{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0001","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0896218ed1f9693ea90ad0e1c7988cf6c59341292719605b1097d597345593cd","family_id":"qf-simple-graph-edge-membership","seed":202610044001,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[],\"V\":[],\"operation\":\"edge_membership\",\"u\":1,\"v\":0}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V=∅ and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {1, 0} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0002","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae0456d2bc837a3756c94e46e5cd8a9fc380b89eb76dd7252213421d9dfd9647","family_id":"qf-simple-graph-edge-membership","seed":202610044002,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-11,-10],[-10,-9],[-9,-8]],\"V\":[-11,-10,-9,-8],\"operation\":\"edge_membership\",\"u\":-11,\"v\":-11}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-11, -10, -9, -8} and E={{-11, -10}, {-10, -9}, {-9, -8}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-11, -11} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0003","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1336a1d6531be0fbafae24e724dc470dcde3e1f4641298701e53d9b44a4b6d47","family_id":"qf-simple-graph-edge-membership","seed":202610044003,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-10,-9],[-10,-8],[-10,-7]],\"V\":[-10,-9,-8,-7,-6],\"operation\":\"edge_membership\",\"u\":-10,\"v\":-5}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-10, -9, -8, -7, -6} and E={{-10, -9}, {-10, -8}, {-10, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-10, -5} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0004","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78994c71fe4a1a23377cb4eb6a57e739ddff16f7b45e446fce5f4887b1d01e76","family_id":"qf-simple-graph-edge-membership","seed":202610044004,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-9,-8],[-9,-7],[-8,-7]],\"V\":[-9,-8,-7,-6,-5,-4],\"operation\":\"edge_membership\",\"u\":-8,\"v\":-9}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-9, -8, -7, -6, -5, -4} and E={{-9, -8}, {-9, -7}, {-8, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-8, -9} an edge in E?","answer":{"kind":"short_text","value":"Yes","explanation":"Undirected edges ignore endpoint order. The answer is Yes; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0005","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"137d7755992e29a0513ae3953479c06bf60537dd98ad8678d6ba61ff93a18bee","family_id":"qf-simple-graph-edge-membership","seed":202610044005,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-8,-7]],\"V\":[-8,-7,-6],\"operation\":\"edge_membership\",\"u\":-8,\"v\":-6}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-8, -7, -6} and E={{-8, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-8, -6} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0006","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66e430ef0fee0c8b14032947b17019c75086af4d2980e0e491b6c8014c0de628","family_id":"qf-simple-graph-edge-membership","seed":202610044006,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-7,-6],[-7,-5],[-7,-4],[-6,-5],[-6,-4],[-5,-4]],\"V\":[-7,-6,-5,-4],\"operation\":\"edge_membership\",\"u\":-7,\"v\":-3}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-7, -6, -5, -4} and E={{-7, -6}, {-7, -5}, {-7, -4}, {-6, -5}, {-6, -4}, {-5, -4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-7, -3} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0007","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"765ce566d091cd3b9f02a4d83a8f8df4f7eb97e88c8aeca167e7034a668cc996","family_id":"qf-simple-graph-edge-membership","seed":202610044007,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[],\"V\":[-6,-5,-4,-3,-2],\"operation\":\"edge_membership\",\"u\":-2,\"v\":-6}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-6, -5, -4, -3, -2} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-2, -6} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0008","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03ef0cf6a2b7aa1b0b3a3967aa512d138c28f9975ec9fa2cb641312cdc96dba9","family_id":"qf-simple-graph-edge-membership","seed":202610044008,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-5,-4],[-4,-3],[-3,-2],[-2,-1],[-1,0]],\"V\":[-5,-4,-3,-2,-1,0],\"operation\":\"edge_membership\",\"u\":-5,\"v\":0}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-5, -4, -3, -2, -1, 0} and E={{-5, -4}, {-4, -3}, {-3, -2}, {-2, -1}, {-1, 0}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-5, 0} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0009","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"817afda2776f1a4c198cf45417bd5541608dc0fbb3f2df0c30b1a31794c6729f","family_id":"qf-simple-graph-edge-membership","seed":202610044009,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-4,-3]],\"V\":[-4,-3,-2],\"operation\":\"edge_membership\",\"u\":-4,\"v\":-1}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-4, -3, -2} and E={{-4, -3}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-4, -1} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0010","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7aee5fdbeca093cd8c53d766b06b4363bfa9746beeb317eb89161442b4fa0e72","family_id":"qf-simple-graph-edge-membership","seed":202610044010,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-3,-2],[-3,-1],[-2,-1]],\"V\":[-3,-2,-1,0],\"operation\":\"edge_membership\",\"u\":-2,\"v\":-3}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-3, -2, -1, 0} and E={{-3, -2}, {-3, -1}, {-2, -1}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-2, -3} an edge in E?","answer":{"kind":"short_text","value":"Yes","explanation":"Undirected edges ignore endpoint order. The answer is Yes; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0011","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e510e0c1942980d6bc4a20c845ee3c187890ca7b42b3580318282f4606242b77","family_id":"qf-simple-graph-edge-membership","seed":202610044011,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-2,-1],[0,1]],\"V\":[-2,-1,0,1,2],\"operation\":\"edge_membership\",\"u\":-2,\"v\":2}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-2, -1, 0, 1, 2} and E={{-2, -1}, {0, 1}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-2, 2} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0012","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74db547d0bda9a055aa140f73a6d08ee0aba9864117f633091591d11c59ac5cc","family_id":"qf-simple-graph-edge-membership","seed":202610044012,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[-1,0],[-1,1],[-1,2],[-1,3],[-1,4],[0,1],[0,2],[0,3],[0,4],[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]],\"V\":[-1,0,1,2,3,4],\"operation\":\"edge_membership\",\"u\":-1,\"v\":5}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-1, 0, 1, 2, 3, 4} and E={{-1, 0}, {-1, 1}, {-1, 2}, {-1, 3}, {-1, 4}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {-1, 5} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0013","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5559b4053465737603beac062035cccc452c70e869106c6bd8e3f41520fca601","family_id":"qf-simple-graph-edge-membership","seed":202610044013,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[],\"V\":[0,1,2],\"operation\":\"edge_membership\",\"u\":2,\"v\":0}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={0, 1, 2} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {2, 0} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0014","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d10c38735fc9c36059e873353f0eb73781e31aa8f0054ae899d70a40494f2b47","family_id":"qf-simple-graph-edge-membership","seed":202610044014,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[1,2],[2,3],[3,4]],\"V\":[1,2,3,4],\"operation\":\"edge_membership\",\"u\":1,\"v\":4}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={1, 