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Optimal 3D Angular Resolution for Low-Degree Graphs

  • David Eppstein
  • Maarten Löffler
  • Elena Mumford
  • Martin Nöllenburg
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6502)

Abstract

We show that every graph of maximum degree three can be drawn in three dimensions with at most two bends per edge, and with 120° angles between any two edge segments meeting at a vertex or a bend. We show that every graph of maximum degree four can be drawn in three dimensions with at most three bends per edge, and with 109.5° angles, i.e., the angular resolution of the diamond lattice, between any two edge segments meeting at a vertex or bend.

Keywords

Maximum Degree Angular Resolution Horizontal Segment Diamond Lattice Edge Segment 
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.
    Angelini, P., Cittadini, L., Di Battista, G., Didimo, W., Frati, F., Kaufmann, M., Symvonis, A.: On the perspectives opened by right angle crossing drawings. In: Eppstein, D., Gansner, E.R. (eds.) GD 2009. LNCS, vol. 5849, pp. 21–32. Springer, Heidelberg (2010), doi:10.1007/978-3-642-11805-0_5CrossRefGoogle Scholar
  2. 2.
    Biedl, T.C., Bose, P., Demaine, E.D., Lubiw, A.: Efficient algorithms for Petersen’s matching theorem. Journal of Algorithms 38(1), 110–134 (2001), doi:10.1006/jagm.2000.1132MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Biedl, T.C., Thiele, T., Wood, D.R.: Three-dimensional orthogonal graph drawing with optimal volume. Algorithmica 44(3), 233–255 (2006), doi:10.1007/s00453-005-1148-zMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Carlson, J., Eppstein, D.: Trees with convex faces and optimal angles. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 77–88. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Clare, B.W., Kepert, D.L.: The closest packing of equal circles on a sphere. Proc. Roy. Soc. London A 405(1829), 329–344 (1986), doi:10.1098/rspa.1986.0056MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Demaine, E.D.: Problem 46: 3D minimum-bend orthogonal graph drawings. The Open Problems Project. Posed by David R. Wood at the CCCG, open-problem session (2002), http://maven.smith.edu/~orourke/TOPP/P46.html
  7. 7.
    Didimo, W., Eades, P., Liotta, G.: Drawing graphs with right angle crossings. In: Dehne, F., Gavrilova, M., Sack, J.-R., Tóth, C.D. (eds.) WADS 2009. LNCS, vol. 5664, pp. 206–217. Springer, Heidelberg (2009), doi:10.1007/978-3-642-03367-4_19CrossRefGoogle Scholar
  8. 8.
    Dujmović, V., Gudmundsson, J., Morin, P., Wolle, T.: Notes on large angle crossing graphs. arXiv:0908.3545 (2009)Google Scholar
  9. 9.
    Duncan, C.A., Eppstein, D., Kobourov, S.G.: The geometric thickness of low degree graphs. In: Proc. 20th ACM Symp. Computational Geometry (SoCG 2004), pp. 340–346 (2004), doi:10.1145/997817.997868, arXiv:cs.CG/0312056Google Scholar
  10. 10.
    Eades, P., Symvonis, A., Whitesides, S.: Three-dimensional orthogonal graph drawing algorithms. Discrete Applied Mathematics 103(1-3), 55–87 (2000), doi:10.1016/S0166-218X(00)00172-4MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Eiglsperger, M., Fekete, S.P., Klau, G.W.: Orthogonal graph drawing. In: Kaufmann, M., Wagner, D. (eds.) Drawing Graphs. LNCS, vol. 2025, pp. 121–171. Springer, Heidelberg (2001), doi:10.1007/3-540-44969-8_6CrossRefGoogle Scholar
  12. 12.
    Eppstein, D.: Isometric diamond subgraphs. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 384–389. Springer, Heidelberg (2009), doi:10.1007/978-3-642-00219-9_37CrossRefGoogle Scholar
  13. 13.
    Garg, A., Tamassia, R.: Planar drawings and angular resolution: algorithms and bounds. In: van Leeuwen, J. (ed.) ESA 1994. LNCS, vol. 855, pp. 12–23. Springer, Heidelberg (1994), doi:10.1007/BFb0049393CrossRefGoogle Scholar
  14. 14.
    Gutwenger, C., Mutzel, P.: Planar polyline drawings with good angular resolution. In: Whitesides, S.H. (ed.) GD 1998. LNCS, vol. 1547, pp. 167–182. Springer, Heidelberg (1999), doi:10.1007/3-540-37623-2_13CrossRefGoogle Scholar
  15. 15.
    Huang, W., Hong, S.-H., Eades, P.: Effects of crossing angles. In: Proc. IEEE Pacific Visualization Symp., pp. 41–46 (2008), doi:10.1109/PACIFICVIS.2008.4475457Google Scholar
  16. 16.
    Malitz, S.: On the angular resolution of planar graphs. In: Proc. 24th ACM Symp. Theory of Computing (STOC 1992), pp. 527–538 (1992), doi:10.1145/129712.129764Google Scholar
  17. 17.
    Petersen, J.: Die Theorie der regulären Graphs. Acta Math. 15(1), 193–220 (1891), doi:10.1007/BF02392606MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tammes, P.M.L.: On the origin of the number and arrangement of the places of exit on the surface of pollen grains. Ree. Trav. Bot. Néerl. 27, 1–82 (1930)Google Scholar
  19. 19.
    Wood, D.R.: Optimal three-dimensional orthogonal graph drawing in the general position model. Theor. Comput. Sci. 299(1-3), 151–178 (2003), doi:10.1016/S0304-3975(02)00044-0MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • David Eppstein
    • 1
  • Maarten Löffler
    • 1
  • Elena Mumford
    • 2
  • Martin Nöllenburg
    • 1
  1. 1.Department of Computer ScienceUniversity of CaliforniaIrvineUSA
  2. 2.EindhovenThe Netherlands

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