Relevant bibliographies by topics / Planarität

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Planarität'

Author: Grafiati
Published: 4 June 2021
Last updated: 18 February 2022

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Journal articles on the topic "Planarität"

1

Schubert, Ulrich, Siegfried Hünig, and Alexander Aumüller. "Zur Frage der Planarität von 9,10-Anthrachinonderivaten." Liebigs Annalen der Chemie 1985, no. 6 (June 12, 1985): 1216–22. http://dx.doi.org/10.1002/jlac.198519850612.

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Araneda, Juan F., Benedikt Neue, and Warren E. Piers. "Erzwungene Planarität: eine Strategie für stabile, borhaltige π-konjugierte Materialien." Angewandte Chemie 124, no. 40 (August 29, 2012): 10117–19. http://dx.doi.org/10.1002/ange.201205265.

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Gerson, Fabian, Javier Lopez, Adolf Krebs, and Wolfgang Rüger. "Radikal-Kationen von sterisch gehinderten Bicycloalkylidenen, ein experimenteller Beitrag zur Frage der Planarität des Ethylen-Radikalkations." Angewandte Chemie 93, no. 1 (January 20, 2006): 106–7. http://dx.doi.org/10.1002/ange.19810930122.

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王, 丹. "G+--的平面性." Advances in Applied Mathematics 07, no. 03 (2018): 237–42. http://dx.doi.org/10.12677/aam.2018.73029.

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Hughes, David W. "Planetary planarity." Nature 337, no. 6203 (January 1989): 113. http://dx.doi.org/10.1038/337113a0.

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Angelini, Patrizio, Giordano Da Lozzo, Giuseppe Di Battista, Valentino Di Donato, Philipp Kindermann, Günter Rote, and Ignaz Rutter. "Windrose Planarity." ACM Transactions on Algorithms 14, no. 4 (October 13, 2018): 1–24. http://dx.doi.org/10.1145/3239561.

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Guha, S., W. Graupner, R. Resel, M. Chandrasekhar, H. R. Chandrasekhar, R. Glaser, and G. Leising. "Planarity ofparaHexaphenyl." Physical Review Letters 82, no. 18 (May 3, 1999): 3625–28. http://dx.doi.org/10.1103/physrevlett.82.3625.

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Barth, Lukas, Guido Brückner, Paul Jungeblut, and Marcel Radermacher. "Multilevel Planarity." Journal of Graph Algorithms and Applications 25, no. 1 (2021): 151–70. http://dx.doi.org/10.7155/jgaa.00554.

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Angelini, Patrizio, Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, Maurizio Patrignani, and Ignaz Rutter. "Beyond level planarity: Cyclic, torus, and simultaneous level planarity." Theoretical Computer Science 804 (January 2020): 161–70. http://dx.doi.org/10.1016/j.tcs.2019.11.024.

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Klemz, Boris, and Günter Rote. "Ordered Level Planarity and Its Relationship to Geodesic Planarity, Bi-Monotonicity, and Variations of Level Planarity." ACM Transactions on Algorithms 15, no. 4 (October 12, 2019): 1–25. http://dx.doi.org/10.1145/3359587.

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Dissertations / Theses on the topic "Planarität"

1

Zschalig, Christian. "Characterizations of Planar Lattices by Left-relations." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2009. http://nbn-resolving.de/urn:nbn:de:bsz:14-ds-1240834941828-67021.