2, 3, 4} and E={{1, 2}, {2, 3}, {3, 4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {1, 4} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0015","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"424c54fd5ebbbb0d919542461229538fd2d6f0247538da2f787b833087151035","family_id":"qf-simple-graph-edge-membership","seed":202610044015,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[2,3],[2,4],[2,5]],\"V\":[2,3,4,5,6],\"operation\":\"edge_membership\",\"u\":2,\"v\":7}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={2, 3, 4, 5, 6} and E={{2, 3}, {2, 4}, {2, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {2, 7} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0016","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0bf15a9ad312fff9e4985ccd9c67a88b20418ea8147ed8962de668db3e332b1","family_id":"qf-simple-graph-edge-membership","seed":202610044016,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[3,4],[3,5],[4,5]],\"V\":[3,4,5,6,7,8],\"operation\":\"edge_membership\",\"u\":4,\"v\":3}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={3, 4, 5, 6, 7, 8} and E={{3, 4}, {3, 5}, {4, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {4, 3} an edge in E?","answer":{"kind":"short_text","value":"Yes","explanation":"Undirected edges ignore endpoint order. The answer is Yes; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0017","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d8e2c8a53d39d6e6496debb1a073025d2d491922ec83836ddcdf7b43d66c2bc","family_id":"qf-simple-graph-edge-membership","seed":202610044017,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[4,5]],\"V\":[4,5,6],\"operation\":\"edge_membership\",\"u\":4,\"v\":6}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={4, 5, 6} and E={{4, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {4, 6} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0018","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6591f18559e19d24941214293ae3336468fe3b79b36ba36de48909834f3c9753","family_id":"qf-simple-graph-edge-membership","seed":202610044018,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[5,6],[5,7],[5,8],[6,7],[6,8],[7,8]],\"V\":[5,6,7,8],\"operation\":\"edge_membership\",\"u\":5,\"v\":9}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={5, 6, 7, 8} and E={{5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {5, 9} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0019","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4888284928309c94a58c98cfb184dac44f892a7650e06a606a6c0c7b3a63a71d","family_id":"qf-simple-graph-edge-membership","seed":202610044019,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[],\"V\":[6,7,8,9,10],\"operation\":\"edge_membership\",\"u\":10,\"v\":6}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={6, 7, 8, 9, 10} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {10, 6} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0020","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e43fb0e08b3f7137f0fbd4ebba7eaff4b8d41799ce95e9a056eba6362407bdb5","family_id":"qf-simple-graph-edge-membership","seed":202610044020,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[7,8],[8,9],[9,10],[10,11],[11,12]],\"V\":[7,8,9,10,11,12],\"operation\":\"edge_membership\",\"u\":7,\"v\":12}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={7, 8, 9, 10, 11, 12} and E={{7, 8}, {8, 9}, {9, 10}, {10, 11}, {11, 12}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {7, 12} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0021","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66d69a45e72539e23f67b6e3ca2bb91901cb788a1f3d6b66ec816e0de676381b","family_id":"qf-simple-graph-edge-membership","seed":202610044021,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[8,9]],\"V\":[8,9,10],\"operation\":\"edge_membership\",\"u\":8,\"v\":11}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={8, 9, 10} and E={{8, 9}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {8, 11} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0022","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b931e512ca8c2ae2e14f65fc2c0fcc13a4bbdac73fae3d726b498383a510fc98","family_id":"qf-simple-graph-edge-membership","seed":202610044022,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[9,10],[9,11],[10,11]],\"V\":[9,10,11,12],\"operation\":\"edge_membership\",\"u\":10,\"v\":9}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={9, 10, 11, 12} and E={{9, 10}, {9, 11}, {10, 11}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {10, 9} an edge in E?","answer":{"kind":"short_text","value":"Yes","explanation":"Undirected edges ignore endpoint order. The answer is Yes; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0023","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9b43265dbdb6668bb6c45413ee76fd52250d8bc701fac0a65bea8d3ec94fd2a","family_id":"qf-simple-graph-edge-membership","seed":202610044023,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[10,11],[12,13]],\"V\":[10,11,12,13,14],\"operation\":\"edge_membership\",\"u\":10,\"v\":14}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={10, 11, 12, 13, 14} and E={{10, 11}, {12, 13}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {10, 14} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0024","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9bf1693449bc30422327ca75a337a6cca559f71fe19626d43596a4749eacc886","family_id":"qf-simple-graph-edge-membership","seed":202610044024,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[[11,12],[11,13],[11,14],[11,15],[11,16],[12,13],[12,14],[12,15],[12,16],[13,14],[13,15],[13,16],[14,15],[14,16],[15,16]],\"V\":[11,12,13,14,15,16],\"operation\":\"edge_membership\",\"u\":11,\"v\":17}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={11, 12, 13, 14, 15, 16} and E={{11, 12}, {11, 13}, {11, 14}, {11, 15}, {11, 16}, {12, 13}, {12, 14}, {12, 15}, {12, 16}, {13, 14}, {13, 15}, {13, 16}, {14, 15}, {14, 16}, {15, 16}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {11, 17} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0025","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dfb0d9fe0182efa8cd49e47924760acb6aff7df695c984c6232c50028f233fa","family_id":"qf-simple-graph-edge-membership","seed":202610044025,"semantic_template":"qf-simple-graph-edge-membership/{\"E\":[],\"V\":[12,13,14],\"operation\":\"edge_membership\",\"u\":14,\"v\":12}"},"fields":{"title":"Edge Membership in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={12, 13, 14} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {14, 12} an edge in E?","answer":{"kind":"short_text","value":"No","explanation":"Undirected edges ignore endpoint order. The answer is No; equal endpoints or endpoints outside V cannot give an edge of this simple graph."},"skills":["Edge Membership in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0026","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88c0399ac554858dff35bb60044d4331da43e36db8dd3a71011a47dc93ae335f","family_id":"qf-simple-graph-neighbors","seed":202610044026,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[],\"V\":[0],\"operation\":\"neighbors\",\"v\":0}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={0} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=0 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Find distinct neighbors using the explicit edge set: N(0)=∅. