{"id":"https://openalex.org/W7131808331","doi":"https://doi.org/10.48550/arxiv.2602.22874","title":"Flip Distance of Triangulations of Convex Polygons / Rotation Distance of Binary Trees is NP-complete","display_name":"Flip Distance of Triangulations of Convex Polygons / Rotation Distance of Binary Trees is NP-complete","publication_year":2026,"publication_date":"2026-02-26","ids":{"openalex":"https://openalex.org/W7131808331","doi":"https://doi.org/10.48550/arxiv.2602.22874"},"language":null,"primary_location":{"id":"pmh:doi:10.48550/arxiv.2602.22874","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"publisher-specific-oa","license_id":"https://openalex.org/licenses/publisher-specific-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":null,"any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5120318647","display_name":"Joseph Dorfer","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Dorfer, Joseph","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5120318647"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.8307999968528748,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.8307999968528748,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.059300001710653305,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.03350000083446503,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.6432999968528748},{"id":"https://openalex.org/keywords/triangulation","display_name":"Triangulation","score":0.583899974822998},{"id":"https://openalex.org/keywords/rotation","display_name":"Rotation (mathematics)","score":0.4957999885082245},{"id":"https://openalex.org/keywords/binary-number","display_name":"Binary number","score":0.4758000075817108},{"id":"https://openalex.org/keywords/point","display_name":"Point (geometry)","score":0.44920000433921814},{"id":"https://openalex.org/keywords/convex-set","display_name":"Convex set","score":0.4478999972343445},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.42160001397132874},{"id":"https://openalex.org/keywords/computational-geometry","display_name":"Computational geometry","score":0.3944000005722046},{"id":"https://openalex.org/keywords/convex-polytope","display_name":"Convex polytope","score":0.34790000319480896},{"id":"https://openalex.org/keywords/spanning-tree","display_name":"Spanning tree","score":0.3379000127315521}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7858999967575073},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.692799985408783},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.6432999968528748},{"id":"https://openalex.org/C135981907","wikidata":"https://www.wikidata.org/wiki/Q188056","display_name":"Triangulation","level":2,"score":0.583899974822998},{"id":"https://openalex.org/C74050887","wikidata":"https://www.wikidata.org/wiki/Q848368","display_name":"Rotation (mathematics)","level":2,"score":0.4957999885082245},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.4758000075817108},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.44920000433921814},{"id":"https://openalex.org/C49870271","wikidata":"https://www.wikidata.org/wiki/Q193657","display_name":"Convex set","level":4,"score":0.4478999972343445},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.42160001397132874},{"id":"https://openalex.org/C29123130","wikidata":"https://www.wikidata.org/wiki/Q874709","display_name":"Computational geometry","level":2,"score":0.3944000005722046},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3538999855518341},{"id":"https://openalex.org/C110943637","wikidata":"https://www.wikidata.org/wiki/Q4129097","display_name":"Convex polytope","level":5,"score":0.34790000319480896},{"id":"https://openalex.org/C64331007","wikidata":"https://www.wikidata.org/wiki/Q831672","display_name":"Spanning tree","level":2,"score":0.3379000127315521},{"id":"https://openalex.org/C197855036","wikidata":"https://www.wikidata.org/wiki/Q380172","display_name":"Binary tree","level":2,"score":0.3361999988555908},{"id":"https://openalex.org/C2780428219","wikidata":"https://www.wikidata.org/wiki/Q16952335","display_name":"Cover (algebra)","level":2,"score":0.32260000705718994},{"id":"https://openalex.org/C128321387","wikidata":"https://www.wikidata.org/wiki/Q5164356","display_name":"Constrained