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Reprezentace a vizualizace grafů
dc.contributor.advisorKratochvíl, Jan
dc.creatorŠtola, Jan
dc.date.accessioned2018-11-30T11:38:40Z
dc.date.available2018-11-30T11:38:40Z
dc.date.issued2010
dc.identifier.urihttp://hdl.handle.net/20.500.11956/23702
dc.description.abstractThe 3D visibility (graph) drawing is a graph drawing in IR3 where vertices are represented by 2D sets placed into planes parallel to xy-plane and the edges correspond to z-parallel visibility among these sets. We continue the study of 3D visibility drawing of complete graphs by rectangles and regular polygons. We show that the maximum size of a complete graph with a 3D visibility drawing by regular n-gons is O(n4). This polynomial bound improves signifficantly the previous best known (exponential) bound 6n3 3n1 3 26n.We also provide several lower bounds. We show that the complete graph K2k+3 (resp. K4k+6) has a 3D visibility drawing by regular 2k-gons (resp.(2k + 1)-gons). We improve the best known upper bound on the size of a complete graph with a 3D visibility drawing by rectangles from 55 to 50. This result is based on the exploration of unimodal sequences of k-tuples of numbers. A sequence of numbers is unimodal if it rst increases and then decreases. A sequence of k-tuples of numbers is unimodal if it is unimodal in each component. We derive tight bounds on the maximum length of a sequence of k-tuples without a unimodal subsequence of length n. We show a connection between these results and Dedekind numbers, i.e., the numbers of antichains of a power set P(1; : : : ; k) ordered by inclusion.en_US
dc.languageEnglishcs_CZ
dc.language.isoen_US
dc.publisherUniverzita Karlova, Matematicko-fyzikální fakultacs_CZ
dc.titleRepresentations and Visualization of Graphsen_US
dc.typedizertační prácecs_CZ
dcterms.created2010
dcterms.dateAccepted2010-07-29
dc.description.departmentKatedra aplikované matematikycs_CZ
dc.description.departmentDepartment of Applied Mathematicsen_US
dc.description.facultyFaculty of Mathematics and Physicsen_US
dc.description.facultyMatematicko-fyzikální fakultacs_CZ
dc.identifier.repId42673
dc.title.translatedReprezentace a vizualizace grafůcs_CZ
dc.contributor.refereeValtr, Pavel
dc.contributor.refereeWood, David
dc.identifier.aleph001389126
thesis.degree.namePh.D.
thesis.degree.leveldoktorskécs_CZ
thesis.degree.disciplineDiskrétní modely a algoritmycs_CZ
thesis.degree.disciplineDiscrete Models and Algorithmsen_US
thesis.degree.programInformaticsen_US
thesis.degree.programInformatikacs_CZ
uk.faculty-name.csMatematicko-fyzikální fakultacs_CZ
uk.faculty-name.enFaculty of Mathematics and Physicsen_US
uk.faculty-abbr.csMFFcs_CZ
uk.degree-discipline.csDiskrétní modely a algoritmycs_CZ
uk.degree-discipline.enDiscrete Models and Algorithmsen_US
uk.degree-program.csInformatikacs_CZ
uk.degree-program.enInformaticsen_US
thesis.grade.csProspěl/acs_CZ
thesis.grade.enPassen_US
uk.abstract.enThe 3D visibility (graph) drawing is a graph drawing in IR3 where vertices are represented by 2D sets placed into planes parallel to xy-plane and the edges correspond to z-parallel visibility among these sets. We continue the study of 3D visibility drawing of complete graphs by rectangles and regular polygons. We show that the maximum size of a complete graph with a 3D visibility drawing by regular n-gons is O(n4). This polynomial bound improves signifficantly the previous best known (exponential) bound 6n3 3n1 3 26n.We also provide several lower bounds. We show that the complete graph K2k+3 (resp. K4k+6) has a 3D visibility drawing by regular 2k-gons (resp.(2k + 1)-gons). We improve the best known upper bound on the size of a complete graph with a 3D visibility drawing by rectangles from 55 to 50. This result is based on the exploration of unimodal sequences of k-tuples of numbers. A sequence of numbers is unimodal if it rst increases and then decreases. A sequence of k-tuples of numbers is unimodal if it is unimodal in each component. We derive tight bounds on the maximum length of a sequence of k-tuples without a unimodal subsequence of length n. We show a connection between these results and Dedekind numbers, i.e., the numbers of antichains of a power set P(1; : : : ; k) ordered by inclusion.en_US
uk.file-availabilityV
uk.publication.placePrahacs_CZ
uk.grantorUniverzita Karlova, Matematicko-fyzikální fakulta, Katedra aplikované matematikycs_CZ
thesis.grade.codeP


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