{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:G5WBXXMSAPZCPJHDKCJB2YQOCQ","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":"c7fff9c210025520be2526991c17f53944a346d70cabe99669581de4bb3c5e5a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-12T19:49:33Z","title_canon_sha256":"f229e8a5c0264f0518d90e283418b1e2420a31b35e04f473631b59ebb0ed1d3c"},"schema_version":"1.0","source":{"id":"1705.04735","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.04735","created_at":"2026-05-18T00:20:44Z"},{"alias_kind":"arxiv_version","alias_value":"1705.04735v2","created_at":"2026-05-18T00:20:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.04735","created_at":"2026-05-18T00:20:44Z"},{"alias_kind":"pith_short_12","alias_value":"G5WBXXMSAPZC","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_16","alias_value":"G5WBXXMSAPZCPJHD","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_8","alias_value":"G5WBXXMS","created_at":"2026-05-18T12:31:15Z"}],"graph_snapshots":[{"event_id":"sha256:d2a1f34387ed6c6c1391d8ece4b195e4fff94a77d531813c830408a09c490b6b","target":"graph","created_at":"2026-05-18T00:20:44Z","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":"A vertex $v$ of a graph $G=(V,E)$ is said to be undefended with respect to a function $f: V \\longrightarrow \\{0,1,2\\}$ if $f(v)=0$ and $f(u)=0$ for every vertex $u$ adjacent to $v$. We call the function $f$ a weak Roman dominating function if for every $v$ such that $f(v)=0$ there exists a vertex $u$ adjacent to $v$ such that $f(u)\\in \\{1,2\\}$ and the function $f': V \\longrightarrow \\{0,1,2\\}$ defined by $f'(v)=1$, $f'(u)=f(u)-1$ and $f'(z)=f(z)$ for every $z\\in V \\setminus\\{u,v\\}$, has no undefended vertices. The weight of $f$ is $w(f)=\\sum_{v\\in V(G) }f(v)$. The weak Roman domination number ","authors_text":"Hebert P\\'erez-Ros\\'es, Juan A. Rodr\\'iguez-Vel\\'azquez, Magdalena Valveny","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-12T19:49:33Z","title":"On the weak Roman domination number of lexicographic product graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.04735","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":"sha256:88a15f5cd2ec4616965eeefc9b2e01712f3420c25f8068c57309843fc488f7b5","target":"record","created_at":"2026-05-18T00:20:44Z","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":"c7fff9c210025520be2526991c17f53944a346d70cabe99669581de4bb3c5e5a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-12T19:49:33Z","title_canon_sha256":"f229e8a5c0264f0518d90e283418b1e2420a31b35e04f473631b59ebb0ed1d3c"},"schema_version":"1.0","source":{"id":"1705.04735","kind":"arxiv","version":2}},"canonical_sha256":"376c1bdd9203f227a4e350921d620e141daec1ae419c0311fa1a4e114fab6db0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"376c1bdd9203f227a4e350921d620e141daec1ae419c0311fa1a4e114fab6db0","first_computed_at":"2026-05-18T00:20:44.672371Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:20:44.672371Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4mZXqZCwSu8CCjce3MqRK6GQeAQfOImXK3zoA3AkI2LMT221f9x1VTXH9lraOSXFJSJJKlS3OitQXy6+usFwDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:20:44.673021Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.04735","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:88a15f5cd2ec4616965eeefc9b2e01712f3420c25f8068c57309843fc488f7b5","sha256:d2a1f34387ed6c6c1391d8ece4b195e4fff94a77d531813c830408a09c490b6b"],"state_sha256":"128a899fd1022e662bb8f2fbb32285e7953e9d59b37327b54d6e02e0111e8c95"}