{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:UUYXPMCHQQKIEZXD4G32NIKXY4","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":"8af96839c5d8d505bf3a56f5a9df7fe85795e4af20c6474625c7615baa2ceb65","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-03-08T09:29:48Z","title_canon_sha256":"14165a3dd77ae05d235032939a0df03dc41f48c2bfcd64646d8d695dddb3765f"},"schema_version":"1.0","source":{"id":"1603.02433","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.02433","created_at":"2026-05-18T00:08:24Z"},{"alias_kind":"arxiv_version","alias_value":"1603.02433v2","created_at":"2026-05-18T00:08:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.02433","created_at":"2026-05-18T00:08:24Z"},{"alias_kind":"pith_short_12","alias_value":"UUYXPMCHQQKI","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"UUYXPMCHQQKIEZXD","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"UUYXPMCH","created_at":"2026-05-18T12:30:46Z"}],"graph_snapshots":[{"event_id":"sha256:eb077c1286cc1ca09202cf50edda9c3c548ac97ad4aa56caec73fc3200ee39d1","target":"graph","created_at":"2026-05-18T00:08:24Z","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":"For $k \\ge 1$ an integer, a set $S$ of vertices in a graph $G$ with minimum degree at least~$k-1$ is a $k$-tuple dominating set of $G$ if every vertex of $S$ is adjacent to at least $k-1$ vertices in $S$ and every vertex of $V(G) \\setminus S$ is adjacent to at least $k$ vertices in $S$; that is, $|N_G[v] \\cap S| \\ge k$ for every vertex $v$ of $G$ where $N_G[v]$ denotes the closed neighborhood of $v$ which consists of $v$ and all neighbors of $v$. A $k$-tuple restrained dominating set of $G$ is a $k$-tuple dominating set $S$ of $G$ with the additional property that every vertex outside $S$ has ","authors_text":"Adel P. Kazemi, M. A. Henning","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-03-08T09:29:48Z","title":"k-Tuple Restrained Domination in Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.02433","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:c9cee92a84d2c4f903664290be61f79b43d2073f9a9f159d27a993a1af62f0e3","target":"record","created_at":"2026-05-18T00:08:24Z","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":"8af96839c5d8d505bf3a56f5a9df7fe85795e4af20c6474625c7615baa2ceb65","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-03-08T09:29:48Z","title_canon_sha256":"14165a3dd77ae05d235032939a0df03dc41f48c2bfcd64646d8d695dddb3765f"},"schema_version":"1.0","source":{"id":"1603.02433","kind":"arxiv","version":2}},"canonical_sha256":"a53177b04784148266e3e1b7a6a157c7362d50c404e4a594128535e3b701d530","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a53177b04784148266e3e1b7a6a157c7362d50c404e4a594128535e3b701d530","first_computed_at":"2026-05-18T00:08:24.082822Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:08:24.082822Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qaStz54DfEbEA/efIfOmwNTag5l/8DC711AaTUdluYN3U8pDGt+SX1m0ZaWJqRaIaOHjzXuvDfG8ryvauuR7BA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:08:24.083441Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.02433","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c9cee92a84d2c4f903664290be61f79b43d2073f9a9f159d27a993a1af62f0e3","sha256:eb077c1286cc1ca09202cf50edda9c3c548ac97ad4aa56caec73fc3200ee39d1"],"state_sha256":"ca183b2b0d19b5d70381be1a831a63d50ac53a6affa870c40200391dbefa7b10"}