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0027","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66d373838610e96a5db83f98a93e88bf1e329722e4b0c3d89aa8954169fd4bdb","family_id":"qf-simple-graph-neighbors","seed":202610044027,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-11,-10],[-10,-9],[-9,-8]],\"V\":[-11,-10,-9,-8],\"operation\":\"neighbors\",\"v\":-9}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-11, -10, -9, -8} and E={{-11, -10}, {-10, -9}, {-9, -8}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-9 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"{-10, -8}","explanation":"Find distinct neighbors using the explicit edge set: N(-9)={-10, -8}. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0028","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64db7142f2271039d43750ea2d2e4220640977bcce3ba2f857c37d9db781cacd","family_id":"qf-simple-graph-neighbors","seed":202610044028,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-10,-9],[-10,-8],[-10,-7]],\"V\":[-10,-9,-8,-7,-6],\"operation\":\"neighbors\",\"v\":-7}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-10, -9, -8, -7, -6} and E={{-10, -9}, {-10, -8}, {-10, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-7 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"{-10}","explanation":"Find distinct neighbors using the explicit edge set: N(-7)={-10}. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0029","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ee40379eb4869b2f7a78cc44e52b71b27ccd28386db046b52cf1d2f40807ad0","family_id":"qf-simple-graph-neighbors","seed":202610044029,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-9,-8],[-9,-7],[-8,-7]],\"V\":[-9,-8,-7,-6,-5,-4],\"operation\":\"neighbors\",\"v\":-5}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-9, -8, -7, -6, -5, -4} and E={{-9, -8}, {-9, -7}, {-8, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-5 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Find distinct neighbors using the explicit edge set: N(-5)=∅. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0030","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5105e8fa04ca3becb0f6ec0433d3fc02a51e387c21f7868d5687bef1b7cc616e","family_id":"qf-simple-graph-neighbors","seed":202610044030,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-8,-7]],\"V\":[-8,-7,-6],\"operation\":\"neighbors\",\"v\":-6}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-8, -7, -6} and E={{-8, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-6 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Find distinct neighbors using the explicit edge set: N(-6)=∅. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0031","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0748c7dca41b518d1a71cc035dbc69f93915db1712920cf522a85c12f5216e20","family_id":"qf-simple-graph-neighbors","seed":202610044031,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-7,-6],[-7,-5],[-7,-4],[-6,-5],[-6,-4],[-5,-4]],\"V\":[-7,-6,-5,-4],\"operation\":\"neighbors\",\"v\":-5}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-7, -6, -5, -4} and E={{-7, -6}, {-7, -5}, {-7, -4}, {-6, -5}, {-6, -4}, {-5, -4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-5 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"{-7, -6, -4}","explanation":"Find distinct neighbors using the explicit edge set: N(-5)={-7, -6, -4}. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0032","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68de9dd7c1f6fd76af98e7781e58eb086ecb33d91e9f2bed673d3c00b9c40da7","family_id":"qf-simple-graph-neighbors","seed":202610044032,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[],\"V\":[-6,-5,-4,-3,-2],\"operation\":\"neighbors\",\"v\":-4}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-6, -5, -4, -3, -2} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-4 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Find distinct neighbors using the explicit edge set: N(-4)=∅. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0033","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12d31b77342d7b14fac9456444fb968dae1d0a402c9c424590e5bf58d02244c1","family_id":"qf-simple-graph-neighbors","seed":202610044033,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-5,-4],[-4,-3],[-3,-2],[-2,-1],[-1,0]],\"V\":[-5,-4,-3,-2,-1,0],\"operation\":\"neighbors\",\"v\":-3}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-5, -4, -3, -2, -1, 0} and E={{-5, -4}, {-4, -3}, {-3, -2}, {-2, -1}, {-1, 0}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-3 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"{-4, -2}","explanation":"Find distinct neighbors using the explicit edge set: N(-3)={-4, -2}. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0034","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8096768a42a65216f0047ee6a54c17139c46391bee6468b85b5b72aed48d3311","family_id":"qf-simple-graph-neighbors","seed":202610044034,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-4,-3]],\"V\":[-4,-3,-2],\"operation\":\"neighbors\",\"v\":-4}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-4, -3, -2} and E={{-4, -3}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-4 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"{-3}","explanation":"Find distinct neighbors using the explicit edge set: N(-4)={-3}. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0035","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b74d85f55c8df6eb101bd7793c982b11521114aaeae5ea94b58f4adb88532fac","family_id":"qf-simple-graph-neighbors","seed":202610044035,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-3,-2],[-3,-1],[-2,-1]],\"V\":[-3,-2,-1,0],\"operation\":\"neighbors\",\"v\":-1}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-3, -2, -1, 0} and E={{-3, -2}, {-3, -1}, {-2, -1}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-1 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"{-3, -2}","explanation":"Find distinct neighbors using the explicit edge set: N(-1)={-3, -2}. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0036","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bea4ffc061f0ef605f438540363a90d2a89a6d885140607e1269a23a37035fd2","family_id":"qf-simple-graph-neighbors","seed":202610044036,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-2,-1],[0,1]],\"V\":[-2,-1,0,1,2],\"operation\":\"neighbors\",\"v\":-1}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-2, -1, 0, 1, 2} and E={{-2, -1}, {0, 1}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-1 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"{-2}","explanation":"Find distinct neighbors using the explicit edge set: N(-1)={-2}. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0037","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d7e253f606b00c0d6879054ede419b59679da25da3a1b5ff9ab9c1eab8e8808","family_id":"qf-simple-graph-neighbors","seed":202610044037,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[-1,0],[-1,1],[-1,2],[-1,3],[-1,4],[0,1],[0,2],[0,3],[0,4],[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]],\"V\":[-1,0,1,2,3,4],\"operation\":\"neighbors\",\"v\":-1}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-1, 0, 1, 2, 3, 4} and E={{-1, 0}, {-1, 1}, {-1, 2}, {-1, 3}, {-1, 4}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-1 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"{0, 1, 2, 3, 4}","explanation":"Find distinct neighbors using the explicit edge set: N(-1)={0, 1, 2, 3, 4}. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0038","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fbfe93d6fba9ce4e0321a0088545a1a0c2708cda9665c24a28d0bf925755c16","family_id":"qf-simple-graph-neighbors","seed":202610044038,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[],\"V\":[0,1,2],\"operation\":\"neighbors\",\"v\":1}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={0, 1, 2} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=1 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Find distinct neighbors using the explicit edge set: N(1)=∅. