{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:YKECDKKDHY4UVXR5LXRZZG2TES","short_pith_number":"pith:YKECDKKD","schema_version":"1.0","canonical_sha256":"c28821a9433e394ade3d5de39c9b5324b73c7301885cc6283179d804b6feacfd","source":{"kind":"arxiv","id":"2503.04228","version":1},"attestation_state":"computed","paper":{"title":"Polynomial Bounds in the Apex Minor Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"David R. Wood, Kevin Hendrey","submitted_at":"2025-03-06T09:08:45Z","abstract_excerpt":"A graph $A$ is \"apex\" if $A-z$ is planar for some vertex $z\\in V(A)$. Eppstein [Algorithmica, 2000] showed that for a minor-closed class $\\mathcal{G}$, the graphs in $\\mathcal{G}$ with bounded radius have bounded treewidth if and only if some apex graph is not in $\\mathcal{G}$. In particular, for every apex graph $A$ and integer $r$, there is a minimum integer $g(A,r)$ such that every $A$-minor-free graph with radius $r$ has treewidth at most $g(A,r)$. We show that if $t=|V(A)|$ then $g(A,r)\\in O^\\ast(r^9t^{18})$ which is the first upper bound on $g(A,r)$ with polynomial dependence on both $r$"},"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":"2503.04228","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-03-06T09:08:45Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"2a6e94fb241ee547d41faf9ab76abe93fe5cdf9199c86efd33891f83bc3e2b3e","abstract_canon_sha256":"880f56106f8ab2be478b3bd4a1f71920f17032169c21e9abed2398cd5a65dd97"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:25:32.835713Z","signature_b64":"ucr9M5BBURwDwdcMgn+V2grO7KVXr8nRXgGi/fQnUixn4Ze7nQ+wOHc1BDnzb/uxmPEwUvl+gvNdSKPjOJ4tDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c28821a9433e394ade3d5de39c9b5324b73c7301885cc6283179d804b6feacfd","last_reissued_at":"2026-07-05T10:25:32.834531Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:25:32.834531Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Polynomial Bounds in the Apex Minor Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"David R. Wood, Kevin Hendrey","submitted_at":"2025-03-06T09:08:45Z","abstract_excerpt":"A graph $A$ is \"apex\" if $A-z$ is planar for some vertex $z\\in V(A)$. Eppstein [Algorithmica, 2000] showed that for a minor-closed class $\\mathcal{G}$, the graphs in $\\mathcal{G}$ with bounded radius have bounded treewidth if and only if some apex graph is not in $\\mathcal{G}$. In particular, for every apex graph $A$ and integer $r$, there is a minimum integer $g(A,r)$ such that every $A$-minor-free graph with radius $r$ has treewidth at most $g(A,r)$. We show that if $t=|V(A)|$ then $g(A,r)\\in O^\\ast(r^9t^{18})$ which is the first upper bound on $g(A,r)$ with polynomial dependence on both $r$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.04228","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2503.04228/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2503.04228","created_at":"2026-07-05T10:25:32.834684+00:00"},{"alias_kind":"arxiv_version","alias_value":"2503.04228v1","created_at":"2026-07-05T10:25:32.834684+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2503.04228","created_at":"2026-07-05T10:25:32.834684+00:00"},{"alias_kind":"pith_short_12","alias_value":"YKECDKKDHY4U","created_at":"2026-07-05T10:25:32.834684+00:00"},{"alias_kind":"pith_short_16","alias_value":"YKECDKKDHY4UVXR5","created_at":"2026-07-05T10:25:32.834684+00:00"},{"alias_kind":"pith_short_8","alias_value":"YKECDKKD","created_at":"2026-07-05T10:25:32.834684+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2607.06869","citing_title":"An Erd\\H{o}s-P\\'osa theorem for cycles and faces of distinct lengths","ref_index":11,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/YKECDKKDHY4UVXR5LXRZZG2TES","json":"https://pith.science/pith/YKECDKKDHY4UVXR5LXRZZG2TES.json","graph_json":"https://pith.science/api/pith-number/YKECDKKDHY4UVXR5LXRZZG2TES/graph.json","events_json":"https://pith.science/api/pith-number/YKECDKKDHY4UVXR5LXRZZG2TES/events.json","paper":"https://pith.science/paper/YKECDKKD"},"agent_actions":{"view_html":"https://pith.science/pith/YKECDKKDHY4UVXR5LXRZZG2TES","download_json":"https://pith.science/pith/YKECDKKDHY4UVXR5LXRZZG2TES.json","view_paper":"https://pith.science/paper/YKECDKKD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2503.04228&json=true","fetch_graph":"https://pith.science/api/pith-number/YKECDKKDHY4UVXR5LXRZZG2TES/graph.json","fetch_events":"https://pith.science/api/pith-number/YKECDKKDHY4UVXR5LXRZZG2TES/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YKECDKKDHY4UVXR5LXRZZG2TES/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YKECDKKDHY4UVXR5LXRZZG2TES/action/storage_attestation","attest_author":"https://pith.science/pith/YKECDKKDHY4UVXR5LXRZZG2TES/action/author_attestation","sign_citation":"https://pith.science/pith/YKECDKKDHY4UVXR5LXRZZG2TES/action/citation_signature","submit_replication":"https://pith.science/pith/YKECDKKDHY4UVXR5LXRZZG2TES/action/replication_record"}},"created_at":"2026-07-05T10:25:32.834684+00:00","updated_at":"2026-07-05T10:25:32.834684+00:00"}