Small-Area Orthogonal Drawings of 3-Connected Graphs

  • Therese Biedl
  • Jens M. SchmidtEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)


It is well-known that every graph with maximum degree 4 has an orthogonal drawing with area at most \(\frac{49}{64}n^2+O(n)\approx 0.76n^2\). In this paper, we show that if the graph is 3-connected, then the area can be reduced even further to \(\frac{9}{16}n^2+O(n) \approx 0.56n^2\). The drawing uses the 3-canonical order for (not necessarily planar) 3-connected graphs, which is a special Mondshein sequence and can hence be computed in linear time. To our knowledge, this is the first application of a Mondshein sequence in graph drawing.



We wish to thank the anonymous reviewers for their constructive comments.


  1. 1.
    Biedl, T.C., Kant, G.: A better heuristic for orthogonal graph drawings. Comput. Geom. 9(3), 159–180 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Cheriyan, J., Maheshwari, S.N.: Finding nonseparating induced cycles and independent spanning trees in 3-connected graphs. J. Algorithms 9(4), 507–537 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Chiang, Y.-T., Lin, C.-C., Lu, H.-I.: Orderly spanning trees with applications. SIAM J. Comput. 34(4), 924–945 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    de Fraysseix, H., de Mendez, P.O.: Regular orientations, arboricity and augmentation. In: Tamassia, R., Tollis, I.G. (eds.) GD 1994. LNCS, vol. 894, pp. 111–118. Springer, Heidelberg (1995) CrossRefGoogle Scholar
  5. 5.
    Dietz, P., Sleator, D.: Two algorithms for maintaining order in a list. In: 19th Annual ACM Symposium on Theory of Computing, pp. 365–372 (1987)Google Scholar
  6. 6.
    de Fraysseix, H., Pollack, R., Pach, J.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16(1), 4–32 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Mondshein, L.F.: Combinatorial ordering and the geometric embedding of graphs. Ph.D. thesis, M.I.T. Lincoln Laboratory / Harvard University (1971)Google Scholar
  9. 9.
    Papakostas, A., Tollis, I.G.: Algorithms for area-efficient orthogonal drawings. Comput. Geom. 9(1–2), 83–110 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Rosenstiehl, P., Tarjan, R.E.: Rectilinear planar layouts and bipolar orientation of planar graphs. Discrete Comput. Geom. 1, 343–353 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Schäffter, M.: Drawing graphs on rectangular grids. Discrete Appl. Math. 63, 75–89 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Schmidt, J.M.: The Mondshein sequence (2013).
  13. 13.
    Schmidt, J.M.: The Mondshein sequence. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 967–978. Springer, Heidelberg (2014) Google Scholar
  14. 14.
    Biedl, T.: Optimal orthogonal drawings of triconnected plane graphs. In: McCune, W., Padmanabhan, R. (eds.) Automated Deduction in Equational Logic and Cubic Curves. LNCS, vol. 1095, pp. 333–344. Springer, Heidelberg (1996) Google Scholar
  15. 15.
    Tamassia, R., Tollis, I.: A unified approach to visibility representations of planargraphs. Discrete Comput. Geom. 1, 321–341 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Tamassia, R., Tollis, I.G., Vitter, J.S.: Lower bounds for planar orthogonal drawings of graphs. Inf. Process. Lett. 39, 35–40 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Valiant, L.G.: Universality considerations in VLSI circuits. IEEE Trans. Comput. C–30(2), 135–140 (1981)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Institute of MathematicsTU IlmenauIlmenauGermany

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