{"?xml":{"@version":"1.0"},"edm:RDF":{"@xmlns:dc":"http://purl.org/dc/elements/1.1/","@xmlns:edm":"http://www.europeana.eu/schemas/edm/","@xmlns:wgs84_pos":"http://www.w3.org/2003/01/geo/wgs84_pos","@xmlns:foaf":"http://xmlns.com/foaf/0.1/","@xmlns:rdaGr2":"http://rdvocab.info/ElementsGr2","@xmlns:oai":"http://www.openarchives.org/OAI/2.0/","@xmlns:owl":"http://www.w3.org/2002/07/owl#","@xmlns:rdf":"http://www.w3.org/1999/02/22-rdf-syntax-ns#","@xmlns:ore":"http://www.openarchives.org/ore/terms/","@xmlns:skos":"http://www.w3.org/2004/02/skos/core#","@xmlns:dcterms":"http://purl.org/dc/terms/","edm:WebResource":[{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-EK0G70ZF/b8674f36-ba9d-4750-b761-37dd8627dc7f/PDF","dcterms:extent":"523 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-EK0G70ZF/33a18669-1091-4592-bb61-919822292dec/TEXT","dcterms:extent":"81 KB"}],"edm:TimeSpan":{"@rdf:about":"2008-2025","edm:begin":{"@xml:lang":"en","#text":"2008"},"edm:end":{"@xml:lang":"en","#text":"2025"}},"edm:ProvidedCHO":{"@rdf:about":"URN:NBN:SI:doc-EK0G70ZF","dcterms:isPartOf":[{"@rdf:resource":"https://www.dlib.si/details/URN:NBN:SI:spr-UP1WMFAR"},{"@xml:lang":"sl","#text":"Ars mathematica contemporanea"}],"dcterms:issued":"2022","dc:creator":["Arroyo, Alan","McQuillan, Dan","Richter, R. Bruce","Salazar, Gelasio"],"dc:format":[{"@xml:lang":"sl","#text":"letnik:22"},{"@xml:lang":"sl","#text":"številka:3"},{"@xml:lang":"sl","#text":"P3.04 (27 str.)"}],"dc:identifier":["DOI:10.26493/1855-3974.2134.ac9","COBISSID_HOST:117983491","ISSN:1855-3966","URN:URN:NBN:SI:doc-EK0G70ZF"],"dc:language":"en","dc:publisher":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije"},"dc:subject":[{"@xml:lang":"en","#text":"complete graphs"},{"@xml:lang":"en","#text":"convex drawings"},{"@xml:lang":"sl","#text":"enostavne risbe"},{"@xml:lang":"sl","#text":"konveksne risbe"},{"@xml:lang":"sl","#text":"polni grafi"},{"@xml:lang":"en","#text":"simple drawings"}],"dcterms:temporal":{"@rdf:resource":"2008-2025"},"dc:title":{"@xml:lang":"sl","#text":"Convex drawings of the complete graph: topology meets geometry|"},"dc:description":[{"@xml:lang":"sl","#text":"In a geometric drawing of ?$K_n$?, trivially each 3-cycle bounds a convex region: if two vertices are in that region, then so is the (geometric) edge between them. We define a topological drawing ?$D$? of ?$K_n$? to be convex if each 3-cycle bounds a closed region ?$R$? such that any two vertices in ?$R$? have the (topological) edge between them contained in ?$R$?. While convex drawings generalize geometric drawings, they specialize topological ones. Therefore it might be surprising if all optimal (that is, crossing-minimal) topological drawings of ?$K_n$? were convex. However, we take a first step to showing that they are convex: we show that if ?$D$? has a non-convex ?$K_5$? all of whose extensions to a ?$K_7$? have no other non-convex ?$K_5$?, then ?$D$? is not optimal (without reference to the conjecture for the crossing number of ?$K_n$?). This is the first example of non-trivial local considerations providing sufficient conditions for suboptimality. At our request, Aichholzer has computationally verified that, up to ?$n = 12$?, every optimal drawing of ?$K_n$? is convex. Convexity naturally lends itself to refinements, including hereditarily convex (?$h$?-convex) and face convex (?$f$?-convex). The hierarchy rectilinear ?$\\subseteq$? ?$f$?-convex ?$\\subseteq$? ?$h$?-convex ?$\\subseteq$? convex ?