{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:37VPV2GJVTJ73WIG4TYSNCLSOJ","short_pith_number":"pith:37VPV2GJ","canonical_record":{"source":{"id":"math/0602428","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2006-02-20T14:15:55Z","cross_cats_sorted":[],"title_canon_sha256":"ad60fc2b11a20916bb1706aee776f95a8e83e1da60c70472026e9fe2d9fd5d39","abstract_canon_sha256":"6229befc4b9b79337f474547a73297fee031da08d22210a8a8b00969800d9a20"},"schema_version":"1.0"},"canonical_sha256":"dfeafae8c9acd3fdd906e4f126897272545b4eb8d7479ec0583cd869a79015e3","source":{"kind":"arxiv","id":"math/0602428","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0602428","created_at":"2026-05-18T03:05:55Z"},{"alias_kind":"arxiv_version","alias_value":"math/0602428v2","created_at":"2026-05-18T03:05:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0602428","created_at":"2026-05-18T03:05:55Z"},{"alias_kind":"pith_short_12","alias_value":"37VPV2GJVTJ7","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"37VPV2GJVTJ73WIG","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"37VPV2GJ","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:37VPV2GJVTJ73WIG4TYSNCLSOJ","target":"record","payload":{"canonical_record":{"source":{"id":"math/0602428","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2006-02-20T14:15:55Z","cross_cats_sorted":[],"title_canon_sha256":"ad60fc2b11a20916bb1706aee776f95a8e83e1da60c70472026e9fe2d9fd5d39","abstract_canon_sha256":"6229befc4b9b79337f474547a73297fee031da08d22210a8a8b00969800d9a20"},"schema_version":"1.0"},"canonical_sha256":"dfeafae8c9acd3fdd906e4f126897272545b4eb8d7479ec0583cd869a79015e3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:05:55.561524Z","signature_b64":"IOoRnmND9MsiYBxPmD71f+yp65aaFIwSMj4XPnQ3kjLRYYYZ9jPY/+ISRsgkhOxWs6Z4Juc04QAw/BCIzQaACw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dfeafae8c9acd3fdd906e4f126897272545b4eb8d7479ec0583cd869a79015e3","last_reissued_at":"2026-05-18T03:05:55.560906Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:05:55.560906Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0602428","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:05:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IJcdZfKsmfO+KrnH/W//De/7MuZST7UKW1jvBPziTC9pu4+TmTGw9oh4V6QyfyflkwIngOMMtmJwjzPPasGYDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T03:30:17.959197Z"},"content_sha256":"29b3c3035ef95a8f62b91ad2e34ec33bac792ca4f0e7c900aeca2dea1be9cdb3","schema_version":"1.0","event_id":"sha256:29b3c3035ef95a8f62b91ad2e34ec33bac792ca4f0e7c900aeca2dea1be9cdb3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:37VPV2GJVTJ73WIG4TYSNCLSOJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Alliance free and alliance cover sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"I. G. Yero, J. A. Rodriguez-Velazquez, J. M. Sigarreta, S. Bermudo","submitted_at":"2006-02-20T14:15:55Z","abstract_excerpt":"A \\emph{defensive} (\\emph{offensive}) $k$-\\emph{alliance} in $\\Gamma=(V,E)$ is a set $S\\subseteq V$ such that every $v$ in $S$ (in the boundary of $S$) has at least $k$ more neighbors in $S$ than it has in $V\\setminus S$. A set $X\\subseteq V$ is \\emph{defensive} (\\emph{offensive}) $k$-\\emph{alliance free,} if for all defensive (offensive) $k$-alliance $S$, $S\\setminus X\\neq\\emptyset$, i.e., $X$ does not contain any defensive (offensive) $k$-alliance as a subset. A set $Y \\subseteq V$ is a \\emph{defensive} (\\emph{offensive}) $k$-\\emph{alliance cover}, if for all defensive (offensive) $k$-allian"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0602428","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:05:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3dPeAKgyEDpWegKHIbqJDkPJLlE/qYDNPaDfB4SyOQTYhE+lJ4Jsx3pSBW8msGyNMpRUVGHzsPvRD9b/DKNRDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T03:30:17.959562Z"},"content_sha256":"c65f1d43229bb4affde539b43b2b5ffb45389b5a994d7230b2a8f53c1d127672","schema_version":"1.0","event_id":"sha256:c65f1d43229bb4affde539b43b2b5ffb45389b5a994d7230b2a8f53c1d127672"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/37VPV2GJVTJ73WIG4TYSNCLSOJ/bundle.json","state_url":"https://pith.science