{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:4FOTTEENZDVWDFZT2QINAHDV6V","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d6276f069dcaae4e26ad7a767c41f40ecad2f8d2adc6682fd16f79a203814fd8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-28T20:07:19Z","title_canon_sha256":"fad9e72e520be0d4c1ad3f83740c22f48c8fc49d5a0ebbe8be8cd097f03df979"},"schema_version":"1.0","source":{"id":"1609.09098","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.09098","created_at":"2026-05-18T01:03:40Z"},{"alias_kind":"arxiv_version","alias_value":"1609.09098v1","created_at":"2026-05-18T01:03:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.09098","created_at":"2026-05-18T01:03:40Z"},{"alias_kind":"pith_short_12","alias_value":"4FOTTEENZDVW","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4FOTTEENZDVWDFZT","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4FOTTEEN","created_at":"2026-05-18T12:29:58Z"}],"graph_snapshots":[{"event_id":"sha256:ee50e512805906540dd139908471d3ba57071485087f99602c7466dffedecc1e","target":"graph","created_at":"2026-05-18T01:03:40Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"A graph has tree-width at most $k$ if it can be obtained from a set of graphs each with at most $k+1$ vertices by a sequence of clique sums. We refine this definition by, for each non-negative integer $\\theta$, defining the $\\theta$-tree-width of a graph to be at most $k$ if it can be obtained from a set of graphs each with at most $k+1$ vertices by a sequence of clique sums on cliques of size less than $\\theta$. We find the unavoidable minors for the graphs with large $\\theta$-tree-width and we obtain Robertson and Seymour's Grid Theorem as a corollary.","authors_text":"Benson Joeris, Jim Geelen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-28T20:07:19Z","title":"A generalization of the Grid Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.09098","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:e8d4d61f25c895cf9fec99f12c2ab73a3adba9d42914a6a73b6a0417121c1832","target":"record","created_at":"2026-05-18T01:03:40Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d6276f069dcaae4e26ad7a767c41f40ecad2f8d2adc6682fd16f79a203814fd8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-28T20:07:19Z","title_canon_sha256":"fad9e72e520be0d4c1ad3f83740c22f48c8fc49d5a0ebbe8be8cd097f03df979"},"schema_version":"1.0","source":{"id":"1609.09098","kind":"arxiv","version":1}},"canonical_sha256":"e15d39908dc8eb619733d410d01c75f544cca84383c0ce17a8e0abd33bc0adb0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e15d39908dc8eb619733d410d01c75f544cca84383c0ce17a8e0abd33bc0adb0","first_computed_at":"2026-05-18T01:03:40.228444Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:03:40.228444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fpjpUQoN9URw27FQW/gQFNFybP/qpz9c0qPKfDCIIytyDfQPFMFC2N0hDEBpN0ZWU8hzmz0vggsMRhFbBDt4Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:03:40.228879Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.09098","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e8d4d61f25c895cf9fec99f12c2ab73a3adba9d42914a6a73b6a0417121c1832","sha256:ee50e512805906540dd139908471d3ba57071485087f99602c7466dffedecc1e"],"state_sha256":"60d2a211420e96950b2205f4c678657609cdf5138dcb8e46085a24014a2158a1"}