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[{"key": "dc.contributor.advisor", "value": "Rajala, Tapio", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Pohjola, Anna-Sofia", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2024-02-09T07:48:05Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2024-02-09T07:48:05Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2024", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/93310", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa todistetaan Kuratowskin lause, joka on verkkoteorian\nkeskeinen tulos. Tutkielma jakautuu kolmeen kappaleeseen, joista ensimm\u00e4isess\u00e4 tutustutaan verkkoteorian perusteisiin ja yleisiin m\u00e4\u00e4ritelmiin alkaen\nverkon k\u00e4rkien sek\u00e4 sivujen m\u00e4\u00e4rittelemisest\u00e4. Ty\u00f6ss\u00e4 keskityt\u00e4\u00e4n yksinkertaisiin suuntaamattomiin verkkoihin. Verkkoteorian perusk\u00e4sitteiden j\u00e4lkeen\nm\u00e4\u00e4ritell\u00e4\u00e4n tasoverkko sellaiseksi verkoksi, joka voidaan piirt\u00e4\u00e4 tasoon niin,\nett\u00e4 sen sivut eiv\u00e4t leikkaa toisiaan. Erilaisista tasoverkoista esitell\u00e4\u00e4n puut\nsek\u00e4 sillat. Siltojen m\u00e4\u00e4ritelm\u00e4\u00e4 sek\u00e4 erityisesti kierrosten siltoja tarvitaan\nKuratowskin lauseen todistamiseen.\nTutkielman kolmannessa osassa m\u00e4\u00e4ritell\u00e4\u00e4n Kuratowskin lausetta varten ensin Kuratowskin verkot K5 ja K3,3. N\u00e4ille verkoille osoitetaan, ett\u00e4 ne\neiv\u00e4t ole tasoverkkoja. Tarvittavien tietojen j\u00e4lkeen esitell\u00e4\u00e4n Kuratowskin\nlause, jonka mukaan verkko on tasoverkko, jos ja vain jos se ei sis\u00e4ll\u00e4 aliverkkonaan verkkoja K5 tai K3,3 eik\u00e4 niiden alijaotuksia. Lis\u00e4ksi esitell\u00e4\u00e4n\nWagnerin lause, joka n\u00e4ytet\u00e4\u00e4n yht\u00e4pit\u00e4v\u00e4ksi Kuratowskin lauseen kanssa.\nWagnerin lauseen mukaan verkko on tasoverkko, jos ja vain jos sill\u00e4 ei ole Kuratowskin minoria. T\u00e4t\u00e4 lausetta varten m\u00e4\u00e4ritell\u00e4\u00e4n verkon minori ja esitet\u00e4\u00e4n minoreihin liittyvi\u00e4 tuloksia. Tarvittavien aputulosten j\u00e4lkeen esitet\u00e4\u00e4n\nWagnerin lauseen todistus, jolloin my\u00f6s yht\u00e4pit\u00e4v\u00e4 Kuratowskin lause on todistettu. Lopuksi esitet\u00e4\u00e4n esimerkki, jossa osoitetaan Kuratowskin lauseen\navulla, ett\u00e4 Petersenin verkko ei ole tasoverkko.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2024-02-09T07:48:05Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2024-02-09T07:48:05Z (GMT). No. of bitstreams: 0\n Previous issue date: 2024", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "32", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.title", "value": "Kuratowskin lause", "language": null, "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202402091792", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.copyright", "value": "\u00a9 The Author(s)", "language": null, "element": "rights", "qualifier": "copyright", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": null, "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "verkkoteoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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