{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:5LBIQGVQ2OUQCB6YGYNAIP7ODK","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":"5ba76a762fb9b51f04fdc5fbb1920c4daff319c25e473abc795ccce16898453b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-12-08T18:18:53Z","title_canon_sha256":"2ee7f7e9047a9e9f4ccbc54106588a7b00329fa95b40ed7dd7469ee6c54108a0"},"schema_version":"1.0","source":{"id":"0812.1528","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0812.1528","created_at":"2026-05-18T03:06:04Z"},{"alias_kind":"arxiv_version","alias_value":"0812.1528v2","created_at":"2026-05-18T03:06:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0812.1528","created_at":"2026-05-18T03:06:04Z"},{"alias_kind":"pith_short_12","alias_value":"5LBIQGVQ2OUQ","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"5LBIQGVQ2OUQCB6Y","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"5LBIQGVQ","created_at":"2026-05-18T12:25:56Z"}],"graph_snapshots":[{"event_id":"sha256:7e65aa3ec6be152d711938732d1498a82e091b4b016206118e0f08cb2024e792","target":"graph","created_at":"2026-05-18T03:06:04Z","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":"We investigate the relationship between global offensive $k$-alliances and some characteristic sets of a graph including $r$-dependent sets and $\\tau$-dominating sets. As a consequence of the study, we obtain bounds on the global offensive $k$-alliance number in terms of several parameters of the graph.","authors_text":"Ismael G. Yero, Jose M. Sigarreta, Juan A. Rodriguez-Velazquez, Sergio Bermudo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-12-08T18:18:53Z","title":"On global offensive k-alliances in graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0812.1528","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:ae0432265d3c5ffc101e66b806b69d92b19c42fbabbaceeabed5d1543e109f1c","target":"record","created_at":"2026-05-18T03:06:04Z","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":"5ba76a762fb9b51f04fdc5fbb1920c4daff319c25e473abc795ccce16898453b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-12-08T18:18:53Z","title_canon_sha256":"2ee7f7e9047a9e9f4ccbc54106588a7b00329fa95b40ed7dd7469ee6c54108a0"},"schema_version":"1.0","source":{"id":"0812.1528","kind":"arxiv","version":2}},"canonical_sha256":"eac2881ab0d3a90107d8361a043fee1a88b91e0bf9c297d78d418559e699756c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eac2881ab0d3a90107d8361a043fee1a88b91e0bf9c297d78d418559e699756c","first_computed_at":"2026-05-18T03:06:04.012907Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:06:04.012907Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XNFAPx1GGsQBaggQwCfhC+GUy3+GRihYplXZS2o4yXaffsP7va2nLtjU+2RX1a8HKV1okUpd3GWX5YnHuja+DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:06:04.013531Z","signed_message":"canonical_sha256_bytes"},"source_id":"0812.1528","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ae0432265d3c5ffc101e66b806b69d92b19c42fbabbaceeabed5d1543e109f1c","sha256:7e65aa3ec6be152d711938732d1498a82e091b4b016206118e0f08cb2024e792"],"state_sha256":"22d98b8d6d23fd3e93e8ac9a43eeec949e6da1e2455e08c3a1be399c92ac60e4"}