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Recently, Formal Concept Analysis has proven to be an efficient method for the analysis and representation of information. However, the possibility to visualize concept hierarchies is being affected by the difficulty of drawing attractive diagrams automatically. Reducing the number of edge crossings seems to increase the readability of those drawings. This dissertation concerns with a mandatory prerequisite of this constraint, namely the characterization and visual representation of planar lattices. The manifold existing approaches and algorithms are thereby considered under a different point of view. It is well known that exactly the planar lattices (or planar posets) possess an additional order ``from left to right''. Our aim in this work is to define left-relations and left-orders more precisely and to describe several aspects of planar lattices with their help. The three approaches employed structure the work in as many parts: Left-relations on lattices allow a more efficient consideration of conjugate orders since they are uniquely determined by the sorting of the meet-irreducibles. Additionally, the restriction on the meet-irreducibles enables us to achieve an intuitive description of standard contexts of planar lattices similar to the consecutive-one property. With the help of left-relations on diagrams, planar lattices can indeed be drawn without edge crossings in the plane. Thereby, lattice-theoretically found left-orders can be detected in the graphical representation again. Furthermore, we modify the left-right-numbering algorithm in order to obtain attribute-additive and plane drawings of planar lattices. Finally, we will consider left-relations on contexts. They turn out to be fairly similar structures to the Ferrers-graphs. Planar lattices can be characterized by a property of these graphs, namely the bipartiteness. We will constructively prove this result. Subsequently, we can design an efficient algorithm that finds all non-similar plane diagrams of a lattice
Die Formale Begriffsanalyse hat sich in den letzten Jahren als effizientes Werkzeug zur Datenanalyse und -repräsentation bewährt. Die Möglichkeit der visuellen Darstellung von Begriffshierarchien wird allerdings durch die Schwierigkeit, ansprechende Diagramme automatisch generieren zu können, beeinträchtigt. Offenbar sind Diagramme mit möglichst wenig Kantenkreuzungen für den menschlichen Anwender leichter lesbar. Diese Arbeit beschäftigt sich mit mit einer diesem Kriterium zugrunde liegenden Vorleistung, nämlich der Charakterisierung und Darstellung planarer Verbände. Die schon existierenden vielfältigen Ansätze und Methoden werden dabei unter einem neuen Gesichtspunkt betrachtet. Bekannterweise besitzen genau die planaren Verbände (bzw. planare geordnete Mengen) eine zusätzliche Ordnung "von links nach rechts". Unser Ziel in dieser Arbeit ist es, solche Links-Relationen bzw. Links-Ordnungen genauer zu definieren und verschiedene Aspekte planarer Verbände mit ihrer Hilfe zu beschreiben. Die insgesamt drei auftretenden Sichtweisen gliedern die Arbeit in ebensoviele Teile: Links-Relationen auf Verbänden erlauben eine effizientere Behandlung konjugierter Ordnungen, da sie durch die Anordnung der Schnitt-Irreduziblen schon eindeutig festgelegt sind. Außerdem erlaubt die Beschränkung auf die Schnitt-Irreduziblen eine anschauliche Beschreibung von Standardkontexten planarer Verbände ähnlich der consecutive-one property. Mit Hilfe der Links-Relationen auf Diagrammen können planare Verbände tatsächlich eben gezeichnet werden. Dabei lassen sich verbandstheoretisch ermittelte Links-Ordnungen in der graphischen Darstellung wieder finden. Weiterhin geben wir in eine Modifikation des left-right-numbering an, mit der planare Verbände merkmaladditiv und eben gezeichnet werden können. Schließlich werden wir Links-Relationen auf Kontexten betrachten. Diese stellen sich als sehr ähnlich zu Ferrers-Graphen heraus. Planare Verbände lassen sich durch eine Eigenschaft dieser Graphen, nämlich die Bipartitheit, charakterisieren. Wir werden dieses Ergebnis konstruktiv beweisen und darauf aufbauend einen effizienten Algorithmus angeben, mit dem alle nicht-ähnlichen ebenen Diagramme eines Verbandes bestimmt werden können
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Zschalig, Christian. "Characterizations of Planar Lattices by Left-relations." Doctoral thesis, Technische Universität Dresden, 2008. https://tud.qucosa.de/id/qucosa%3A23687.