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0039","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7381abe864b44ef2b68446e14b521729861f7bec2eec78e8f34acac1a47daf3","family_id":"qf-simple-graph-neighbors","seed":202610044039,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[1,2],[2,3],[3,4]],\"V\":[1,2,3,4],\"operation\":\"neighbors\",\"v\":3}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={1, 2, 3, 4} and E={{1, 2}, {2, 3}, {3, 4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=3 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"{2, 4}","explanation":"Find distinct neighbors using the explicit edge set: N(3)={2, 4}. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0040","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e8dfb4d4c7cc340da5faaf96cacbd25bc42b87cfa097a59b15147a4a5f4068a","family_id":"qf-simple-graph-neighbors","seed":202610044040,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[2,3],[2,4],[2,5]],\"V\":[2,3,4,5,6],\"operation\":\"neighbors\",\"v\":2}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={2, 3, 4, 5, 6} and E={{2, 3}, {2, 4}, {2, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=2 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"{3, 4, 5}","explanation":"Find distinct neighbors using the explicit edge set: N(2)={3, 4, 5}. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0041","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1df457b268be72424e1824a8643a8e3cff0d5dd55d4e63fc78210b08db555d3","family_id":"qf-simple-graph-neighbors","seed":202610044041,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[3,4],[3,5],[4,5]],\"V\":[3,4,5,6,7,8],\"operation\":\"neighbors\",\"v\":7}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={3, 4, 5, 6, 7, 8} and E={{3, 4}, {3, 5}, {4, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=7 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Find distinct neighbors using the explicit edge set: N(7)=∅. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0042","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd765c26bb2ec006457e54d5ebcdd93d8ebae3f9916d866b1596235c2d24d337","family_id":"qf-simple-graph-neighbors","seed":202610044042,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[4,5]],\"V\":[4,5,6],\"operation\":\"neighbors\",\"v\":6}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={4, 5, 6} and E={{4, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=6 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Find distinct neighbors using the explicit edge set: N(6)=∅. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0043","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3cfc8d5c276f8b26d51edf8ead212b28b08e8d927020e5c2d3dd741e743a7ad","family_id":"qf-simple-graph-neighbors","seed":202610044043,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[5,6],[5,7],[5,8],[6,7],[6,8],[7,8]],\"V\":[5,6,7,8],\"operation\":\"neighbors\",\"v\":7}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={5, 6, 7, 8} and E={{5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=7 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"{5, 6, 8}","explanation":"Find distinct neighbors using the explicit edge set: N(7)={5, 6, 8}. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0044","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e39f18c8db385b797bc529083b125e0f1ff32971aa9386dc772139455c4dfb66","family_id":"qf-simple-graph-neighbors","seed":202610044044,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[],\"V\":[6,7,8,9,10],\"operation\":\"neighbors\",\"v\":10}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={6, 7, 8, 9, 10} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=10 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Find distinct neighbors using the explicit edge set: N(10)=∅. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0045","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10236415910342c9460cbf3656a0837d2090bf11da6cb803c5a978d216f1dcb7","family_id":"qf-simple-graph-neighbors","seed":202610044045,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[7,8],[8,9],[9,10],[10,11],[11,12]],\"V\":[7,8,9,10,11,12],\"operation\":\"neighbors\",\"v\":9}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={7, 8, 9, 10, 11, 12} and E={{7, 8}, {8, 9}, {9, 10}, {10, 11}, {11, 12}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=9 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"{8, 10}","explanation":"Find distinct neighbors using the explicit edge set: N(9)={8, 10}. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0046","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1521eeb8f96b64a861b137e8b7fe8a5cf485219c7b12f6f24f8a877553fa7ec","family_id":"qf-simple-graph-neighbors","seed":202610044046,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[8,9]],\"V\":[8,9,10],\"operation\":\"neighbors\",\"v\":8}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={8, 9, 10} and E={{8, 9}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=8 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"{9}","explanation":"Find distinct neighbors using the explicit edge set: N(8)={9}. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0047","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92d17611e0f0c3e0799c3ac70e432e8be3d362a2b9f8f9d3955a8175e3fdff05","family_id":"qf-simple-graph-neighbors","seed":202610044047,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[9,10],[9,11],[10,11]],\"V\":[9,10,11,12],\"operation\":\"neighbors\",\"v\":11}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={9, 10, 11, 12} and E={{9, 10}, {9, 11}, {10, 11}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=11 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"{9, 10}","explanation":"Find distinct neighbors using the explicit edge set: N(11)={9, 10}. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0048","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43c03162dd084ec48a0e19ce7655df31f206899d42f62864a7a141274f419c4b","family_id":"qf-simple-graph-neighbors","seed":202610044048,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[10,11],[12,13]],\"V\":[10,11,12,13,14],\"operation\":\"neighbors\",\"v\":13}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={10, 11, 12, 13, 14} and E={{10, 11}, {12, 13}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=13 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"{12}","explanation":"Find distinct neighbors using the explicit edge set: N(13)={12}. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0049","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83234241a169cb47380a00f63c36461c0c144c41b8eae254a941fedd84dcc054","family_id":"qf-simple-graph-neighbors","seed":202610044049,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[[11,12],[11,13],[11,14],[11,15],[11,16],[12,13],[12,14],[12,15],[12,16],[13,14],[13,15],[13,16],[14,15],[14,16],[15,16]],\"V\":[11,12,13,14,15,16],\"operation\":\"neighbors\",\"v\":11}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={11, 12, 13, 14, 15, 16} and E={{11, 12}, {11, 13}, {11, 14}, {11, 15}, {11, 16}, {12, 13}, {12, 14}, {12, 15}, {12, 16}, {13, 14}, {13, 15}, {13, 16}, {14, 15}, {14, 16}, {15, 16}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=11 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"{12, 13, 14, 15, 16}","explanation":"Find distinct neighbors using the explicit edge set: N(11)={12, 13, 14, 15, 16}. