{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:7K4NP66QIC6CTBROAC6MVY32UQ","short_pith_number":"pith:7K4NP66Q","schema_version":"1.0","canonical_sha256":"fab8d7fbd040bc29862e00bccae37aa4038372fabe398e77e3061d2986b54d02","source":{"kind":"arxiv","id":"1108.5949","version":1},"attestation_state":"computed","paper":{"title":"Equality in a Linear Vizing-Like Relation that Relates the Size and Total Domination Number of a Graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ernst J. Joubert, Michael A. Henning","submitted_at":"2011-08-30T12:54:07Z","abstract_excerpt":"Let $G$ be a graph each component of which has order at least 3, and let $G$ have order $n$, size $m$, total domination number $\\gamma_t$ and maximum degree $\\Delta(G)$. Let $\\Delta = 3$ if $\\Delta(G) = 2$ and $\\Delta = \\Delta (G)$ if $\\Delta(G) \\ge 3$. It is known [J. Graph Theory 49 (2005), 285--290; J. Graph Theory 54 (2007), 350--353] that $m \\le \\Delta (n- \\gamma_t)$. In this paper we characterize the extremal graphs $G$ satisfying $m = \\Delta (n- \\gamma_t)$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1108.5949","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-08-30T12:54:07Z","cross_cats_sorted":[],"title_canon_sha256":"125634e37635fbf74ce201f873e68a542f6a628dfb8807e3017c5fa28dbae58d","abstract_canon_sha256":"566a30eb8b343bc32b19d87a5901d4f2ddc0fa582bd3e6c2c0c4a7e552708b9c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:14:28.188387Z","signature_b64":"Fhb1fZjwqMjTUJlM7R/aJtOM6QXhVqfQY3ik9oadsnPHJANNMbT3NMPCu5yhwLEQ1dh0ZRvi7LqTE8APKLP5Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fab8d7fbd040bc29862e00bccae37aa4038372fabe398e77e3061d2986b54d02","last_reissued_at":"2026-05-18T04:14:28.187215Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:14:28.187215Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Equality in a Linear Vizing-Like Relation that Relates the Size and Total Domination Number of a Graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ernst J. Joubert, Michael A. Henning","submitted_at":"2011-08-30T12:54:07Z","abstract_excerpt":"Let $G$ be a graph each component of which has order at least 3, and let $G$ have order $n$, size $m$, total domination number $\\gamma_t$ and maximum degree $\\Delta(G)$. Let $\\Delta = 3$ if $\\Delta(G) = 2$ and $\\Delta = \\Delta (G)$ if $\\Delta(G) \\ge 3$. It is known [J. Graph Theory 49 (2005), 285--290; J. Graph Theory 54 (2007), 350--353] that $m \\le \\Delta (n- \\gamma_t)$. In this paper we characterize the extremal graphs $G$ satisfying $m = \\Delta (n- \\gamma_t)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.5949","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1108.5949","created_at":"2026-05-18T04:14:28.187349+00:00"},{"alias_kind":"arxiv_version","alias_value":"1108.5949v1","created_at":"2026-05-18T04:14:28.187349+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.5949","created_at":"2026-05-18T04:14:28.187349+00:00"},{"alias_kind":"pith_short_12","alias_value":"7K4NP66QIC6C","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_16","alias_value":"7K4NP66QIC6CTBRO","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_8","alias_value":"7K4NP66Q","created_at":"2026-05-18T12:26:22.705136+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7K4NP66QIC6CTBROAC6MVY32UQ","json":"https://pith.science/pith/7K4NP66QIC6CTBROAC6MVY32UQ.json","graph_json":"https://pith.science/api/pith-number/7K4NP66QIC6CTBROAC6MVY32UQ/graph.json","events_json":"https://pith.science/api/pith-number/7K4NP66QIC6CTBROAC6MVY32UQ/events.json","paper":"https://pith.science/paper/7K4NP66Q"},"agent_actions":{"view_html":"https://pith.science/pith/7K4NP66QIC6CTBROAC6MVY32UQ","download_json":"https://pith.science/pith/7K4NP66QIC6CTBROAC6MVY32UQ.json","view_paper":"https://pith.science/paper/7K4NP66Q","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1108.5949&json=true","fetch_graph":"https://pith.science/api/pith-number/7K4NP66QIC6CTBROAC6MVY32UQ/graph.json","fetch_events":"https://pith.science/api/pith-number/7K4NP66QIC6CTBROAC6MVY32UQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7K4NP66QIC6CTBROAC6MVY32UQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7K4NP66QIC6CTBROAC6MVY32UQ/action/storage_attestation","attest_author":"https://pith.science/pith/7K4NP66QIC6CTBROAC6MVY32UQ/action/author_attestation","sign_citation":"https://pith.science/pith/7K4NP66QIC6CTBROAC6MVY32UQ/action/citation_signature","submit_replication":"https://pith.science/pith/7K4NP66QIC6CTBROAC6MVY32UQ/action/replication_record"}},"created_at":"2026-05-18T04:14:28.187349+00:00","updated_at":"2026-05-18T04:14:28.187349+00:00"}