{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:T6PLPOAFR2DJKXSAW3WQB5TBXU","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":"fb3101eb3a0b735673015bba5f8d54915617cf7344abdd94614c708b4b077a7b","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-04-21T16:58:00Z","title_canon_sha256":"9c27daa907e1704a9ac7c2aad7d5dfb9a66e201957e8f74a917627c79ba6e4a5"},"schema_version":"1.0","source":{"id":"1204.4827","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.4827","created_at":"2026-05-18T03:57:15Z"},{"alias_kind":"arxiv_version","alias_value":"1204.4827v1","created_at":"2026-05-18T03:57:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.4827","created_at":"2026-05-18T03:57:15Z"},{"alias_kind":"pith_short_12","alias_value":"T6PLPOAFR2DJ","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"T6PLPOAFR2DJKXSA","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"T6PLPOAF","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:86ded5323847b90f789de30b1166de4b734c67fb589e251fb0a5052be2c9f887","target":"graph","created_at":"2026-05-18T03:57:15Z","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 $k$ be a positive integer and $G=(V,E)$ be a graph of minimum degree at least $k-1$. A function $f:V\\rightarrow \\{-1,1\\}$ is called a \\emph{signed $k$-dominating function} of $G$ if $\\sum_{u\\in N_G[v]}f(u)\\geq k$ for all $v\\in V$. The \\emph{signed $k$-domination number} of $G$ is the minimum value of $\\sum_{v\\in V}f(v)$ taken over all signed $k$-dominating functions of $G$. The \\emph{signed total $k$-dominating function} and \\emph{signed total $k$-domination number} of $G$ can be similarly defined by changing the closed neighborhood $N_G[v]$ to the open neighborhood $N_G(v)$ in the definit","authors_text":"Hongyu Liang","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-04-21T16:58:00Z","title":"On the Signed (Total) $k$-Domination Number of a Graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.4827","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":"sha256:17443464c3ef775f14fa929d20c24357cd176a7a4dca78de26d0b1c985c86b01","target":"record","created_at":"2026-05-18T03:57:15Z","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":"fb3101eb3a0b735673015bba5f8d54915617cf7344abdd94614c708b4b077a7b","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-04-21T16:58:00Z","title_canon_sha256":"9c27daa907e1704a9ac7c2aad7d5dfb9a66e201957e8f74a917627c79ba6e4a5"},"schema_version":"1.0","source":{"id":"1204.4827","kind":"arxiv","version":1}},"canonical_sha256":"9f9eb7b8058e86955e40b6ed00f661bd227ee990d54d02ba4dad64f6044aec39","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9f9eb7b8058e86955e40b6ed00f661bd227ee990d54d02ba4dad64f6044aec39","first_computed_at":"2026-05-18T03:57:15.306295Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:57:15.306295Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0up1OrAx4P+xb7PMi+BAohzhmTbFSbcmnA1lND0Wf7IEvaQRZ1kixfHcn9dOM5+Arfx2J0F+hodm2Ijw+bZ8Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:57:15.306753Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.4827","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:17443464c3ef775f14fa929d20c24357cd176a7a4dca78de26d0b1c985c86b01","sha256:86ded5323847b90f789de30b1166de4b734c67fb589e251fb0a5052be2c9f887"],"state_sha256":"c4b12ac2c3df22ea080fe555c870d7d6bc623eea8c9fb9615ca29c35069b372e"}