The Topology of Bendless Three-Dimensional Orthogonal Graph Drawing

  • David Eppstein
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5417)


We define an xyz graph to be a spatial embedding of a 3-regular graph such that the edges at each vertex are mutually perpendicular and no three points lie on an axis-parallel line. We describe an equivalence between xyz graphs and 3-face-colored polyhedral maps, under which bipartiteness of the graph is equivalent to orientability of the map. We show that planar graphs are xyz graphs if and only if they are bipartite, cubic, and three-connected. It is NP-complete to recognize xyz graphs, but we show how to do this in time O(n2 n/2).


Planar Graph Cayley Graph Graph Drawing Symmetric Graph Split Vertex 
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.


  1. 1.
    Annexstein, F., Baumslag, M., Rosenberg, A.L.: Group action graphs and parallel architectures. SIAM J. Comput. 19(3), 544–569 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Biedl, T., Shermer, T.C., Whitesides, S., Wismath, S.K.: Bounds for orthogonal 3-D graph drawing. J. Graph Alg. Appl. 3(4), 63–79 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Biedl, T., Thiele, T., Wood, D.R.: Three-dimensional orthogonal graph drawing with optimal volume. Algorithmica 44(3), 233–255 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bonnington, C.P., Little, C.H.C.: The Foundations of Topological Graph Theory. Springer, Heidelberg (1995)CrossRefzbMATHGoogle Scholar
  5. 5.
    Calamoneri, T., Massini, A.: Optimal three-dimensional layout of interconnection networks. Theor. Comput. Sci. 255(1-2), 263–279 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Closson, M., Gartshore, S., Johansen, J.R., Wismath, S.K.: Fully dynamic 3-dimensional orthogonal graph drawing. J. Graph Alg. Appl. 5(2), 1–34 (2001)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Craft, D.L., White, A.T.: 3-maps. Discrete Math. (2008)Google Scholar
  8. 8.
    Eades, P., Stirk, C., Whitesides, S.: The techniques of Komolgorov and Bardzin for three-dimensional orthogonal graph drawings. Inf. Proc. Lett. 60(2), 97–103 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Eades, P., Symvonis, A., Whitesides, S.: Two algorithms for three dimensional orthogonal graph drawing. In: North, S.C. (ed.) GD 1996. LNCS, vol. 1190, pp. 139–154. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  10. 10.
    Eppstein, D.: Dynamic generators of topologically embedded graphs. In: Proc. 14th Symp. Discrete Algorithms, pp. 599–608. ACM and SIAM (January 2003)Google Scholar
  11. 11.
    Eppstein, D.: Isometric diamond subgraphs. In: Proc. 16th Int. Symp. Graph Drawing (2008)Google Scholar
  12. 12.
    Even, S., Tarjan, R.E.: Computing an st-numbering. Theor. Comput. Sci. 2(3), 339–344 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Gaiha, P., Gupta, S.K.: Adjacent vertices on a permutohedron. SIAM J. Appl. Math. 32(2), 323–327 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Kochol, M.: 3-regular non 3-edge-colorable graphs with polyhedral embeddings in orientable surfaces. In: Proc. 16th Int. Symp. Graph Drawing (2008)Google Scholar
  15. 15.
    Papakostas, A., Tollis, I.G.: Algorithms for incremental orthogonal graph drawing in three dimensions. J. Graph Alg. Appl. 3(4), 81–115 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Preparata, F.P., Vuillemin, J.: The cube-connected cycles: a versatile network for parallel computation. Commun. ACM 24(5), 300–309 (1981)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Royle, G., Conder, M., McKay, B., Dobscanyi, P.: Cubic symmetric graphs (The Foster Census). Web page (2001),
  18. 18.
    Wood, D.R.: An algorithm for three-dimensional orthogonal graph drawing. In: Whitesides, S.H. (ed.) GD 1998. LNCS, vol. 1547, pp. 332–346. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  19. 19.
    Wood, D.R.: Bounded degree book embeddings and three-dimensional orthogonal graph drawing. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, pp. 312–327. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  20. 20.
    Wood, D.R.: Optimal three-dimensional orthogonal graph drawing in the general position model. Theor. Comput. Sci. 299(1-3), 151–178 (2003)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • David Eppstein
    • 1
  1. 1.Computer Science DepartmentUniversity of CaliforniaIrvineUSA

Personalised recommendations