/pith/37VPV2GJVTJ73WIG4TYSNCLSOJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/37VPV2GJVTJ73WIG4TYSNCLSOJ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-02T03:30:17Z","links":{"resolver":"https://pith.science/pith/37VPV2GJVTJ73WIG4TYSNCLSOJ","bundle":"https://pith.science/pith/37VPV2GJVTJ73WIG4TYSNCLSOJ/bundle.json","state":"https://pith.science/pith/37VPV2GJVTJ73WIG4TYSNCLSOJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/37VPV2GJVTJ73WIG4TYSNCLSOJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:37VPV2GJVTJ73WIG4TYSNCLSOJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"6229befc4b9b79337f474547a73297fee031da08d22210a8a8b00969800d9a20","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2006-02-20T14:15:55Z","title_canon_sha256":"ad60fc2b11a20916bb1706aee776f95a8e83e1da60c70472026e9fe2d9fd5d39"},"schema_version":"1.0","source":{"id":"math/0602428","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0602428","created_at":"2026-05-18T03:05:55Z"},{"alias_kind":"arxiv_version","alias_value":"math/0602428v2","created_at":"2026-05-18T03:05:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0602428","created_at":"2026-05-18T03:05:55Z"},{"alias_kind":"pith_short_12","alias_value":"37VPV2GJVTJ7","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"37VPV2GJVTJ73WIG","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"37VPV2GJ","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:c65f1d43229bb4affde539b43b2b5ffb45389b5a994d7230b2a8f53c1d127672","target":"graph","created_at":"2026-05-18T03:05:55Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"A \\emph{defensive} (\\emph{offensive}) $k$-\\emph{alliance} in $\\Gamma=(V,E)$ is a set $S\\subseteq V$ such that every $v$ in $S$ (in the boundary of $S$) has at least $k$ more neighbors in $S$ than it has in $V\\setminus S$. A set $X\\subseteq V$ is \\emph{defensive} (\\emph{offensive}) $k$-\\emph{alliance free,} if for all defensive (offensive) $k$-alliance $S$, $S\\setminus X\\neq\\emptyset$, i.e., $X$ does not contain any defensive (offensive) $k$-alliance as a subset. A set $Y \\subseteq V$ is a \\emph{defensive} (\\emph{offensive}) $k$-\\emph{alliance cover}, if for all defensive (offensive) $k$-allian","authors_text":"I. G. Yero, J. A. Rodriguez-Velazquez, J. M. Sigarreta, S. Bermudo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2006-02-20T14:15:55Z","title":"Alliance free and alliance cover sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0602428","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:29b3c3035ef95a8f62b91ad2e34ec33bac792ca4f0e7c900aeca2dea1be9cdb3","target":"record","created_at":"2026-05-18T03:05:55Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"6229befc4b9b79337f474547a73297fee031da08d22210a8a8b00969800d9a20","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2006-02-20T14:15:55Z","title_canon_sha256":"ad60fc2b11a20916bb1706aee776f95a8e83e1da60c70472026e9fe2d9fd5d39"},"schema_version":"1.0","source":{"id":"math/0602428","kind":"arxiv","version":2}},"canonical_sha256":"dfeafae8c9acd3fdd906e4f126897272545b4eb8d7479ec0583cd869a79015e3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dfeafae8c9acd3fdd906e4f126897272545b4eb8d7479ec0583cd869a79015e3","first_computed_at":"2026-05-18T03:05:55.560906Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:05:55.560906Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IOoRnmND9MsiYBxPmD71f+yp65aaFIwSMj4XPnQ3kjLRYYYZ9jPY/+ISRsgkhOxWs6Z4Juc04QAw/BCIzQaACw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:05:55.561524Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0602428","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:29b3c3035ef95a8f62b91ad2e34ec33bac792ca4f0e7c900aeca2dea1be9cdb3","sha256:c65f1d43229bb4affde539b43b2b5ffb45389b5a994d7230b2a8f53c1d127672"],"state_sha256":"4451a54daf81643a3b13bdba55a67ae4dbd018b8f1e86a859363fab46ee3b2fe"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BA1fjZUW5gCkd3JSwn+PVh9QGwk9YdRj3l3XGvm4uI/ioYI+rs+UynoQ6BFrq2+7Dp8B2+D6td/79oJruu+9CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T03:30:17.961644Z","bundle_sha256":"a366730a78707d8ad2b064302a473d15006fb57a0b7ec64985c2586d469e7940"}}