{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:3ZZSNJARBYKSZQLF6MD2ATO4RC","short_pith_number":"pith:3ZZSNJAR","canonical_record":{"source":{"id":"1112.2068","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-09T10:45:17Z","cross_cats_sorted":[],"title_canon_sha256":"bfd3bbceccea0cc5bffe5c362bd1949a9a5ffa9bcc00b8d5b7bb3c45e5c1b709","abstract_canon_sha256":"b58ba99cd5bf19f51becce5e2f69abf7bf79ddbd3c7792576157e33eec95c310"},"schema_version":"1.0"},"canonical_sha256":"de7326a4110e152cc165f307a04ddc888b55defa6e1f67ac5ebd8c9b2d6ff05a","source":{"kind":"arxiv","id":"1112.2068","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.2068","created_at":"2026-05-18T04:06:44Z"},{"alias_kind":"arxiv_version","alias_value":"1112.2068v1","created_at":"2026-05-18T04:06:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.2068","created_at":"2026-05-18T04:06:44Z"},{"alias_kind":"pith_short_12","alias_value":"3ZZSNJARBYKS","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"3ZZSNJARBYKSZQLF","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"3ZZSNJAR","created_at":"2026-05-18T12:26:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:3ZZSNJARBYKSZQLF6MD2ATO4RC","target":"record","payload":{"canonical_record":{"source":{"id":"1112.2068","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-09T10:45:17Z","cross_cats_sorted":[],"title_canon_sha256":"bfd3bbceccea0cc5bffe5c362bd1949a9a5ffa9bcc00b8d5b7bb3c45e5c1b709","abstract_canon_sha256":"b58ba99cd5bf19f51becce5e2f69abf7bf79ddbd3c7792576157e33eec95c310"},"schema_version":"1.0"},"canonical_sha256":"de7326a4110e152cc165f307a04ddc888b55defa6e1f67ac5ebd8c9b2d6ff05a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:06:44.167836Z","signature_b64":"jRzsnS8OkPuXPIf4QMB9aYOoqy8Th+dFdFD781QJg0MW84GUlflPgi763ypGz8S1S9Sj4zBVM0AJYxLvx2RWBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"de7326a4110e152cc165f307a04ddc888b55defa6e1f67ac5ebd8c9b2d6ff05a","last_reissued_at":"2026-05-18T04:06:44.167256Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:06:44.167256Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1112.2068","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:06:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1GuD8bNzgHtr03vJ5ZR08BZnt86R9+RWHWI5f44JB/5cUk8MI/ydz4ebgknzaie24++453trzp96PYDot8JZDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T21:24:17.498386Z"},"content_sha256":"5d112b041f74165fd92a9b46a660450afbdb9943f6ce86abcc5614eee6427015","schema_version":"1.0","event_id":"sha256:5d112b041f74165fd92a9b46a660450afbdb9943f6ce86abcc5614eee6427015"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:3ZZSNJARBYKSZQLF6MD2ATO4RC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Alliance free sets in Cartesian product graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ismael G. Yero, Juan A. Rodriguez-Velazquez, Sergio Bermudo","submitted_at":"2011-12-09T10:45:17Z","abstract_excerpt":"Let $G=(V,E)$ be a graph. For a non-empty subset of vertices $S\\subseteq V$, and vertex $v\\in V$, let $\\delta_S(v)=|\\{u\\in S:uv\\in E\\}|$ denote the cardinality of the set of neighbors of $v$ in $S$, and let $\\bar{S}=V-S$. Consider the following condition: {equation}\\label{alliancecondition} \\delta_S(v)\\ge \\delta_{\\bar{S}}(v)+k, \\{equation} which states that a vertex $v$ has at least $k$ more neighbors in $S$ than it has in $\\bar{S}$. A set $S\\subseteq V$ that satisfies Condition (\\ref{alliancecondition}) for every vertex $v \\in S$ is called a \\emph{defensive} $k$-\\emph{alliance}; for every ver"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.2068","kind":"arxiv","version":1},"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":"2026-05-18T04:06:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZMfVAxuwI4x1+y2R4v4QuBn986PXoCD6/BTYqKcirLoaUEnJSRe5eZjSDPJc0kZaVAuyQjZSS6pVtjQ2gwvLBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T21:24:17.498796Z"},"content_sha256":"22568afdcaf982c9249f73d04552304f1a05b89915300e4987a695d2fb54920a","schema_version":"1.0","event_id":"sha256:22568afdcaf982c9249f73d04552304f1a05b89915300e4987a695d2fb54920a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3ZZSNJARBYKSZQLF6MD2ATO4RC/bundle.json","state_url":"https://pith.science/pith/3ZZSNJARBYKSZQLF6MD2ATO4RC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3ZZSNJARBYKSZQLF6MD2ATO4RC/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":"2026-06-26T21:24:17Z","links":{"resolver":"https://pith.science/pith/3ZZSNJARBYKSZQLF6MD2ATO4RC","bundle":"https://pith.science/pith/3ZZSNJARBYKSZQLF6MD2ATO4RC/bundle.json","state":"https://pith.science/pith/3ZZSNJARBYKSZQLF6MD2ATO4RC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3ZZSNJARBYKSZQLF6MD2ATO4RC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:3ZZSNJARBYKSZQLF6MD2ATO4RC","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":"b58ba99cd5bf19f51becce5e2f69abf7bf79ddbd3c7792576157e33eec95c310","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-09T10:45:17Z","title_canon_sha256":"bfd3bbceccea0cc5bffe5c362bd1949a9a5ffa9bcc00b8d5b7bb3c45e5c1b709"},"schema_version":"1.0","source":{"id":"1112.2068","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.2068","created_at":"2026-05-18T04:06:44Z"},{"alias_kind":"arxiv_version","alias_value":"1112.2068v1","created_at":"2026-05-18T04:06:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.2068","created_at":"2026-05-18T04:06:44Z"},{"alias_kind":"pith_short_12","alias_value":"3ZZSNJARBYKS","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"3ZZSNJARBYKSZQLF","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"3ZZSNJAR","created_at":"2026-05-18T12:26:20Z"}],"graph_snapshots":[{"event_id":"sha256:22568afdcaf982c9249f73d04552304f1a05b89915300e4987a695d2fb54920a","target":"graph","created_at":"2026-05-18T04:06:44Z","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":"Let $G=(V,E)$ be a graph. For a non-empty subset of vertices $S\\subseteq V$, and vertex $v\\in V$, let $\\delta_S(v)=|\\{u\\in S:uv\\in E\\}|$ denote the cardinality of the set of neighbors of $v$ in $S$, and let $\\bar{S}=V-S$. Consider the following condition: {equation}\\label{alliancecondition} \\delta_S(v)\\ge \\delta_{\\bar{S}}(v)+k, \\{equation} which states that a vertex $v$ has at least $k$ more neighbors in $S$ than it has in $\\bar{S}$. A set $S\\subseteq V$ that satisfies Condition (\\ref{alliancecondition}) for every vertex $v \\in S$ is called a \\emph{defensive} $k$-\\emph{alliance}; for every ver","authors_text":"Ismael G. Yero, Juan A. Rodriguez-Velazquez, Sergio Bermudo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-09T10:45:17Z","title":"Alliance free sets in Cartesian product graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.2068","kind":"arxiv","version":1},"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":"sha256:5d112b041f74165fd92a9b46a660450afbdb9943f6ce86abcc5614eee6427015","target":"record","created_at":"2026-05-18T04:06:44Z","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":"b58ba99cd5bf19f51becce5e2f69abf7bf79ddbd3c7792576157e33eec95c310","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-09T10:45:17Z","title_canon_sha256":"bfd3bbceccea0cc5bffe5c362bd1949a9a5ffa9bcc00b8d5b7bb3c45e5c1b709"},"schema_version":"1.0","source":{"id":"1112.2068","kind":"arxiv","version":1}},"canonical_sha256":"de7326a4110e152cc165f307a04ddc888b55defa6e1f67ac5ebd8c9b2d6ff05a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"de7326a4110e152cc165f307a04ddc888b55defa6e1f67ac5ebd8c9b2d6ff05a","first_computed_at":"2026-05-18T04:06:44.167256Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:06:44.167256Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jRzsnS8OkPuXPIf4QMB9aYOoqy8Th+dFdFD781QJg0MW84GUlflPgi763ypGz8S1S9Sj4zBVM0AJYxLvx2RWBA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:06:44.167836Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.2068","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5d112b041f74165fd92a9b46a660450afbdb9943f6ce86abcc5614eee6427015","sha256:22568afdcaf982c9249f73d04552304f1a05b89915300e4987a695d2fb54920a"],"state_sha256":"5a1527bfbffd0738c9f831ba98e4c6ad93e1642008b80fb3ca78f2389b084142"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jlSolJyS04HbJ3YUzLLwa0pFb8/CSZPt4z/7HgGcLngqHgq17YNrlewqoQr/A+QPgwF8AcHjseCif5rmR5TFBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T21:24:17.500894Z","bundle_sha256":"2f92850aebd4dda942e974be776882b6fba909337c48e94d3f51f44d8bfdd953"}}