{"?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-5DK09PKV/EB88C5ED-61AB-4268-B91D-462F2BBBF610/PDF","dcterms:extent":"0 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-5DK09PKV/eeffeb74-d560-4717-94d9-f35cc4cf6dcb/PDF","dcterms:extent":"455 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-5DK09PKV/e0b32546-76df-465f-9728-1d679906841d/TEXT","dcterms:extent":"77 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-5DK09PKV","dcterms:isPartOf":[{"@rdf:resource":"https://www.dlib.si/details/URN:NBN:SI:spr-UP1WMFAR"},{"@xml:lang":"sl","#text":"Ars mathematica contemporanea"}],"dcterms:issued":"2011","dc:creator":["Liu, Hong","Pelsmajer, Michael J."],"dc:format":[{"@xml:lang":"sl","#text":"številka:1"},{"@xml:lang":"sl","#text":"letnik:4"},{"@xml:lang":"sl","#text":"str. 177-204"}],"dc:identifier":["COBISSID:16266329","ISSN:1855-3966","URN:URN:NBN:SI:doc-5DK09PKV"],"dc:language":"en","dc:publisher":{"@xml:lang":"sl","#text":"Društvo matematikov, fizikov in astronomov Slovenije"},"dc:subject":[{"@xml:lang":"sl","#text":"dominacijska množica"},{"@xml:lang":"en","#text":"dominating set"},{"@xml:lang":"sl","#text":"grafi na ploskvah"},{"@xml:lang":"en","#text":"graph theory"},{"@xml:lang":"en","#text":"graphs on surfaces"},{"@xml:lang":"sl","#text":"neorientabilna ploskev"},{"@xml:lang":"sl","#text":"nestisljiv cikel"},{"@xml:lang":"en","#text":"non-contractible cycle"},{"@xml:lang":"en","#text":"non-orientable surface"},{"@xml:lang":"sl","#text":"teorija grafov"},{"@xml:lang":"sl","#text":"triangulacija"},{"@xml:lang":"en","#text":"triangulation"}],"dcterms:temporal":{"@rdf:resource":"2008-2025"},"dc:title":{"@xml:lang":"sl","#text":"Dominating sets in triangulations on surfaces|"},"dc:description":{"@xml:lang":"sl","#text":"A dominating set ?$D \\subset V(G)$? of a graph ?$G$? is a set such that each vertex ?$v \\in V(G)$? is either in the set or adjacent to a vertex in the set. Matheson and Tarjan (1996) proved that any ?$n$?-vertex plane triangulation has a dominating set of size at most ?$n/3$?, and conjectured a bound of ?$n/4$? for ?$n$? sufficiently large. King and Pelsmajer recently proved this for graphs with maximum degree at most 6. Plummer and Zha (2009) and Honjo, Kawarabayashi, and Nakamoto (2009) extended the ?$n/3$? bound to triangulations on surfaces. We prove two related results: (i) There is a constant ?$c_1$? such that any ?$n$?-vertex plane triangulation with maximum degree at most 6 has a dominating set of size at most ?$n/6 + c_1$?. (ii) For any surface ?$S$?, ?$t \\ge 0$?, and ?$\\epsilon > 0$?, there exists ?$c_2$? such that for any ?$n$?-vertex triangulation on ?$S$? with at most ?$t$? vertices of degree other than 6, there is a dominating set of size at most ?$n(1/6 + \\epsilon) + c_2$?. As part of the proof, we also show that any ?$n$?-vertex triangulation of a non-orientable surface has a non-contractible cycle of length at most ?$2\\sqrt{n}$?. Albertson and Hutchinson (1986) proved that for ?$n$?-vertex triangulation of an orientable surface other than a sphere has a non-contractible cycle of length ?$2\\sqrt{2n}$?, but no similar result was known for non-orientable surfaces"},"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-5DK09PKV","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:doc-5DK09PKV"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:doc-5DK09PKV/EB88C5ED-61AB-4268-B91D-462F2BBBF610/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-5DK09PKV/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:doc-5DK09PKV"}}}}