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0050","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d67ab882fc979bbab4e71e8f70a327e59cc1cf4c21b77d153129a76a5e955ca2","family_id":"qf-simple-graph-neighbors","seed":202610044050,"semantic_template":"qf-simple-graph-neighbors/{\"E\":[],\"V\":[12,13,14],\"operation\":\"neighbors\",\"v\":13}"},"fields":{"title":"Neighbors in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={12, 13, 14} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=13 belongs to V. List N(v) in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Find distinct neighbors using the explicit edge set: N(13)=∅. Do not infer adjacency from a sketch or drawn-line crossings."},"skills":["Neighbors in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0051","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d29c846fe0b65fc990480040abbabca8f70e4b77e2c23f2bb041669c4746a6f","family_id":"qf-simple-graph-degree","seed":202610044051,"semantic_template":"qf-simple-graph-degree/{\"E\":[],\"V\":[0],\"operation\":\"degree\",\"v\":0}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={0} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=0 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"0","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(0)=0. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0052","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26a2bc0a1ea1aad09b49986df241b32bfbc864cd413401ce7f540d9d17697923","family_id":"qf-simple-graph-degree","seed":202610044052,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-11,-10],[-10,-9],[-9,-8]],\"V\":[-11,-10,-9,-8],\"operation\":\"degree\",\"v\":-9}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-11, -10, -9, -8} and E={{-11, -10}, {-10, -9}, {-9, -8}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-9 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"2","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(-9)=2. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0053","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88fb18c97d865b05d216c60eb59fba48a7192127543a192d3e185a43dbca9223","family_id":"qf-simple-graph-degree","seed":202610044053,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-10,-9],[-10,-8],[-10,-7]],\"V\":[-10,-9,-8,-7,-6],\"operation\":\"degree\",\"v\":-7}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-10, -9, -8, -7, -6} and E={{-10, -9}, {-10, -8}, {-10, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-7 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"1","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(-7)=1. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0054","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14b0fe30910db59fc2601f875dddd6f5a0243c4a6f8d9a827cf9fbcbed4543bb","family_id":"qf-simple-graph-degree","seed":202610044054,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-9,-8],[-9,-7],[-8,-7]],\"V\":[-9,-8,-7,-6,-5,-4],\"operation\":\"degree\",\"v\":-5}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-9, -8, -7, -6, -5, -4} and E={{-9, -8}, {-9, -7}, {-8, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-5 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"0","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(-5)=0. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0055","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa2344ddfd3c722e6fa53705c2b02f33283e1cbcb6bf23eb3909be4d1ec48410","family_id":"qf-simple-graph-degree","seed":202610044055,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-8,-7]],\"V\":[-8,-7,-6],\"operation\":\"degree\",\"v\":-6}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-8, -7, -6} and E={{-8, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-6 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"0","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(-6)=0. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0056","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4e55fc3e359c1b59dbb486747bd082e3a09ec4cc735dc25913dfbaeec8c5cd7","family_id":"qf-simple-graph-degree","seed":202610044056,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-7,-6],[-7,-5],[-7,-4],[-6,-5],[-6,-4],[-5,-4]],\"V\":[-7,-6,-5,-4],\"operation\":\"degree\",\"v\":-5}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-7, -6, -5, -4} and E={{-7, -6}, {-7, -5}, {-7, -4}, {-6, -5}, {-6, -4}, {-5, -4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-5 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"3","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(-5)=3. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0057","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7553aed08d829a19aeae1f5ab8581791e249696fe18489b4c66b6b5ad98d81a0","family_id":"qf-simple-graph-degree","seed":202610044057,"semantic_template":"qf-simple-graph-degree/{\"E\":[],\"V\":[-6,-5,-4,-3,-2],\"operation\":\"degree\",\"v\":-4}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-6, -5, -4, -3, -2} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-4 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"0","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(-4)=0. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0058","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e91f87027f383a52bccf0a914661e76e9449899f19b573697058e504758b026","family_id":"qf-simple-graph-degree","seed":202610044058,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-5,-4],[-4,-3],[-3,-2],[-2,-1],[-1,0]],\"V\":[-5,-4,-3,-2,-1,0],\"operation\":\"degree\",\"v\":-3}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-5, -4, -3, -2, -1, 0} and E={{-5, -4}, {-4, -3}, {-3, -2}, {-2, -1}, {-1, 0}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-3 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"2","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(-3)=2. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0059","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86970bd3f98dea05602cf6221a3e83d427a0afd3cb90148ca06d1740dc94fd35","family_id":"qf-simple-graph-degree","seed":202610044059,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-4,-3]],\"V\":[-4,-3,-2],\"operation\":\"degree\",\"v\":-4}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-4, -3, -2} and E={{-4, -3}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-4 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"1","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(-4)=1. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0060","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"913e915087d69d26e7cfc03c85beb34896d374b419a99c104945d15ea9ac2c8e","family_id":"qf-simple-graph-degree","seed":202610044060,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-3,-2],[-3,-1],[-2,-1]],\"V\":[-3,-2,-1,0],\"operation\":\"degree\",\"v\":-1}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-3, -2, -1, 0} and E={{-3, -2}, {-3, -1}, {-2, -1}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-1 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"2","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(-1)=2. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0061","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6222c593e3f514115a80574c76bc0b846e697971a11835ab50a2175f69aec31","family_id":"qf-simple-graph-degree","seed":202610044061,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-2,-1],[0,1]],\"V\":[-2,-1,0,1,2],\"operation\":\"degree\",\"v\":-1}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-2, -1, 0, 1, 2} and E={{-2, -1}, {0, 1}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-1 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"1","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(-1)=1. