{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:LDXFUKXDT7IHYGDKZGMXEJEMJH","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":"ddd5ac32e2243b06ba5311ac24d9bcd23345440787261baf32cacd7b4123ac17","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-13T09:47:29Z","title_canon_sha256":"3ef507274681bb6b93a8382f66f33cb5e797f17f11a241e5fe5a80fc123091c1"},"schema_version":"1.0","source":{"id":"1510.03599","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.03599","created_at":"2026-05-17T23:59:04Z"},{"alias_kind":"arxiv_version","alias_value":"1510.03599v2","created_at":"2026-05-17T23:59:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.03599","created_at":"2026-05-17T23:59:04Z"},{"alias_kind":"pith_short_12","alias_value":"LDXFUKXDT7IH","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"LDXFUKXDT7IHYGDK","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"LDXFUKXD","created_at":"2026-05-18T12:29:29Z"}],"graph_snapshots":[{"event_id":"sha256:a766e2a95965ec06b9b22ed98b30e3b3f5f65ace28686832a7501a61ee5efe50","target":"graph","created_at":"2026-05-17T23:59:04Z","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":"In this paper we characterise the graphs containing a $\\mathbb{Z} \\times \\mathbb{Z}$ grid minor in a similar way as it has been done by Halin for graphs with an $\\mathbb{N} \\times \\mathbb{Z}$ grid minor. Using our characterisation, we describe the structure of graphs without $\\mathbb{Z} \\times \\mathbb{Z}$ grid minors in terms of tree-decompositions.","authors_text":"Karl Heuer","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-13T09:47:29Z","title":"Excluding a full grid minor"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.03599","kind":"arxiv","version":2},"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":"sha256:34cc490933f7520b31f44be3306d323ee794418dac9ff307795e9bf795a60cc6","target":"record","created_at":"2026-05-17T23:59:04Z","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":"ddd5ac32e2243b06ba5311ac24d9bcd23345440787261baf32cacd7b4123ac17","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-13T09:47:29Z","title_canon_sha256":"3ef507274681bb6b93a8382f66f33cb5e797f17f11a241e5fe5a80fc123091c1"},"schema_version":"1.0","source":{"id":"1510.03599","kind":"arxiv","version":2}},"canonical_sha256":"58ee5a2ae39fd07c186ac99972248c49fa2be06d06d29a0e5d23cded8ed91b6f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"58ee5a2ae39fd07c186ac99972248c49fa2be06d06d29a0e5d23cded8ed91b6f","first_computed_at":"2026-05-17T23:59:04.003953Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:59:04.003953Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SxMqKI+l/xXwIymci/Jw2x8iM6yO44i0r9iY96+nip/eaJw+76TqBtZfMJjxl3T8guZvhAYVcRTQsJxvpPenDg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:59:04.004357Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.03599","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:34cc490933f7520b31f44be3306d323ee794418dac9ff307795e9bf795a60cc6","sha256:a766e2a95965ec06b9b22ed98b30e3b3f5f65ace28686832a7501a61ee5efe50"],"state_sha256":"5caa02e65552050f95e290084b4418cde3bd88a201dd886faf11727023428736"}