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Second, $G$ is strongly $H$-free if $G$ is $H$-free or else has no bipartition $(B_G,W_G)$ with $B_H\\subseteq B_G$ and $W_H\\subseteq W_G$. Third, $G$ is weakly $H$-free if $G$ is $H$-free or else has at least one bipartition $(B_G,W_G)$ with $B_H\\not\\subseteq B_G$ or $W_H\\not\\subseteq W_G$. Lozin and Volz characterized all bipartite"},"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":"1402.7060","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2014-02-27T20:41:54Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"7a8b4e93b8086ba6ac5c4fed2cc038689b6b5b28bf217ee87d56c108da5d8545","abstract_canon_sha256":"84d6dca6ac1dfc0bb80423dc3c7b8a68b726341344e04f19afa212ff6ea7fd5b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:36.359673Z","signature_b64":"RKhVpGwSgIQgznpSDIVvWXVNsoZHvWg+oMcQo4HWHXCN4bzbtziWDc4wPgUx6Hi2AbwWWoh1cnkZ5UxYcbF4BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e48c91f23bd67770633a95b3fed380f70e7c17fbc37fc70d999151c3e57fdf49","last_reissued_at":"2026-05-18T02:57:36.359189Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:36.359189Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Classifying the Clique-Width of $H$-Free Bipartite Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Dani\\\"el Paulusma, Konrad K. Dabrowski","submitted_at":"2014-02-27T20:41:54Z","abstract_excerpt":"Let $G$ be a bipartite graph, and let $H$ be a bipartite graph with a fixed bipartition $(B_H,W_H)$. We consider three different, natural ways of forbidding $H$ as an induced subgraph in $G$. First, $G$ is $H$-free if it does not contain $H$ as an induced subgraph. Second, $G$ is strongly $H$-free if $G$ is $H$-free or else has no bipartition $(B_G,W_G)$ with $B_H\\subseteq B_G$ and $W_H\\subseteq W_G$. Third, $G$ is weakly $H$-free if $G$ is $H$-free or else has at least one bipartition $(B_G,W_G)$ with $B_H\\not\\subseteq B_G$ or $W_H\\not\\subseteq W_G$. Lozin and Volz characterized all bipartite"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.7060","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":"1402.7060","created_at":"2026-05-18T02:57:36.359260+00:00"},{"alias_kind":"arxiv_version","alias_value":"1402.7060v1","created_at":"2026-05-18T02:57:36.359260+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.7060","created_at":"2026-05-18T02:57:36.359260+00:00"},{"alias_kind":"pith_short_12","alias_value":"4SGJD4R32Z3X","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_16","alias_value":"4SGJD4R32Z3XAYZ2","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_8","alias_value":"4SGJD4R3","created_at":"2026-05-18T12:28:14.216126+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/4SGJD4R32Z3XAYZ2SWZ75U4A64","json":"https://pith.science/pith/4SGJD4R32Z3XAYZ2SWZ75U4A64.json","graph_json":"https://pith.science/api/pith-number/4SGJD4R32Z3XAYZ2SWZ75U4A64/graph.json","events_json":"https://pith.science/api/pith-number/4SGJD4R32Z3XAYZ2SWZ75U4A64/events.json","paper":"https://pith.science/paper/4SGJD4R3"},"agent_actions":{"view_html":"https://pith.science/pith/4SGJD4R32Z3XAYZ2SWZ75U4A64","download_json":"https://pith.science/pith/4SGJD4R32Z3XAYZ2SWZ75U4A64.json","view_paper":"https://pith.science/paper/4SGJD4R3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1402.7060&json=true","fetch_graph":"https://pith.science/api/pith-number/4SGJD4R32Z3XAYZ2SWZ75U4A64/graph.json","fetch_events":"https://pith.science/api/pith-number/4SGJD4R32Z3XAYZ2SWZ75U4A64/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4SGJD4R32Z3XAYZ2SWZ75U4A64/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4SGJD4R32Z3XAYZ2SWZ75U4A64/action/storage_attestation","attest_author":"https://pith.science/pith/4SGJD4R32Z3XAYZ2SWZ75U4A64/action/author_attestation","sign_citation":"https://pith.science/pith/4SGJD4R32Z3XAYZ2SWZ75U4A64/action/citation_signature","submit_replication":"https://pith.science/pith/4SGJD4R32Z3XAYZ2SWZ75U4A64/action/replication_record"}},"created_at":"2026-05-18T02:57:36.359260+00:00","updated_at":"2026-05-18T02:57:36.359260+00:00"}