Delaunay triangulation","level":3,"score":0.3059999942779541},{"id":"https://openalex.org/C2776055716","wikidata":"https://www.wikidata.org/wiki/Q7316893","display_name":"Retract","level":2,"score":0.30219998955726624},{"id":"https://openalex.org/C64051489","wikidata":"https://www.wikidata.org/wiki/Q7104528","display_name":"Orthogonal convex hull","level":5,"score":0.29980000853538513},{"id":"https://openalex.org/C206194317","wikidata":"https://www.wikidata.org/wiki/Q1138624","display_name":"Convex hull","level":3,"score":0.29679998755455017},{"id":"https://openalex.org/C2780034980","wikidata":"https://www.wikidata.org/wiki/Q3259505","display_name":"Convex polygon","level":3,"score":0.2928999960422516},{"id":"https://openalex.org/C68010082","wikidata":"https://www.wikidata.org/wiki/Q192445","display_name":"Delaunay triangulation","level":2,"score":0.2912999987602234},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.2863999903202057},{"id":"https://openalex.org/C167117609","wikidata":"https://www.wikidata.org/wiki/Q17078161","display_name":"Point set triangulation","level":3,"score":0.2775000035762787},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.2768000066280365},{"id":"https://openalex.org/C111110010","wikidata":"https://www.wikidata.org/wiki/Q2627315","display_name":"Convex combination","level":4,"score":0.27250000834465027},{"id":"https://openalex.org/C186399060","wikidata":"https://www.wikidata.org/wiki/Q959962","display_name":"Diagram","level":2,"score":0.2711000144481659},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.26440000534057617},{"id":"https://openalex.org/C65180967","wikidata":"https://www.wikidata.org/wiki/Q130901","display_name":"Binary relation","level":2,"score":0.26109999418258667},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.26010000705718994},{"id":"https://openalex.org/C178609930","wikidata":"https://www.wikidata.org/wiki/Q3538712","display_name":"Pitteway triangulation","level":3,"score":0.2558000087738037},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.25209999084472656}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:doi:10.48550/arxiv.2602.22874","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"publisher-specific-oa","license_id":"https://openalex.org/licenses/publisher-specific-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},{"id":"doi:10.48550/arxiv.2602.22874","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2602.22874","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:doi:10.48550/arxiv.2602.22874","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"publisher-specific-oa","license_id":"https://openalex.org/licenses/publisher-specific-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Flips":[0],"in":[1,7,16,21,30],"triangulations":[2,74,100],"of":[3,24,36,41,47,75,85,99,108,111],"convex":[4,49,76],"polygons":[5],"arise":[6],"many":[8,60],"different":[9],"settings.":[10],"They":[11],"are":[12],"isomorphic":[13],"to":[14],"rotations":[15],"binary":[17,86],"trees,":[18,87],"define":[19],"edges":[20],"the":[22,25,31,38,82,106],"1-skeleton":[23],"Associahedron":[26],"and":[27,78,119,121,126],"cover":[28],"relations":[29],"Tamari":[32],"Lattice.":[33],"The":[34],"complexity":[35],"determining":[37],"minimum":[39],"number":[40],"flips":[42],"that":[43,68],"transform":[44],"one":[45],"triangulation":[46],"a":[48,55],"point":[50],"set":[51],"into":[52],"another":[53],"remained":[54],"tantalizing":[56],"open":[57],"question":[58,65],"for":[59,96,105],"decades.":[61],"We":[62],"settle":[63],"this":[64],"by":[66,115],"proving":[67],"computing":[69,81],"shortest":[70],"flip":[71,97,109],"sequences":[72,98,110],"between":[73],"polygons,":[77],"therefore":[79],"also":[80],"rotation":[83],"distance":[84],"is":[88],"NP-hard.":[89],"For":[90],"our":[91],"proof":[92],"we":[93],"develop":[94],"techniques":[95],"whose":[101],"counterparts":[102],"were":[103],"introduced":[104],"study":[107],"non-crossing":[112],"spanning":[113],"trees":[114],"Bjerkevik,":[116,122],"Kleist,":[117,124],"Ueckerdt,":[118,125],"Vogtenhuber~[SODA25]":[120],"Dorfer,":[123],"Vogtenhuber~[SoCG26].":[127]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-02-28T00:00:00"}