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Recently, Formal Concept Analysis has proven to be an efficient method for the analysis and representation of information. However, the possibility to visualize concept hierarchies is being affected by the difficulty of drawing attractive diagrams automatically. Reducing the number of edge crossings seems to increase the readability of those drawings. This dissertation concerns with a mandatory prerequisite of this constraint, namely the characterization and visual representation of planar lattices. The manifold existing approaches and algorithms are thereby considered under a different point of view. It is well known that exactly the planar lattices (or planar posets) possess an additional order ``from left to right''. Our aim in this work is to define left-relations and left-orders more precisely and to describe several aspects of planar lattices with their help. The three approaches employed structure the work in as many parts: Left-relations on lattices allow a more efficient consideration of conjugate orders since they are uniquely determined by the sorting of the meet-irreducibles. Additionally, the restriction on the meet-irreducibles enables us to achieve an intuitive description of standard contexts of planar lattices similar to the consecutive-one property. With the help of left-relations on diagrams, planar lattices can indeed be drawn without edge crossings in the plane. Thereby, lattice-theoretically found left-orders can be detected in the graphical representation again. Furthermore, we modify the left-right-numbering algorithm in order to obtain attribute-additive and plane drawings of planar lattices. Finally, we will consider left-relations on contexts. They turn out to be fairly similar structures to the Ferrers-graphs. Planar lattices can be characterized by a property of these graphs, namely the bipartiteness. We will constructively prove this result. Subsequently, we can design an efficient algorithm that finds all non-similar plane diagrams of a lattice.
Die Formale Begriffsanalyse hat sich in den letzten Jahren als effizientes Werkzeug zur Datenanalyse und -repräsentation bewährt. Die Möglichkeit der visuellen Darstellung von Begriffshierarchien wird allerdings durch die Schwierigkeit, ansprechende Diagramme automatisch generieren zu können, beeinträchtigt. Offenbar sind Diagramme mit möglichst wenig Kantenkreuzungen für den menschlichen Anwender leichter lesbar. Diese Arbeit beschäftigt sich mit mit einer diesem Kriterium zugrunde liegenden Vorleistung, nämlich der Charakterisierung und Darstellung planarer Verbände. Die schon existierenden vielfältigen Ansätze und Methoden werden dabei unter einem neuen Gesichtspunkt betrachtet. Bekannterweise besitzen genau die planaren Verbände (bzw. planare geordnete Mengen) eine zusätzliche Ordnung "von links nach rechts". Unser Ziel in dieser Arbeit ist es, solche Links-Relationen bzw. Links-Ordnungen genauer zu definieren und verschiedene Aspekte planarer Verbände mit ihrer Hilfe zu beschreiben. Die insgesamt drei auftretenden Sichtweisen gliedern die Arbeit in ebensoviele Teile: Links-Relationen auf Verbänden erlauben eine effizientere Behandlung konjugierter Ordnungen, da sie durch die Anordnung der Schnitt-Irreduziblen schon eindeutig festgelegt sind. Außerdem erlaubt die Beschränkung auf die Schnitt-Irreduziblen eine anschauliche Beschreibung von Standardkontexten planarer Verbände ähnlich der consecutive-one property. Mit Hilfe der Links-Relationen auf Diagrammen können planare Verbände tatsächlich eben gezeichnet werden. Dabei lassen sich verbandstheoretisch ermittelte Links-Ordnungen in der graphischen Darstellung wieder finden. Weiterhin geben wir in eine Modifikation des left-right-numbering an, mit der planare Verbände merkmaladditiv und eben gezeichnet werden können. Schließlich werden wir Links-Relationen auf Kontexten betrachten. Diese stellen sich als sehr ähnlich zu Ferrers-Graphen heraus. Planare Verbände lassen sich durch eine Eigenschaft dieser Graphen, nämlich die Bipartitheit, charakterisieren. Wir werden dieses Ergebnis konstruktiv beweisen und darauf aufbauend einen effizienten Algorithmus angeben, mit dem alle nicht-ähnlichen ebenen Diagramme eines Verbandes bestimmt werden können.
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Fowler, Joe. "Unlabled Level Planarity." Diss., The University of Arizona, 2009. http://hdl.handle.net/10150/195812.

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Consider a graph G with vertex set V in which each of the n vertices is assigned a number from the set {1, ..., k} for some positive integer k. This assignment phi is a labeling if all k numbers are used. If phi does not assign adjacent vertices the same label, then phi partitions V into k levels. In a level drawing, the y-coordinate of each vertex matches its label and the edges are drawn strictly y-monotone. This leads to level drawings in the xy-plane where all vertices with label j lie along the line lj = {(x, j) : x in Reals} and where each edge crosses any of the k horizontal lines lj for j in [1..k] at most once. A graph with such a labeling forms a level graph and is level planar if it has a level drawing without crossings.We first consider the class of level trees that are level planar regardless of their labeling. We call such trees unlabeled level planar (ULP). We describe which trees are ULP and provide linear-time level planar drawing algorithms for any labeling. We characterize ULP trees in terms of two forbidden subdivisions so that any other tree must contain a subtree homeomorphic to one of these. We also provide linear-time recognition algorithms for ULP trees. We then extend this characterization to all ULP graphs with five additional forbidden subdivisions, and provide linear-time recogntion and drawing algorithms for any given labeling.
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Hayer, Matthias. "Testing planarity in linear time." Thesis, Georgia Institute of Technology, 1994. http://hdl.handle.net/1853/30483.