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0062","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21fe35e3dc571eff55b74e37d5fdd6c141474af21e3b3cf2496e4760ff28cf44","family_id":"qf-simple-graph-degree","seed":202610044062,"semantic_template":"qf-simple-graph-degree/{\"E\":[[-1,0],[-1,1],[-1,2],[-1,3],[-1,4],[0,1],[0,2],[0,3],[0,4],[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]],\"V\":[-1,0,1,2,3,4],\"operation\":\"degree\",\"v\":-1}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-1, 0, 1, 2, 3, 4} and E={{-1, 0}, {-1, 1}, {-1, 2}, {-1, 3}, {-1, 4}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=-1 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"5","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(-1)=5. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0063","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38b21c26e923541ab8b539a34efad85b57c4aa8fb5be288b2a7c98f4d9a336bd","family_id":"qf-simple-graph-degree","seed":202610044063,"semantic_template":"qf-simple-graph-degree/{\"E\":[],\"V\":[0,1,2],\"operation\":\"degree\",\"v\":1}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={0, 1, 2} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=1 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"0","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(1)=0. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0064","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2dd4575ffc8b069e6dc59574b14541055059b669bf82cc7b1e397176cadb504","family_id":"qf-simple-graph-degree","seed":202610044064,"semantic_template":"qf-simple-graph-degree/{\"E\":[[1,2],[2,3],[3,4]],\"V\":[1,2,3,4],\"operation\":\"degree\",\"v\":3}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={1, 2, 3, 4} and E={{1, 2}, {2, 3}, {3, 4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=3 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"2","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(3)=2. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0065","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ccc282bf419fb368d9ba7a6ff96dcd687f274619f4700a49403f0cd4af94b05","family_id":"qf-simple-graph-degree","seed":202610044065,"semantic_template":"qf-simple-graph-degree/{\"E\":[[2,3],[2,4],[2,5]],\"V\":[2,3,4,5,6],\"operation\":\"degree\",\"v\":2}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={2, 3, 4, 5, 6} and E={{2, 3}, {2, 4}, {2, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=2 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"3","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(2)=3. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0066","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1955c70bfe6e80f556ffb3ed98efc505b85ca96e3c9f25a35b10904918e40142","family_id":"qf-simple-graph-degree","seed":202610044066,"semantic_template":"qf-simple-graph-degree/{\"E\":[[3,4],[3,5],[4,5]],\"V\":[3,4,5,6,7,8],\"operation\":\"degree\",\"v\":7}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={3, 4, 5, 6, 7, 8} and E={{3, 4}, {3, 5}, {4, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=7 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"0","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(7)=0. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0067","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af6708e74f6669a41d348c758049f520b354af8d9a60b1140bde789efec33000","family_id":"qf-simple-graph-degree","seed":202610044067,"semantic_template":"qf-simple-graph-degree/{\"E\":[[4,5]],\"V\":[4,5,6],\"operation\":\"degree\",\"v\":6}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={4, 5, 6} and E={{4, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=6 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"0","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(6)=0. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0068","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ef0783001108aec2e334a7c73dcbb299e221a3975f268b8a5118253084b06af","family_id":"qf-simple-graph-degree","seed":202610044068,"semantic_template":"qf-simple-graph-degree/{\"E\":[[5,6],[5,7],[5,8],[6,7],[6,8],[7,8]],\"V\":[5,6,7,8],\"operation\":\"degree\",\"v\":7}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={5, 6, 7, 8} and E={{5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=7 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"3","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(7)=3. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0069","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9cd8a02f31edb782aac7b3f3b6ce96dedd0506c942438e271c999fd96d2429df","family_id":"qf-simple-graph-degree","seed":202610044069,"semantic_template":"qf-simple-graph-degree/{\"E\":[],\"V\":[6,7,8,9,10],\"operation\":\"degree\",\"v\":10}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={6, 7, 8, 9, 10} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=10 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"0","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(10)=0. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0070","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05880558066f46aa78c97de503784f8256e224e87d0bc5db844c83a66a78f4af","family_id":"qf-simple-graph-degree","seed":202610044070,"semantic_template":"qf-simple-graph-degree/{\"E\":[[7,8],[8,9],[9,10],[10,11],[11,12]],\"V\":[7,8,9,10,11,12],\"operation\":\"degree\",\"v\":9}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={7, 8, 9, 10, 11, 12} and E={{7, 8}, {8, 9}, {9, 10}, {10, 11}, {11, 12}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=9 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"2","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(9)=2. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0071","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9020ca46ad75ccb4c26a81a3d1112860e22a69ddba19c53b9d94535b2197630","family_id":"qf-simple-graph-degree","seed":202610044071,"semantic_template":"qf-simple-graph-degree/{\"E\":[[8,9]],\"V\":[8,9,10],\"operation\":\"degree\",\"v\":8}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={8, 9, 10} and E={{8, 9}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=8 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"1","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(8)=1. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0072","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"660c496511344dd9eb717500df7ed7f5f7e43d351c651f542a7b6837f177ca80","family_id":"qf-simple-graph-degree","seed":202610044072,"semantic_template":"qf-simple-graph-degree/{\"E\":[[9,10],[9,11],[10,11]],\"V\":[9,10,11,12],\"operation\":\"degree\",\"v\":11}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={9, 10, 11, 12} and E={{9, 10}, {9, 11}, {10, 11}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=11 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"2","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(11)=2. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0073","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"264ace35ea1a7b93f52f364e165b6ca06054864fc26f46de87a22638ed6a2edf","family_id":"qf-simple-graph-degree","seed":202610044073,"semantic_template":"qf-simple-graph-degree/{\"E\":[[10,11],[12,13]],\"V\":[10,11,12,13,14],\"operation\":\"degree\",\"v\":13}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={10, 11, 12, 13, 14} and E={{10, 11}, {12, 13}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=13 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"1","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(13)=1. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0074","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23afc632b30487ee417ab438dea452462964ddef4fb24ff34d9a0b281ddcb659","family_id":"qf-simple-graph-degree","seed":202610044074,"semantic_template":"qf-simple-graph-degree/{\"E\":[[11,12],[11,13],[11,14],[11,15],[11,16],[12,13],[12,14],[12,15],[12,16],[13,14],[13,15],[13,16],[14,15],[14,16],[15,16]],\"V\":[11,12,13,14,15,16],\"operation\":\"degree\",\"v\":11}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={11, 12, 13, 14, 15, 16} and E={{11, 12}, {11, 13}, {11, 14}, {11, 15}, {11, 16}, {12, 13}, {12, 14}, {12, 15}, {12, 16}, {13, 14}, {13, 15}, {13, 16}, {14, 15}, {14, 16}, {15, 16}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=11 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"5","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(11)=5. