minimum."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "V={0,1,2,3}，E={01,02,03}；C={1,2,3}，D={0}。",
          "en": "V={0,1,2,3}, E={01,02,03}; C={1,2,3}, D={0}."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "C=\\{1,2,3\\}\\subseteq V",
        "after_tex": "C\\text{ is a unique-vertex candidate subset}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先核候選確為V子集且無重複。",
          "en": "First qualify C as a duplicate-free subset of V."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "cover選頂點不是選邊。",
            "en": "A vertex cover selects vertices, not edges."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "01\\cap C=\\{1\\}",
        "after_tex": "01\\text{ covered}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第一邊由端點1覆蓋。",
          "en": "Endpoint1 covers the first edge."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "只需至少一端，不需兩端。",
            "en": "At least one endpoint is required, not both."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "02\\cap C=\\{2\\}",
        "after_tex": "02\\text{ covered}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二邊由端點2覆蓋。",
          "en": "Endpoint2 covers the second edge."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "各edge有自己的覆蓋義務。",
            "en": "Every edge has its own coverage obligation."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "03\\cap C=\\{3\\}",
        "after_tex": "03\\text{ covered; }C\\text{ is a cover}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第三邊也覆蓋，已完成全部E。",
          "en": "The third edge is covered too, exhausting E."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "大多數邊覆蓋不夠，這裡是全部。",
            "en": "Covering most edges is insufficient; all are covered here."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "C\\setminus\\{1\\},C\\setminus\\{2\\},C\\setminus\\{3\\}",
        "after_tex": "01,02,03\\text{ respectively become uncovered}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "每個單點刪除都有對應uncovered edge，故C inclusion-minimal。",
          "en": "Each one-vertex deletion has an uncovered-edge witness, so C is inclusion-minimal."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "三個刪除案例全列，不只試一點。",
            "en": "All three deletion cases are listed."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "D=\\{0\\}",
        "after_tex": "01\\cap D=02\\cap D=03\\cap D=\\{0\\}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "較小D也覆蓋全部邊，否定C為minimum。",
          "en": "The smaller D covers all edges, refuting minimum cardinality of C."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "C與D不需互為子集。",
            "en": "C and D need not be nested."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "C\\subseteq C'\\subseteq V,\\quad e\\cap C\\ne\\varnothing",
        "after_tex": "e\\cap C'\\ne\\varnothing",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "任一cover超集仍保有原覆蓋端點。",
          "en": "Every cover superset retains the original covered endpoint."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "超集可能不minimal，兩性質分開。",
            "en": "A superset may cease to be minimal; keep the properties separate."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "E=\\varnothing",
        "after_tex": "\\varnothing\\text{ is a cover}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "沒有邊時all-edge條件空真；isolated vertices無須加入。",
          "en": "With no edges the all-edge condition is vacuous; isolated vertices need not be selected."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "本star有邊，不能套空圖結論稱空集cover。",
            "en": "This star has edges, so the edgeless boundary does not make its empty set a cover."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "C是inclusion-minimal cover，但D={0}更小，故C非minimum。",
      "en": "C is an inclusion-minimal cover, but D={0} is smaller, so C is not minimum."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "C minimal故minimum。",
          "en": "C is minimal, hence minimum."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "D大小1小於C大小3且合法。",
          "en": "Valid D has size 1 versus C's3."
        }
      },
      {
        "claim": {
          "zh-Hant": "{1,2}覆蓋此star。",
          "en": "{1,2} covers the star."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "03兩端皆不被選。",
          "en": "Neither endpoint of 03 is selected."
        }
      },
      {
        "claim": {
          "zh-Hant": "cover每邊須兩端都選。",
          "en": "A cover must select both endpoints of every edge."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "D只選中心0即覆蓋三邊。",
          "en": "D selects only center0 and covers all three edges."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "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。",
          "en": "Declare finite unique vertices and loop-free duplicate-free unordered edges. Audit every edge; one edge with neither endpoint in C refutes coverage. The empty set covers an edgeless graph, isolated vertices are optional and any superset of a cover remains a cover. Inclusion-minimal means every single vertex deletion destroys coverage; minimum means least cardinality among all covers. Do not conflate those notions or substitute edge cover, independent set or directed-cover conventions. Minimum size is not automatically solved by the basic definition."
        },
        "source": {
          "mko_id": "mko-vertex-cover",
          "sha256": "4e6ca5a6921189d4aac44783f8135dd0f613679ec348ac53ddf8b3fd648ff22b",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "有限無向簡單圖的邊為無loop、無重複unordered pairs；圖性質須對完整明示V、E逐一檢查。 vertex cover為C⊆V且每條邊至少一端在C；inclusion-minimal表示刪任一選點會失效，minimum表示全部covers中最小基數。",
          "en": "A finite simple undirected graph has loop-free duplicate-free unordered edges; graph properties are audited on all declared V and E. A vertex cover is C contained in V with at least one endpoint of every edge in C; inclusion-minimal means every selected-vertex deletion fails, while minimum means least cardinality over all covers."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "3",
        "rhs": "1",
        "relation": ">"
      },
      {
        "id": "N02",
        "lhs": "1+1+1",
        "rhs": "3"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "basic cover責任，不自動使用König theorem。",
      "en": "Basic cover obligations, with no automatic Konig theorem."
    },
    "md_symbols": [
      [
        "V,E",
        "完整頂點與邊集合",
        "complete vertex and edge sets"
      ],
      [
        "C,D",
        "頂點子集合",
        "vertex subsets"
      ]
    ],
    "md_error_controls": [
      {
        "title": "反例edge / Counterexample edge",
        "zh": "每失敗cover提供一條兩端皆不在集合的邊。",
        "en": "For each failed cover give an edge with both endpoints outside."
      },
      {
        "title": "刪點全列 / All deletions",
        "zh": "C三個元素各刪一次，與全域更小witness分開。",
        "en": "Delete each of C's three elements once, separately from the global smaller witness."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "3e1cb818fce0186d9bc40d27d6db5c6f3685c62c296183c10027a8f6e28ed321",
    "original_source_sha256": "bcf513dc1646f8e2bb4552b8f00156931aa83edb1d0ffa53ca9e3fd932a3f728",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
      "BR142-S204",
      "BR142-S205"
    ],
    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}