{"paper":{"title":"Min-1-Planarity is NP-Hard","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Deciding whether a graph admits a min-1-planar drawing is NP-hard.","cross_cats":["cs.DS"],"primary_cat":"cs.CG","authors_text":"Yuto Okada","submitted_at":"2026-05-14T13:40:04Z","abstract_excerpt":"In this paper, we show that it is NP-hard to determine whether a given graph admits a min-1-planar drawing. A drawing of a graph is min-$k$-planar if, for every crossing in the drawing, at least one of the two crossing edges involves at most $k$ crossings. This notion of min-$k$-planarity was introduced by Binucci, B\\\"{u}ngener, Di Battista, Didimo, Dujmovi\\'c, Hong, Kaufmann, Liotta, Morin, and Tappini [GD 2023; JGAA, 2024] as a generalization of $k$-planarity."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we show that it is NP-hard to determine whether a given graph admits a min-1-planar drawing.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The polynomial-time reduction from a known NP-complete problem to min-1-planarity testing is correct and preserves the yes/no answer.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Deciding if a graph admits a min-1-planar drawing is NP-hard.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Deciding whether a graph admits a min-1-planar drawing is NP-hard.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"fa8f8ea068e9b18f69adaf7dbdc9e13b9723032d13feb89b09b36abf50ab9f05"},"source":{"id":"2605.14834","kind":"arxiv","version":1},"verdict":{"id":"b63bf3d7-b4eb-428f-81b8-eb11ec6253d8","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T03:13:31.806385Z","strongest_claim":"we show that it is NP-hard to determine whether a given graph admits a min-1-planar drawing.","one_line_summary":"Deciding if a graph admits a min-1-planar drawing is NP-hard.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The polynomial-time reduction from a known NP-complete problem to min-1-planarity testing is correct and preserves the yes/no answer.","pith_extraction_headline":"Deciding whether a graph admits a min-1-planar drawing is NP-hard."},"references":{"count":15,"sample":[{"doi":"","year":2021,"title":"2-level quasi-planarity or how caterpillars climb (spqr-)trees","work_id":"5b5b8512-bfbe-4553-acb9-1f3d726c7393","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1137/1","year":2014,"title":"More asymmetry yields faster matrix multiplication","work_id":"5ec132ea-8bab-4415-b2af-677e9db16f00","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/s00453-016-0200-5","year":2023,"title":"5 Carla Binucci, Aaron Büngener, Giuseppe Di Battista, Walter Didimo, Vida Dujmović, Seok- Hee Hong, Michael Kaufmann, Giuseppe Liotta, Pat Morin, and Alessandra Tappini","work_id":"c510fbd5-ed5e-4546-8854-57d15a1b4140","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.7155/jgaa.v28i2.2925","year":null,"title":"7 Carla Binucci, Emilio Di Giacomo, Walter Didimo, Fabrizio Montecchiani, Maurizio Patrignani, Antonios Symvonis, and Ioannis G","work_id":"6b790ed2-f7f0-4e6a-8c89-1e4c756b0bcc","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/j.tcs.2015.04.020","year":2015,"title":"URL:https://doi.org/10.1016/j.tcs.2015.04.020, doi: 10.1016/J.TCS.2015.04.020. 8 Franz J. Brandenburg. Recognizing optimal 1-planar graphs in linear time.Algorith- mica, 80(1):1–28,","work_id":"c8bdc92e-4000-4e34-a1bc-19043d923923","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":15,"snapshot_sha256":"a409903c2bc0432299fa7a77b6d29b51e6f9f9626036c38c00072b46ed9fd8ad","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"}