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Bachmaier, Christian. "Circle planarity of level graphs." [S.l.] : [s.n.], 2004. http://deposit.ddb.de/cgi-bin/dokserv?idn=973953985.

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Estrella, Balderrama Alejandro. "Simultaneous Embedding and Level Planarity." Diss., The University of Arizona, 2009. http://hdl.handle.net/10150/195738.

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Graphs are a common model for representing information consisting of a set of objects or entities and a set of connections or relations between them. Graph Drawing is concerned with the automatic visualization of graphs in order to make the information useful. That is, a good drawing should be helpful in the application domain where it is used by capturing the relationships in the underlying data. We consider two important problems in automated graph drawing: simultaneous embedding and level planarity. Simultaneous embedding is the problem of drawing multiple graphs while maintaining the readability of each graph independently and preserving the mental map when going from one graph to another. In this case, each graph has the same vertex set (same entities) but different edge sets (different relationships). Level planarity arises in the layout of graphs that contain hierarchical relationships. When drawing graphs in the plane, this translates to a restricted form of planarity where the vertical order of the entities is pre-determined. We consider the computational complexity of the simultaneous embedding problem. In particular, we show that in its generality the simultaneous embedding problem is NP-hard if the edges are drawn as straight-lines. We present algorithms for drawing graphs on predetermined levels, which allow the simultaneous embedding of restricted types of graphs, such as outerplanar graphs, trees and paths. Finally, our practical contribution is a tool that implements known and novel algorithms related to simultaneous embedding and level planarity and can be used both as a visualization software and as an aid to study theoretical problems.
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Taylor, Martyn G. "Planarity testing by path addition." Thesis, University of Kent, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.580367.

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The first linear-time planarity testing algorithm was developed in 1974 by Hopcroft and Tarjan (H&T) [32] using a method to split a biconnected graph up into edge disjoint paths and then sequentially embed them to test for planarity (a path addition method). Shortly afterwards Booth and Leuker [5] developed an alternative vertex addition linear-time planarity test, based on the earlier work of Lempel, Evan and Cederbaum [47], using a new PQ-Tree data structure. Since then there have been many developments in PQ- Tree vertex addition (and related PC-Tree edge addition) methods including authors such as: Chiba et al. [14]; Shih & Hsu [35, 69]; Boyer and Myrvold [10, 11]; and Haeupler and Tarjan [29]. In comparison, path addition has changed very little from the original algorithm. In 1984, Williamson [84] showed how H&T's algorithm can be extended to find Kuratowski sub-graphs in the event of a non-planar graph; and, in 1993, Mehlhorn, Mutzel and Naher [53] produced an implementation (in C) of H&T's algorithm and extended it to create a planar embedding of a graph. This has remained the state-of-the-art in path addition algorithms for over a decade. Recently", de Fraysseix formulated an algorithm [15, 17], based on Tremaux Trees and a characterisation of planarity by W. Wu [87]; this may prove to be a highly optimised version of H&T's algorithm but is difficult to definitively prove as only an outline of its planarity testing phase is provided. These authors represent the majority of the work on path addition methods of planarity testing and embedding; indicating that it receives little attention compared to vertex or edge addition methods This thesis attempts to reinvigorate the field of path addition and demonstrates: • How Trernaux Trees, which allow undirected connected graphs to be represented as a simple partial order relationship are fundamentally related to H&T-‘^planarity testing algorithm and includes some related invariant properties of these trees; • That the restriction on H&T's planarity testing algorithm to test undirected biconnected graphs can be relaxed to undirected connected graphs; • How to generate all possible embeddings of a biconnected component and how to extend this to generate all possible embeddings of separable graphs; and • Empirical Testing of various graph types and sizes to validate these results.
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Tassion, Vincent. "Planarité et Localité en Percolation." Phd thesis, Ecole normale supérieure de lyon - ENS LYON, 2014. http://tel.archives-ouvertes.fr/tel-01061007.