$\\subseteq$? topological provides links between geometric and topological drawings. It is known that ?$f$?-convex is equivalent to pseudolinear (generalizing rectilinear) and ?$h$?-convex is equivalent to pseudospherical (generalizing spherical geodesic). We characterize ?$h$?-convexity by three forbidden (topological) subdrawings. This hierarchy provides a framework to consider generalizations of other geometric questions for point sets in the plane. We provide two examples of such questions, namely numbers of empty triangles and existence of convex ?$k$?-gons"},{"@xml:lang":"sl","#text":"V geometrijski risbi grafa ?$K_n$?, trivialno vsak 3-cikel omejuje konveksno območje: če sta dva vozlišča v tem območju, potem to velja tudi za (geometrijsko) povezavo med njima. Za topološko risbo ?$D$? grafa ?$K_n$? na sferi rečemo, da je konveksna, če vsak 3-cikel omejuje zaprto območje ?$R$? (na katerikoli od dveh strani 3-cikla), pri tem pa imata poljubna dva vozlišča v ?$R$? (topološko) povezavo med njima vsebovano v ?$R$?. Konveksne risbe po eni strani posplošujejo geometrijske risbe, po drugi strani pa so poddružina topoloških risb. Zato bi bilo lahko presenetljivo, če bi bile vse optimalne (to je, z minimalnim številom presečišč) topološke risbe grafa ?$K_n$? konveksne. Vseeno pa storimo prvi korak k dokazu, da so konveksne: pokažemo, da če ?$D$? vsebuje nekonveksen ?$K_5$?, vse njegove razširitve do ?$K_7$? pa ne vsebujejo nobenega drugega nekonveksnega ?$K_5$?, potem ?$D$? ni optimalen (brez sklicevanja na domnevo o številu presečišč grafa ?$K_n$)?. To je prvi primer netrivialnih lokalnih argumentov, ki dajejo zadostne pogoje za suboptimalnost. Na našo prošnjo je Aichholzer računalniško potrdil, da je, vse do ?$n = 12$?, vsaka optimalna risba grafa ?$K_n$? konveksna. Konveksnost naravno dopušča izpopolnitve, kot sta npr. lastnosti hereditarno konveksen (?$h$?-konveksen) in lično konveksen (?$f$?-konveksen). Hierarhija premočrten ?$\\subseteq$? ?$f$?-konveksen ?$\\subseteq$? ?$h$?-konveksen ?$\\subseteq$? konveksen ?$\\subseteq$? topološki opisuje relacije med geometrijskimi in topološkimi risbami. Znano je, da je ?$f$?-konveksnost ekvivalentna psevdolinearnosti (ki posplošuje premočrtnost) in da je ?$h$?-konveksnost ekvivalentna psevdosferičnosti (ki posplošuje sferično geodetskost). Karakteriziramo ?$h$?-konveksnost s tremi prepovedanimi (topološkimi) podrisbami. Ta hierarhija predstavlja okvir za obravnavo posplošitev tudi drugih geometrijskih problemov v zvezi z množicami točk v ravnini. Predstavimo dva primera takšnih problemov, in sicer o številu odprtih trikotnikov ter o obstoju konveksnih ?$k$?-kotnikov"}],"edm:type":"TEXT","dc:type":[{"@xml:lang":"sl","#text":"znanstveno časopisje"},{"@xml:lang":"en","#text":"journals"},{"@rdf:resource":"http://www.wikidata.org/entity/Q361785"}]},"ore:Aggregation":{"@rdf:about":"http://www.dlib.si/?URN=URN:NBN:SI:doc-EK0G70ZF","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:doc-EK0G70ZF"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:doc-EK0G70ZF/b8674f36-ba9d-4750-b761-37dd8627dc7f/PDF"},"edm:rights":{"@rdf:resource":"http://creativecommons.org/licenses/by/4.0/"},"edm:provider":"Slovenian National E-content Aggregator","edm:intermediateProvider":{"@xml:lang":"en","#text":"National and University Library of Slovenia"},"edm:dataProvider":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije"},"edm:object":{"@rdf:resource":"http://www.dlib.si/streamdb/URN:NBN:SI:doc-EK0G70ZF/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:doc-EK0G70ZF"}}}}