{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:I2KCDVZLTCKN3QSLAVKTABEWHC","short_pith_number":"pith:I2KCDVZL","schema_version":"1.0","canonical_sha256":"469421d72b9894ddc24b0555300496389c47e0046e692744d8311d8187ef5ce6","source":{"kind":"arxiv","id":"2001.08860","version":2},"attestation_state":"computed","paper":{"title":"Notes on Graph Product Structure Theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","cs.DM"],"primary_cat":"math.CO","authors_text":"Chun-Hung Liu, David R. Wood, Gwena\\\"el Joret, Tony Huynh, Zden\\v{e}k Dvo\\v{r}\\'ak","submitted_at":"2020-01-24T01:45:42Z","abstract_excerpt":"It was recently proved that every planar graph is a subgraph of the strong product of a path and a graph with bounded treewidth. This paper surveys generalisations of this result for graphs on surfaces, minor-closed classes, various non-minor-closed classes, and graph classes with polynomial growth. We then explore how graph product structure might be applicable to more broadly defined graph classes. In particular, we characterise when a graph class defined by a cartesian or strong product has bounded or polynomial expansion. We then explore graph product structure theorems for various geometr"},"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":"2001.08860","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-01-24T01:45:42Z","cross_cats_sorted":["cs.CG","cs.DM"],"title_canon_sha256":"5647808166a3ba67433f416f8fec03c16f701537a3203de2741755f655d8e3e3","abstract_canon_sha256":"99bef6b2fe67ff61151418f300bdc6d7263a28591c39ae3269fc23f3db80ade7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:15:56.879753Z","signature_b64":"BvGk+9b8afulIo58lAYCjh7pbZIocyPHwfozoPypzxwOeYSQRFUSC7ckw/bFKM1v0fsyWe0ZBkslE1oHy57lDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"469421d72b9894ddc24b0555300496389c47e0046e692744d8311d8187ef5ce6","last_reissued_at":"2026-07-05T02:15:56.879248Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:15:56.879248Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Notes on Graph Product Structure Theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","cs.DM"],"primary_cat":"math.CO","authors_text":"Chun-Hung Liu, David R. Wood, Gwena\\\"el Joret, Tony Huynh, Zden\\v{e}k Dvo\\v{r}\\'ak","submitted_at":"2020-01-24T01:45:42Z","abstract_excerpt":"It was recently proved that every planar graph is a subgraph of the strong product of a path and a graph with bounded treewidth. This paper surveys generalisations of this result for graphs on surfaces, minor-closed classes, various non-minor-closed classes, and graph classes with polynomial growth. We then explore how graph product structure might be applicable to more broadly defined graph classes. In particular, we characterise when a graph class defined by a cartesian or strong product has bounded or polynomial expansion. We then explore graph product structure theorems for various geometr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2001.08860","kind":"arxiv","version":2},"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/2001.08860/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":"2001.08860","created_at":"2026-07-05T02:15:56.879303+00:00"},{"alias_kind":"arxiv_version","alias_value":"2001.08860v2","created_at":"2026-07-05T02:15:56.879303+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2001.08860","created_at":"2026-07-05T02:15:56.879303+00:00"},{"alias_kind":"pith_short_12","alias_value":"I2KCDVZLTCKN","created_at":"2026-07-05T02:15:56.879303+00:00"},{"alias_kind":"pith_short_16","alias_value":"I2KCDVZLTCKN3QSL","created_at":"2026-07-05T02:15:56.879303+00:00"},{"alias_kind":"pith_short_8","alias_value":"I2KCDVZL","created_at":"2026-07-05T02:15:56.879303+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2608.09495","citing_title":"Row pathwidth of complete binary trees","ref_index":3,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/I2KCDVZLTCKN3QSLAVKTABEWHC","json":"https://pith.science/pith/I2KCDVZLTCKN3QSLAVKTABEWHC.json","graph_json":"https://pith.science/api/pith-number/I2KCDVZLTCKN3QSLAVKTABEWHC/graph.json","events_json":"https://pith.science/api/pith-number/I2KCDVZLTCKN3QSLAVKTABEWHC/events.json","paper":"https://pith.science/paper/I2KCDVZL"},"agent_actions":{"view_html":"https://pith.science/pith/I2KCDVZLTCKN3QSLAVKTABEWHC","download_json":"https://pith.science/pith/I2KCDVZLTCKN3QSLAVKTABEWHC.json","view_paper":"https://pith.science/paper/I2KCDVZL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2001.08860&json=true","fetch_graph":"https://pith.science/api/pith-number/I2KCDVZLTCKN3QSLAVKTABEWHC/graph.json","fetch_events":"https://pith.science/api/pith-number/I2KCDVZLTCKN3QSLAVKTABEWHC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I2KCDVZLTCKN3QSLAVKTABEWHC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I2KCDVZLTCKN3QSLAVKTABEWHC/action/storage_attestation","attest_author":"https://pith.science/pith/I2KCDVZLTCKN3QSLAVKTABEWHC/action/author_attestation","sign_citation":"https://pith.science/pith/I2KCDVZLTCKN3QSLAVKTABEWHC/action/citation_signature","submit_replication":"https://pith.science/pith/I2KCDVZLTCKN3QSLAVKTABEWHC/action/replication_record"}},"created_at":"2026-07-05T02:15:56.879303+00:00","updated_at":"2026-07-05T02:15:56.879303+00:00"}