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Cette thèse s'inscrit dans l'étude mathématique de la percolation, qui regroupe une famille de modèles présentant une transition de phase. Des avancées majeures au cours des quinze dernières années, notamment l'invention du SLE et la preuve de l'invariance conforme de la percolation de Bernoulli critique, nous permettent aujourd'hui d'avoir une image très complète de la percolation de Bernoulli sur le réseau triangulaire. Cependant, de nombreuses questions demeurent ouvertes, et ont motivé notre travail.La première d'entre elle est l'universalité de la percolation plane, qui affirme que les propriétés macroscopiques de la percolation plane critique ne devraient pas dépendre du réseau sous-jacent à sa définition. Nous montrons, dans le cadre de la percolation Divide and Color, un résultat qui va dans le sens de cette universalité et identifions, dans ce contexte, des phénomènes macroscopiques indépendants du réseau microscopique. Une version plus faible d'universalité est donnée par la théorie de Russo-Seymour-Welsh (RSW), et sa validité est connue pour la percolation de Bernoulli (sans dépendance) sur les réseaux plans suffisamment symétriques. Nous étudions de nouveaux arguments de type RSW pour des modèles de percolation avec dépendance. La deuxième question que nous avons abordée est celle de l'absence d'une composante connexe ouverte infinie au point critique, une question importante du point de vue physique, puisqu'elle traduit la continuité de la transition de phase. Dans deux travaux en collaboration avec Hugo Duminil-Copin et Vladas Sidoravicius, nous montrons que la transition de phase est continue pour la percolation de Bernoulli sur le graphe Z^2x{0,...,k}, et pour la percolation FK sur le réseau carré avec paramètre q inférieur ou égal à 4. Enfin, la dernière question qui nous a guidés est la localité du point critique : la donnée des boules de grands rayons d'un graphe suffit-elle à identifier avec une bonne précision la valeur du point critique? Dans un travail en collaboration avec Sébastien Martineau, nous répondons de manière affirmative à cette question dans le cadre des graphes de Cayley de groupes abéliens.
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Klein, Philip N. (Philip Nathan). "An efficient parallel algorithm for planarity." Thesis, Massachusetts Institute of Technology, 1986. http://hdl.handle.net/1721.1/34303.

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Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1986.
MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING
Bibliography: leaves 56-57.
by Philip Nathan Klein.
M.S.
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Heinz, Adrian. "Planarity testing and drawing in Jedit 4.0." Virtual Press, 2001. http://liblink.bsu.edu/uhtbin/catkey/1204201.

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In this project, an enhanced version of Jedit is presented. Jedit is a Graph Editor developed at Ball State University by a group of students under the direction of Dr. Jay Bagga. The following paper describes the new version, named Jedit 4.0.In this version two new algorithms are implemented. These are: Planarity Testing and Planarity Drawing. The first algorithm tests a graph for planarity and the second one makes a planar embedding of the graph in a grid of size (n-2) x (n-2), where n is the order of the graph. Planar graphs have important applications in the fields of computer engineering, architecture, and many others.Jedit 4.0 also includes new features that were not available in earlier versions. The new features include: graph rotation operation, graph complement, drawing of well-known graphs, and credits window. Several modifications and additions to existing features and algorithms have also been carried out.Jedit 4.0 uses swing java technology what provides a more elegant look. Drop down menus have also been added to provide the user an easier way to use Jedit.
Department of Computer Science
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Books on the topic "Planarität"

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Cai, Jiazhen. An interative version of Hopcroft and Tarjan's planarity testing algorithm. New York: Courant Institute of Mathematical Sciences, New York University, 1987.

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Cai, Jiazhen. An interative version of Hopcroft and Tarjan's planarity testing algorithm. New York: Courant Institute of Mathematical Sciences, New York University, 1987.

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Planarity in ROMDD's of Multiple-Valued Symmetric Functions. Storming Media, 1996.

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Graphs: Block 3: Graphs 3 - Planarity and Colouring (Mathematics and Computing/technology: a Third Level Course). Open University Worldwide, 1995.

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Book chapters on the topic "Planarität"

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Läuchli, Peter. "Planarität." In Algorithmische Graphentheorie, 83–98. Basel: Birkhäuser Basel, 1991. http://dx.doi.org/10.1007/978-3-0348-5635-5_11.

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Büsing, Christina. "Planarität." In Graphen- und Netzwerkoptimierung, 91–109. Heidelberg: Spektrum Akademischer Verlag, 2010. http://dx.doi.org/10.1007/978-3-8274-2423-5_6.

