{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:2W6B2YCDYZ6ZTAMVRRQG4D7UCA","short_pith_number":"pith:2W6B2YCD","schema_version":"1.0","canonical_sha256":"d5bc1d6043c67d9981958c606e0ff4101bb20a503e3c85ac2065906866897ab3","source":{"kind":"arxiv","id":"1102.3760","version":1},"attestation_state":"computed","paper":{"title":"Rooted $K_4$-Minors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David R. Wood, Ruy Fabila-Monroy","submitted_at":"2011-02-18T06:02:14Z","abstract_excerpt":"Let $a,b,c,d$ be four vertices in a graph $G$. A \\emph{$K_4$-minor rooted} at $a,b,c,d$ consists of four pairwise-disjoint pairwise-adjacent connected subgraphs of $G$, respectively containing $a,b,c,d$. We characterise precisely when $G$ contains a $K_4$-minor rooted at $a,b,c,d$ by describing six classes of obstructions, which are the edge-maximal graphs containing no $K_4$-minor rooted at $a,b,c,d$. The following two special cases illustrate the full characterisation: (1) A 4-connected non-planar graph contains a $K_4$-minor rooted at $a,b,c,d$ for every choice of $a,b,c,d$. (2) A 3-connect"},"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":"1102.3760","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-02-18T06:02:14Z","cross_cats_sorted":[],"title_canon_sha256":"d315c4a23e55c3a5d6f9cdc0905c341e40970637f4cfbaa5fd0e6ff212593a91","abstract_canon_sha256":"8b7c2d97026e1787a2278089adbdd2189563d971f9abf90a7c0ba0648b1387a4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:11:47.451492Z","signature_b64":"MU8TF3xiYZs72UHpBbOCx/ujufAq2OjjD/mobyEoQaA3n/o7Mk1OmaKeARD1Ll9QN8dbGspXLxDxa+jFBAZnBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d5bc1d6043c67d9981958c606e0ff4101bb20a503e3c85ac2065906866897ab3","last_reissued_at":"2026-05-18T03:11:47.450918Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:11:47.450918Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rooted $K_4$-Minors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David R. Wood, Ruy Fabila-Monroy","submitted_at":"2011-02-18T06:02:14Z","abstract_excerpt":"Let $a,b,c,d$ be four vertices in a graph $G$. A \\emph{$K_4$-minor rooted} at $a,b,c,d$ consists of four pairwise-disjoint pairwise-adjacent connected subgraphs of $G$, respectively containing $a,b,c,d$. We characterise precisely when $G$ contains a $K_4$-minor rooted at $a,b,c,d$ by describing six classes of obstructions, which are the edge-maximal graphs containing no $K_4$-minor rooted at $a,b,c,d$. The following two special cases illustrate the full characterisation: (1) A 4-connected non-planar graph contains a $K_4$-minor rooted at $a,b,c,d$ for every choice of $a,b,c,d$. (2) A 3-connect"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.3760","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":"1102.3760","created_at":"2026-05-18T03:11:47.451006+00:00"},{"alias_kind":"arxiv_version","alias_value":"1102.3760v1","created_at":"2026-05-18T03:11:47.451006+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.3760","created_at":"2026-05-18T03:11:47.451006+00:00"},{"alias_kind":"pith_short_12","alias_value":"2W6B2YCDYZ6Z","created_at":"2026-05-18T12:26:18.847500+00:00"},{"alias_kind":"pith_short_16","alias_value":"2W6B2YCDYZ6ZTAMV","created_at":"2026-05-18T12:26:18.847500+00:00"},{"alias_kind":"pith_short_8","alias_value":"2W6B2YCD","created_at":"2026-05-18T12:26:18.847500+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/2W6B2YCDYZ6ZTAMVRRQG4D7UCA","json":"https://pith.science/pith/2W6B2YCDYZ6ZTAMVRRQG4D7UCA.json","graph_json":"https://pith.science/api/pith-number/2W6B2YCDYZ6ZTAMVRRQG4D7UCA/graph.json","events_json":"https://pith.science/api/pith-number/2W6B2YCDYZ6ZTAMVRRQG4D7UCA/events.json","paper":"https://pith.science/paper/2W6B2YCD"},"agent_actions":{"view_html":"https://pith.science/pith/2W6B2YCDYZ6ZTAMVRRQG4D7UCA","download_json":"https://pith.science/pith/2W6B2YCDYZ6ZTAMVRRQG4D7UCA.json","view_paper":"https://pith.science/paper/2W6B2YCD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1102.3760&json=true","fetch_graph":"https://pith.science/api/pith-number/2W6B2YCDYZ6ZTAMVRRQG4D7UCA/graph.json","fetch_events":"https://pith.science/api/pith-number/2W6B2YCDYZ6ZTAMVRRQG4D7UCA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2W6B2YCDYZ6ZTAMVRRQG4D7UCA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2W6B2YCDYZ6ZTAMVRRQG4D7UCA/action/storage_attestation","attest_author":"https://pith.science/pith/2W6B2YCDYZ6ZTAMVRRQG4D7UCA/action/author_attestation","sign_citation":"https://pith.science/pith/2W6B2YCDYZ6ZTAMVRRQG4D7UCA/action/citation_signature","submit_replication":"https://pith.science/pith/2W6B2YCDYZ6ZTAMVRRQG4D7UCA/action/replication_record"}},"created_at":"2026-05-18T03:11:47.451006+00:00","updated_at":"2026-05-18T03:11:47.451006+00:00"}