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0075","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e31a8fca0990d88e4eda164f64a1b68b85ecac55691f1c08f980b2eea614561","family_id":"qf-simple-graph-degree","seed":202610044075,"semantic_template":"qf-simple-graph-degree/{\"E\":[],\"V\":[12,13,14],\"operation\":\"degree\",\"v\":13}"},"fields":{"title":"Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={12, 13, 14} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. The vertex v=13 belongs to V. Find deg(v).","answer":{"kind":"numeric","value":"0","explanation":"Count distinct neighbors from the explicit edge set to obtain deg(13)=0. An isolated vertex has degree zero."},"skills":["Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0076","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4188567b59150a5ba3e0cc5c1b63a915a3525812ee2c964744b9294e9a9d4bcc","family_id":"qf-simple-graph-handshaking","seed":202610044076,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[],\"V\":[],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V=∅ and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"0","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=0. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0077","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75e9b1a21e58e7c698980f76019e6273ebfa6144963f1b556f59494cc4ca5bd0","family_id":"qf-simple-graph-handshaking","seed":202610044077,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-11,-10],[-10,-9],[-9,-8]],\"V\":[-11,-10,-9,-8],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-11, -10, -9, -8} and E={{-11, -10}, {-10, -9}, {-9, -8}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"6","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=6. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0078","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3172907ed1f0d0e686bd44408f6328ca8b7c423443d4e5d12f1d2af499466a7","family_id":"qf-simple-graph-handshaking","seed":202610044078,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-10,-9],[-10,-8],[-10,-7]],\"V\":[-10,-9,-8,-7,-6],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-10, -9, -8, -7, -6} and E={{-10, -9}, {-10, -8}, {-10, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"6","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=6. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0079","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df6920ce85981ccfc8134ce9d5415a3a471649cd17b21d176d5bb3b13041fff4","family_id":"qf-simple-graph-handshaking","seed":202610044079,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-9,-8],[-9,-7],[-8,-7]],\"V\":[-9,-8,-7,-6,-5,-4],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-9, -8, -7, -6, -5, -4} and E={{-9, -8}, {-9, -7}, {-8, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"6","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=6. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0080","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df27742278b29075be116f6ce7df687b0646afbc303cc95b9750fdc0bb29e1d4","family_id":"qf-simple-graph-handshaking","seed":202610044080,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-8,-7]],\"V\":[-8,-7,-6],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-8, -7, -6} and E={{-8, -7}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"2","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=2. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0081","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c74a334d85037911ae2376ec09bc7d9cc0d21882677cbf0ce9bbbdfecffd61d","family_id":"qf-simple-graph-handshaking","seed":202610044081,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-7,-6],[-7,-5],[-7,-4],[-6,-5],[-6,-4],[-5,-4]],\"V\":[-7,-6,-5,-4],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-7, -6, -5, -4} and E={{-7, -6}, {-7, -5}, {-7, -4}, {-6, -5}, {-6, -4}, {-5, -4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"12","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=12. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0082","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f19d516878fcef26281cb6a93124a046984f7c48d70dee5ba19efdec4636fc89","family_id":"qf-simple-graph-handshaking","seed":202610044082,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[],\"V\":[-6,-5,-4,-3,-2],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-6, -5, -4, -3, -2} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"0","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=0. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0083","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3224e95ecbc3d1ef50a40e72a5bd5b5e36eadd079e1a4935606b7b8079cb6b8","family_id":"qf-simple-graph-handshaking","seed":202610044083,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-5,-4],[-4,-3],[-3,-2],[-2,-1],[-1,0]],\"V\":[-5,-4,-3,-2,-1,0],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-5, -4, -3, -2, -1, 0} and E={{-5, -4}, {-4, -3}, {-3, -2}, {-2, -1}, {-1, 0}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"10","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=10. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0084","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0490dcffb9a50938c35953ae4afc312404b79b448f48181c86ba8558f339177","family_id":"qf-simple-graph-handshaking","seed":202610044084,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-4,-3]],\"V\":[-4,-3,-2],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-4, -3, -2} and E={{-4, -3}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"2","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=2. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0085","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f07ee1d9cfc6ec3001ff8c600156d7fa2c1ca16c29f7769fd56d6d99ef3eeec3","family_id":"qf-simple-graph-handshaking","seed":202610044085,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-3,-2],[-3,-1],[-2,-1]],\"V\":[-3,-2,-1,0],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-3, -2, -1, 0} and E={{-3, -2}, {-3, -1}, {-2, -1}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"6","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=6. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0086","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09e327fc66b04133e826404727ecebccf78b7c25ef0d200dcaaa06f40f070abc","family_id":"qf-simple-graph-handshaking","seed":202610044086,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-2,-1],[0,1]],\"V\":[-2,-1,0,1,2],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-2, -1, 0, 1, 2} and E={{-2, -1}, {0, 1}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"4","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=4. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0087","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecb3a64b14e0f4a760b412f7b39ea6eb72f165a231518748a1a6067e8381bbfe","family_id":"qf-simple-graph-handshaking","seed":202610044087,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[-1,0],[-1,1],[-1,2],[-1,3],[-1,4],[0,1],[0,2],[0,3],[0,4],[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]],\"V\":[-1,0,1,2,3,4],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={-1, 0, 1, 2, 3, 4} and E={{-1, 0}, {-1, 1}, {-1, 2}, {-1, 3}, {-1, 4}, {0, 1}, {0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"30","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=30. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0088","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1be8ec38ecdfe71e136adfd564e70c48e3425ce84eea0e21b6ffbe94bfef3f7","family_id":"qf-simple-graph-handshaking","seed":202610044088,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[],\"V\":[0,1,2],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={0, 1, 2} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"0","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=0. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0089","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25d6121e15aa56dc75217ab177bd5b1b5ba01d9ab36f31b33edfe8b6b416db99","family_id":"qf-simple-graph-handshaking","seed":202610044089,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[1,2],[2,3],[3,4]],\"V\":[1,2,3,4],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={1, 2, 3, 4} and E={{1, 2}, {2, 3}, {3, 4}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"6","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=6. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0090","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38cbce3b80f04ced5648f80d36d850aba0d9b5fe563879c6f375c40d744648ea","family_id":"qf-simple-graph-handshaking","seed":202610044090,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[2,3],[2,4],[2,5]],\"V\":[2,3,4,5,6],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={2, 3, 4, 5, 6} and E={{2, 3}, {2, 4}, {2, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"6","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=6. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0091","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69e82c4dd244fe6f66cd37cb49eb5b3206bc20355359dc0a013e0bb3e337147c","family_id":"qf-simple-graph-handshaking","seed":202610044091,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[3,4],[3,5],[4,5]],\"V\":[3,4,5,6,7,8],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={3, 4, 5, 6, 7, 8} and E={{3, 4}, {3, 5}, {4, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"6","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=6. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0092","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eeaffd15c434547bad00330b3ca035137f372e6bb2653620ebcfa9079b44701d","family_id":"qf-simple-graph-handshaking","seed":202610044092,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[4,5]],\"V\":[4,5,6],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={4, 5, 6} and E={{4, 5}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"2","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=2. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0093","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d49e5ab70d39e1dfc037904ec08cd0dfef6e8742e01ef11b624dd32406c6f05","family_id":"qf-simple-graph-handshaking","seed":202610044093,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[5,6],[5,7],[5,8],[6,7],[6,8],[7,8]],\"V\":[5,6,7,8],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={5, 6, 7, 8} and E={{5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"12","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=12. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0094","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4e83102db67dec01c556507c585f722aea49ac4badaebe437ff357998ce0cf0","family_id":"qf-simple-graph-handshaking","seed":202610044094,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[],\"V\":[6,7,8,9,10],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={6, 7, 8, 9, 10} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"0","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=0. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0095","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc894005a7e4125e4d1ae4263ac48cdc6d791945b372ffc825959e3ea2caae10","family_id":"qf-simple-graph-handshaking","seed":202610044095,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[7,8],[8,9],[9,10],[10,11],[11,12]],\"V\":[7,8,9,10,11,12],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={7, 8, 9, 10, 11, 12} and E={{7, 8}, {8, 9}, {9, 10}, {10, 11}, {11, 12}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"10","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=10. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0096","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"512ba13b464504d8b798caed0eba03e750fa17aac6996f5fb38e5919d56d2a3f","family_id":"qf-simple-graph-handshaking","seed":202610044096,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[8,9]],\"V\":[8,9,10],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={8, 9, 10} and E={{8, 9}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"2","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=2. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0097","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f336a3a3d384f26447441fba63e7c42e615cf05638c7365f66a850b6267489dc","family_id":"qf-simple-graph-handshaking","seed":202610044097,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[9,10],[9,11],[10,11]],\"V\":[9,10,11,12],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={9, 10, 11, 12} and E={{9, 10}, {9, 11}, {10, 11}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"6","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=6. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0098","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4519a2ab3e9ced59cae5f6d06a08e252077b818705569050300e6b445ece8226","family_id":"qf-simple-graph-handshaking","seed":202610044098,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[10,11],[12,13]],\"V\":[10,11,12,13,14],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={10, 11, 12, 13, 14} and E={{10, 11}, {12, 13}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"4","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=4. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0099","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd9809a0681354eee0018f0a6a215d7eafec3fa3d40bd9450cb7ffc16e942834","family_id":"qf-simple-graph-handshaking","seed":202610044099,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[[11,12],[11,13],[11,14],[11,15],[11,16],[12,13],[12,14],[12,15],[12,16],[13,14],[13,15],[13,16],[14,15],[14,16],[15,16]],\"V\":[11,12,13,14,15,16],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={11, 12, 13, 14, 15, 16} and E={{11, 12}, {11, 13}, {11, 14}, {11, 15}, {11, 16}, {12, 13}, {12, 14}, {12, 15}, {12, 16}, {13, 14}, {13, 15}, {13, 16}, {14, 15}, {14, 16}, {15, 16}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"30","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=30. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r04-graph-0100","batch_id":"batch-20261003-advance-20261004-r04-graph-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82a2b22005fed4a8b8f39f15cd9fe9ce535d59cb6efc1ec5bdbf4f670b421178","family_id":"qf-simple-graph-handshaking","seed":202610044100,"semantic_template":"qf-simple-graph-handshaking/{\"E\":[],\"V\":[12,13,14],\"operation\":\"handshaking\"}"},"fields":{"title":"Total Vertex Degree in a Simple Undirected Graph","stem":"The finite simple undirected graph has V={12, 13, 14} and E=∅. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Find the sum of degrees over all vertices in V.","answer":{"kind":"numeric","value":"0","explanation":"Every unordered edge contributes once at each of its two endpoints, so the degree sum is 2|E|=0. Isolated vertices contribute zero, and the empty graph also gives zero."},"skills":["Total Vertex Degree in a Simple Undirected Graph"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}