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Wallis, W. D. "Planarity." In A Beginner’s Guide to Graph Theory, 113–22. Boston, MA: Birkhäuser Boston, 2007. http://dx.doi.org/10.1007/978-0-8176-4580-9_8.

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Foulds, L. R. "Planarity." In Universitext, 53–73. New York, NY: Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4612-0933-1_5.

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Balakrishnan, R., and K. Ranganathan. "Planarity." In A Textbook of Graph Theory, 175–205. New York, NY: Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-4529-6_8.

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Melnikov, O., V. Sarvanov, R. Tyshkevich, V. Yemelichev, and I. Zverovich. "Planarity." In Kluwer Texts in the Mathematical Sciences, 93–110. Dordrecht: Springer Netherlands, 1998. http://dx.doi.org/10.1007/978-94-017-1514-0_7.

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Balakrishnan, R., and K. Ranganathan. "Planarity." In A Textbook of Graph Theory, 152–84. New York, NY: Springer New York, 2000. http://dx.doi.org/10.1007/978-1-4419-8505-7_8.

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Wallis, W. D. "Planarity." In A Beginner’s Guide to Graph Theory, 105–14. Boston, MA: Birkhäuser Boston, 2000. http://dx.doi.org/10.1007/978-1-4757-3134-7_8.

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Aldous, Joan M., and Robin J. Wilson. "Planarity." In Graphs and Applications, 242–76. London: Springer London, 2000. http://dx.doi.org/10.1007/978-1-4471-0467-4_11.

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Saoub, Karin R. "Planarity." In Graph Theory, 339–72. Boca Raton: CRC Press, 2021. | Series: Textbooks in mathematics: Chapman and Hall/CRC, 2021. http://dx.doi.org/10.1201/9781138361416-7.

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Conference papers on the topic "Planarität"

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Cortese, Pier Francesco, and Giuseppe Di Battista. "Clustered planarity." In the twenty-first annual symposium. New York, New York, USA: ACM Press, 2005. http://dx.doi.org/10.1145/1064092.1064093.

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Didimo, W., F. Giordano, and G. Liotta. "Overlapping cluster planarity." In Asia-Pacific Symposium on Visualisation 2007. IEEE, 2007. http://dx.doi.org/10.1109/apvis.2007.329278.

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Di Battista, G., and R. Tamassia. "Incremental planarity testing." In 30th Annual Symposium on Foundations of Computer Science. IEEE, 1989. http://dx.doi.org/10.1109/sfcs.1989.63515.

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Brückner, Guido, and Ignaz Rutter. "Partial and Constrained Level Planarity." In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2017. http://dx.doi.org/10.1137/1.9781611974782.130.

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Vannoni, M., and G. Molesini. "Calibration of horizontally-placed planarity standards." In 2008 Conference on Precision Electromagnetic Measurements (CPEM 2008). IEEE, 2008. http://dx.doi.org/10.1109/cpem.2008.4574718.

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Galil, Zvi, Giuseppe F. Italiano, and Neil Sarnak. "Fully dynamic planarity testing (extended abstract)." In the twenty-fourth annual ACM symposium. New York, New York, USA: ACM Press, 1992. http://dx.doi.org/10.1145/129712.129761.

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Klein, Philip N., and John H. Reif. "An efficient parallel algorithm for planarity." In 27th Annual Symposium on Foundations of Computer Science (sfcs 1986). IEEE, 1986. http://dx.doi.org/10.1109/sfcs.1986.6.

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Weibel, Thomas, Christian Daul, Didier Wolf, and Ronald Rosch. "Planarity-enforcing higher-order graph cut." In 2011 18th IEEE International Conference on Image Processing (ICIP 2011). IEEE, 2011. http://dx.doi.org/10.1109/icip.2011.6116539.

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9

Angelini, Patrizio, Giuseppe Di Battista, Fabrizio Frati, Vít Jelínek, Jan Kratochvíl, Maurizio Patrignani, and Ignaz Rutter. "Testing Planarity of Partially Embedded Graphs." In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2010. http://dx.doi.org/10.1137/1.9781611973075.19.

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10

Vannoni, Maurizio, and Giuseppe Molesini. "Paired interferometric measurement of planarity and parallelism." In Optical Metrology, edited by Wolfgang Osten, Malgorzata Kujawinska, and Katherine Creath. SPIE, 2003. http://dx.doi.org/10.1117/12.501263.

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