Euler Diagram Decomposition

  • Andrew Fish
  • Jean Flower
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5223)


Euler diagrams are a common visual representation of set-theoretic statements, and they have been used for visualising the results of database search queries or as the basis of diagrammatic logical constraint languages for use in software specification. Such applications rely upon the ability to automatically generate diagrams from an abstract description. However, this problem is difficult and is known to be NP-complete under certain wellformedness conditions. Therefore methods to identify when and how one can decompose abstract Euler diagrams into simpler components provide a vital step in improving the efficiency of tools which implement a generation process. One such decomposition, called diagram nesting, has previously been identified and exploited. In this paper, we make substantial progress, defining the notion of a disconnecting contour and identifying the conditions on an abstract Euler diagram that allow us to identify disconnecting contours. If a diagram has a disconnecting contour, we can draw it more easily, by combining the results of drawing smaller diagrams. The drawing problem is just one context which benefits from such diagram decomposition - we can also use the disconnecting contour to provide a more natural semantic interpretation of the Euler diagram.


Intersection Graph Dual Graph Euler Diagram Abstract Diagram Semantic Predicate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Chow, S., Ruskey, F.: Drawing area-proportional Venn and Euler diagrams. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 466–477. Springer, Heidelberg (2004)Google Scholar
  2. 2.
    Chow, S.C.: Generating and Drawing Area-Proportional Euler and Venn Diagrams. Ph.D thesis, University of Victoria (2007)Google Scholar
  3. 3.
    DeChiara, R., Erra, U., Scarano, V.: VennFS: A Venn diagram file manager. In: Proceedings of Information Visualisation, pp. 120–126. IEEE Computer Society, Los Alamitos (2003)Google Scholar
  4. 4.
    DeChiara, R., Erra, U., Scarano, V.: A system for virtual directories using Euler diagrams. In: Proceedings of Euler Diagrams 2004. Electronic Notes in Theoretical Computer Science, vol. 134, pp. 33–53 (2005)Google Scholar
  5. 5.
    Euler, L.: Letters a une princesse dallemagne sur divers sujets de physique et de philosophie. Letters 2, 102–108 (1775)Google Scholar
  6. 6.
    Fish, A., Flower, J.: Abstractions of Euler diagrams. In: Proceedings of Euler Diagrams 2004, Brighton, UK. ENTCS, vol. 134, pp. 77–101 (2005)Google Scholar
  7. 7.
    Fish, A., Flower, J.: Investigating reasoning with constraint diagrams. In: Visual Language and Formal Methods 2004, Rome, Italy. ENTCS, vol. 127, pp. 53–69. Elsevier, Amsterdam (2005)Google Scholar
  8. 8.
    Flower, J., Fish, A., Howse, J.: Euler diagram generation. Journal of Visual Languages and Computing (2008),
  9. 9.
    Flower, J., Howse, J.: Generating Euler diagrams. In: Proceedings of 2nd International Conference on the Theory and Application of Diagrams, Georgia, USA, April 2002, pp. 61–75. Springer, Heidelberg (2002)Google Scholar
  10. 10.
    Flower, J., Howse, J., Taylor, J.: Nesting in Euler diagrams. In: International Workshop on Graph Transformation and Visual Modeling Techniques, pp. 99–108 (2002)Google Scholar
  11. 11.
    Flower, J., Howse, J., Taylor, J.: Nesting in Euler diagrams: syntax, semantics and construction. Software and Systems Modelling 3, 55–67 (2004)CrossRefGoogle Scholar
  12. 12.
    Hammer, E., Shin, S.J.: Euler’s visual logic. History and Philosophy of Logic, 1–29 (1998)Google Scholar
  13. 13.
    Harel, D.: On visual formalisms. In: Glasgow, J., Narayan, N.H., Chandrasekaran, B. (eds.) Diagrammatic Reasoning, pp. 235–271. MIT Press, Cambridge (1998)Google Scholar
  14. 14.
    Howse, J., Molina, F., Shin, S.-J., Taylor, J.: Type-syntax and token-syntax in diagrammatic systems. In: Proceedings FOIS-2001: 2nd International Conference on Formal Ontology in Information Systems, Maine, USA, pp. 174–185. ACM Press, New York (2001)CrossRefGoogle Scholar
  15. 15.
    Howse, J., Stapleton, G., Flower, J., Taylor, J.: Corresponding regions in Euler diagrams. In: Proceedings of 2nd International Conference on the Theory and Application of Diagrams, Georgia, USA, April 2002, pp. 76–90. Springer, Heidelberg (2002)Google Scholar
  16. 16.
    Howse, J., Stapleton, G., Taylor, J.: Spider diagrams. LMS Journal of Computation and Mathematics 8, 145–194 (2005)MATHMathSciNetGoogle Scholar
  17. 17.
    Kent, S.: Constraint diagrams: Visualizing invariants in object oriented modelling. In: Proceedings of OOPSLA 1997, October 1997, pp. 327–341. ACM Press, New York (1997)CrossRefGoogle Scholar
  18. 18.
    Kestler, H., Muller, A., Gress, T., Buchholz, M.: Generalized Venn diagrams: A new method for visualizing complex genetic set relations. Journal of Bioinformatics 21(8), 1592–1595 (2005)CrossRefGoogle Scholar
  19. 19.
    Ruskey, F.: A survey of Venn diagrams. Electronic Journal of Combinatorics (1997),
  20. 20.
    Shimojima, A.: Inferential and expressive capacities of graphical representations: Survey and some generalizations. In: Blackwell, A.F., Marriott, K., Shimojima, A. (eds.) Diagrams 2004. LNCS (LNAI), vol. 2980, pp. 18–21. Springer, Heidelberg (2004)Google Scholar
  21. 21.
    Shin, S.-J.: The Logical Status of Diagrams. Cambridge University Press, Cambridge (1994)MATHGoogle Scholar
  22. 22.
    Stapleton, G., Thompson, S., Howse, J., Taylor, J.: The expressiveness of spider diagrams. Journal of Logic and Computation 14(6), 857–880 (2004)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Swoboda, N., Allwein, G.: Using DAG transformations to verify Euler/Venn homogeneous and Euler/Venn FOL heterogeneous rules of inference. Journal on Software and System Modeling 3(2), 136–149 (2004)CrossRefGoogle Scholar
  24. 24.
    Venn, J.: On the diagrammatic and mechanical representation of propositions and reasonings. Phil. Mag. (1880)Google Scholar
  25. 25.
    Verroust, A., Viaud, M.-L.: Ensuring the drawability of Euler diagrams for up to eight sets. In: Blackwell, A.F., Marriott, K., Shimojima, A. (eds.) Diagrams 2004. LNCS (LNAI), vol. 2980, pp. 128–141. Springer, Heidelberg (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Andrew Fish
    • 1
  • Jean Flower
    • 1
  1. 1.Visual Modelling Group, School of Computing, Mathematical and Information SciencesUniversity of BrightonBrightonUK

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