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In the language of crossing numbers, these are the only 1-crossing-critical graphs: they each have crossing number at least one, and every proper subgraph has crossing number less than one. In 1987, Kochol exhibited an infinite family of 3-connected, simple 2-crossing-critical graphs. In this work, we: (i) determine all the 3-connected 2-crossing-critical graphs that contain a subdivision of the M\\\"obius Ladder $V_{10}$; (ii) show how to obtain"},"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":"1312.3712","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-13T06:31:32Z","cross_cats_sorted":[],"title_canon_sha256":"c558840e2b3555c595eeabd6c50354ea5746dece0d6de5e8bd2fec9cc462d732","abstract_canon_sha256":"aef14aa96495989a6706e17f296a76c580d37f3d35bb1c36eca7ab2b5b778f31"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:04:47.869385Z","signature_b64":"6u7Rs+icVkGZ79YkFh1ZIiWSZ7KdiuyTecxqR8SC51iBtCVdim6+qYQRrnU4TubgTzlzcUob3LYvQy9ph/2DDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ea2d9f5c2bff0957a5534a98ffac100934449e8bb450e9b4f463152ad25f8116","last_reissued_at":"2026-05-18T03:04:47.868544Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:04:47.868544Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Characterizing 2-crossing-critical graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bogdan Oporowski, Drago Bokal, Gelasio Salazar, R. 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