Advertisement

Orthogonal 3D Shapes of Theta Graphs

  • Emilio Di Giacomo
  • Giuseppe Liotta
  • Maurizio Patrignani
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2528)

Abstract

The recent interest in three dimensional graph drawing has been motivating studies on how to extend two dimensional techniques to higher dimensions. A common approach for computing a 2D orthogonal drawing ofa graph separates the task of defining the shape ofthe drawing from the task of computing its coordinates. First results towards finding a three-dimensional counterpart ofthis approach are presented in [8],[9], where characterizations oforthogonal representations ofpaths and cycles are studied. In this note we show that the known characterization for cycles does not immediately extend to even seemingly simple graphs such as theta graphs. A sufficient condition for recognizing three-dimensional orthogonal representations oftheta graphs is also presented.

Keywords

Disjoint Path Direction Label Graph Drawing Permutation Graph Chromatic Polynomial 
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.

References

  1. 1.
    D. S. Archdeacon and J. Širáň. Characterizing planarity using theta graphs. J. Graph Theory, 27:17–20, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    J. Brown, C. Hickman, A. Sokal, and D. Wagner. On the chromatic roots of generalized theta graphs. J. of Combinatorial Theory, Series B, to appear.Google Scholar
  3. 3.
    I. Bruβ and A. Frick. Fast interactive 3-D graph visualization. In F. J. Brandenburg,editor, Graph Drawing (Proc. GD’ 95), volume 1027 of Lecture Notes Comput. Sci., pages 99–110. Springer-Verlag, 1996.Google Scholar
  4. 4.
    G. Chartrand and F. Harary. Planar permutation graphs. Ann. Inst. H. Poincaré, Sect B, 3:433–438, 1967.MathSciNetzbMATHGoogle Scholar
  5. 5.
    I. F. Cruz and J. P. Twarog. 3D graph drawing with simulated annealing. In F. J. Brandenburg, editor, Graph Drawing (Proc. GD’ 95), volume 1027 of Lecture Notes Comput. Sci., pages 162–165. Springer-Verlag, 1996.Google Scholar
  6. 6.
    G. Di Battista, P. Eades, R. Tamassia, and I. G. Tollis. Graph Drawing. Prentice Hall, Upper Saddle River, NJ, 1999.zbMATHCrossRefGoogle Scholar
  7. 7.
    G. Di Battista, G. Liotta, A. Lubiw, and S. Whitesides. Embedding problems for paths with direction constrained edges. In Annual International Computing and Combinatorics Conference, (COCOON 2000), volume 1858 of Lecture Notes Comput. Sci., pages 64–73. Springer-Verlag, 2000.Google Scholar
  8. 8.
    G. Di Battista, G. Liotta, A. Lubiw, and S. Whitesides. Orthogonal drawings of cycles in 3d space. In J. Marks, editor, Graph Drawing (Proc. GD’ 00), volume 1984 of Lecture Notes Comput. Sci. Springer-Verlag, 2001.Google Scholar
  9. 9.
    G. Di Battista, G. Liotta, A. Lubiw, and S. Whitesides. Embedding problems for paths with direction constrained edges. J. of Theor. Comp. Sci., 2002. to appear.Google Scholar
  10. 10.
    E. Di Giacomo, G. Liotta, and M. Patrignani. On orthogonal 3d shapes oftheta graphs. Tech. Report RT-DIA-71-2002, Dept. of Computer Sci., Univ. di Roma Tre, 2002. http://web.dia.uniroma3.it/research/.
  11. 11.
    D. Dodson. COMAIDE: Information visualization using cooperative 3D diagram layout. In F. J. Brandenburg, editor, Graph Drawing (Proc. GD’ 95), volume 1027 of Lecture Notes Comput. Sci., pages 190–201. Springer-Verlag, 1996.Google Scholar
  12. 12.
    D. Eichhorn, D. Mubayi, K. O’Bryant, and D. B. West. Edge-bandwidth oftheta graphs. J. Graph Theory, 35:89–98, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    F. Harary. Graph Theory. Addison-Wesley, Reading, Mass., 1969.Google Scholar
  14. 14.
    T. Jiang, D. Mubayi, A. Shastri, and D. B. West. Edge-bandwidth ofgraphs. SIAM Journal on Discrete Mathematics, 12(3):307–316, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    A. Papakostas and I. G. Tollis. Algorithms for incremental orthogonal graph drawing in three dimensions. Journal of Graph Algorithms and Applications, 3(4):81–115, 1999.zbMATHMathSciNetGoogle Scholar
  16. 16.
    G. Peck and A. Shastri. Bandwidth oftheta graphs with short paths. Discrete Mathematics, 103, 1992.Google Scholar
  17. 17.
    I. Sciriha and S. Fiorini. On the characteristic polynomial ofhomeomorphic images ofa graph. Discrete Mathematics, 174:293–308, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    R. Tamassia. On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput., 16(3):421–444, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    G. Vijayan and A. Wigderson. Rectilinear graphs and their embeddings. SIAM J. Comput., 14:355–372, 1985.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Emilio Di Giacomo
    • 1
  • Giuseppe Liotta
    • 1
  • Maurizio Patrignani
    • 2
  1. 1.Università di PerugiaItaly
  2. 2.Università di Roma TreItaly

Personalised recommendations