{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:VV2GQW4BWLVSYLZRXSX72TCNK7","short_pith_number":"pith:VV2GQW4B","canonical_record":{"source":{"id":"1103.2415","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-03-12T03:38:26Z","cross_cats_sorted":[],"title_canon_sha256":"c6ec2c2eb06c03a85a088c856d9580f3438b6432fb5bd6199fbc321e98ca801a","abstract_canon_sha256":"eec4fe4158fdd3b560604918cabf8584a6367f15d873ebce53a55dd59f5cc0ec"},"schema_version":"1.0"},"canonical_sha256":"ad74685b81b2eb2c2f31bcaffd4c4d57dfb929ebce548db7404a06721730089f","source":{"kind":"arxiv","id":"1103.2415","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.2415","created_at":"2026-05-18T04:26:55Z"},{"alias_kind":"arxiv_version","alias_value":"1103.2415v1","created_at":"2026-05-18T04:26:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.2415","created_at":"2026-05-18T04:26:55Z"},{"alias_kind":"pith_short_12","alias_value":"VV2GQW4BWLVS","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"VV2GQW4BWLVSYLZR","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"VV2GQW4B","created_at":"2026-05-18T12:26:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:VV2GQW4BWLVSYLZRXSX72TCNK7","target":"record","payload":{"canonical_record":{"source":{"id":"1103.2415","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-03-12T03:38:26Z","cross_cats_sorted":[],"title_canon_sha256":"c6ec2c2eb06c03a85a088c856d9580f3438b6432fb5bd6199fbc321e98ca801a","abstract_canon_sha256":"eec4fe4158fdd3b560604918cabf8584a6367f15d873ebce53a55dd59f5cc0ec"},"schema_version":"1.0"},"canonical_sha256":"ad74685b81b2eb2c2f31bcaffd4c4d57dfb929ebce548db7404a06721730089f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:26:55.321169Z","signature_b64":"1gwAl/sGX1rN1Ud59A9yuaLsB+jnF9RR3RqGUoAjwDoj7V+1D84cMPHQ9oQLXfl+EQdAYIw8s4xweHRkkSd9Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ad74685b81b2eb2c2f31bcaffd4c4d57dfb929ebce548db7404a06721730089f","last_reissued_at":"2026-05-18T04:26:55.320583Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:26:55.320583Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1103.2415","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":"2026-05-18T04:26:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zwBX039UUYbaeBZ6F/al5etVrjTzrX1TkoSfrwoGFCJMes+jHSE+CTM+mb4NM85z6PMT2hwo9OtqOJOLVswpBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T15:04:26.393253Z"},"content_sha256":"7459dcc2296c69bd3e3c49b03c0601fc54877accd580e33a4a404561ea50fadf","schema_version":"1.0","event_id":"sha256:7459dcc2296c69bd3e3c49b03c0601fc54877accd580e33a4a404561ea50fadf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:VV2GQW4BWLVSYLZRXSX72TCNK7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the 3-$\\gamma_t$-Critical Graphs of Order $\\Delta(G)+3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Haoli Wang, Lei Wang, Xirong Xu, Yang Yuansheng","submitted_at":"2011-03-12T03:38:26Z","abstract_excerpt":"Let $\\gamma_t(G)$ be the total domination number of graph $G$, a graph $G$ is $k$-total domination vertex critical (or\\ just\\ $k$-$\\gamma_t$-critical) if $\\gamma_t(G)=k$, and for any vertex $v$ of $G$ that is not adjacent to a vertex of degree one, $\\gamma_t(G-v)=k-1$. Mojdeh and Rad \\cite{MR06} proposed an open problem: Does there exist a 3-$\\gamma_t$-critical graph $G$ of order $\\Delta(G)+3$ with $\\Delta(G)$ odd? In this paper, we prove that there exists a 3-$\\gamma_t$-critical graph $G$ of order $\\Delta(G)+3$ with odd $\\Delta(G)\\geq 9$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.2415","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":"2026-05-18T04:26:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZWni6EclMYHCa2X75g++P9LlUzkuB8Vv7+45s6KvP5aamDb9ezXscdVgYgDzidCkt+d7H/LQYx4obgTIIuYtDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T15:04:26.393618Z"},"content_sha256":"f45c2c9a2f30e496911bb8f73de0f7775b2b2c94022eeccd30b9b6c3a424000a","schema_version":"1.0","event_id":"sha256:f45c2c9a2f30e496911bb8f73de0f7775b2b2c94022eeccd30b9b6c3a424000a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VV2GQW4BWLVSYLZRXSX72TCNK7/bundle.json","state_url":"https://pith.science/pith/VV2GQW4BWLVSYLZRXSX72TCNK7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VV2GQW4BWLVSYLZRXSX72TCNK7/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":"2026-06-28T15:04:26Z","links":{"resolver":"https://pith.science/pith/VV2GQW4BWLVSYLZRXSX72TCNK7","bundle":"https://pith.science/pith/VV2GQW4BWLVSYLZRXSX72TCNK7/bundle.json","state":"https://pith.science/pith/VV2GQW4BWLVSYLZRXSX72TCNK7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VV2GQW4BWLVSYLZRXSX72TCNK7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:VV2GQW4BWLVSYLZRXSX72TCNK7","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":"eec4fe4158fdd3b560604918cabf8584a6367f15d873ebce53a55dd59f5cc0ec","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-03-12T03:38:26Z","title_canon_sha256":"c6ec2c2eb06c03a85a088c856d9580f3438b6432fb5bd6199fbc321e98ca801a"},"schema_version":"1.0","source":{"id":"1103.2415","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.2415","created_at":"2026-05-18T04:26:55Z"},{"alias_kind":"arxiv_version","alias_value":"1103.2415v1","created_at":"2026-05-18T04:26:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.2415","created_at":"2026-05-18T04:26:55Z"},{"alias_kind":"pith_short_12","alias_value":"VV2GQW4BWLVS","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"VV2GQW4BWLVSYLZR","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"VV2GQW4B","created_at":"2026-05-18T12:26:44Z"}],"graph_snapshots":[{"event_id":"sha256:f45c2c9a2f30e496911bb8f73de0f7775b2b2c94022eeccd30b9b6c3a424000a","target":"graph","created_at":"2026-05-18T04:26:55Z","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 $\\gamma_t(G)$ be the total domination number of graph $G$, a graph $G$ is $k$-total domination vertex critical (or\\ just\\ $k$-$\\gamma_t$-critical) if $\\gamma_t(G)=k$, and for any vertex $v$ of $G$ that is not adjacent to a vertex of degree one, $\\gamma_t(G-v)=k-1$. Mojdeh and Rad \\cite{MR06} proposed an open problem: Does there exist a 3-$\\gamma_t$-critical graph $G$ of order $\\Delta(G)+3$ with $\\Delta(G)$ odd? In this paper, we prove that there exists a 3-$\\gamma_t$-critical graph $G$ of order $\\Delta(G)+3$ with odd $\\Delta(G)\\geq 9$.","authors_text":"Haoli Wang, Lei Wang, Xirong Xu, Yang Yuansheng","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-03-12T03:38:26Z","title":"On the 3-$\\gamma_t$-Critical Graphs of Order $\\Delta(G)+3$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.2415","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":"sha256:7459dcc2296c69bd3e3c49b03c0601fc54877accd580e33a4a404561ea50fadf","target":"record","created_at":"2026-05-18T04:26:55Z","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":"eec4fe4158fdd3b560604918cabf8584a6367f15d873ebce53a55dd59f5cc0ec","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-03-12T03:38:26Z","title_canon_sha256":"c6ec2c2eb06c03a85a088c856d9580f3438b6432fb5bd6199fbc321e98ca801a"},"schema_version":"1.0","source":{"id":"1103.2415","kind":"arxiv","version":1}},"canonical_sha256":"ad74685b81b2eb2c2f31bcaffd4c4d57dfb929ebce548db7404a06721730089f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ad74685b81b2eb2c2f31bcaffd4c4d57dfb929ebce548db7404a06721730089f","first_computed_at":"2026-05-18T04:26:55.320583Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:26:55.320583Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1gwAl/sGX1rN1Ud59A9yuaLsB+jnF9RR3RqGUoAjwDoj7V+1D84cMPHQ9oQLXfl+EQdAYIw8s4xweHRkkSd9Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:26:55.321169Z","signed_message":"canonical_sha256_bytes"},"source_id":"1103.2415","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7459dcc2296c69bd3e3c49b03c0601fc54877accd580e33a4a404561ea50fadf","sha256:f45c2c9a2f30e496911bb8f73de0f7775b2b2c94022eeccd30b9b6c3a424000a"],"state_sha256":"17cafa4460cc4ce913351a1baacc69dbc42f362444f0eae828a91ea374e7975a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RL7rQZoY1Cs0bDNCaggRFqZZe10/XLBM69x2+O7ecm/BksRp/AoYPPZnnd9sjq/Vf3T97FoO/gWF81zjbsfxAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T15:04:26.395516Z","bundle_sha256":"1b8d25be91b2995d0ca7a86dad8c29a4fcfd45f96e7a5a48337179071ea534b0"}}