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A Linear-Time Algorithm for Symmetric Convex Drawings of Internally Triconnected Plane Graphs

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Abstract

Symmetry is one of the most important aesthetic criteria in Graph Drawing which can reveal the hidden structure in the graph. Convex drawing is a straight-line drawing where every facial cycle is drawn as a convex polygon.

In this paper, we prove that given an internally triconnected plane graph with symmetries, there exists a convex drawing of the graph which displays the given symmetries. We present a linear-time algorithm for constructing symmetric convex drawings of internally triconnected planar graphs.

This is an extension of a classical result due to Tutte (Proc. Lond. Math. Soc. 10(3):304–320, 1960; Proc. Lond. Math. Soc. 13:743–768, 1963) who proved that every triconnected plane graph with a given convex polygon as a boundary admits a convex drawing. Note that Tutte’s barycenter mapping method can be implemented in O(n 1.5) time and O(nlog n) space at best (Lipton et al. in SIAM J. Numer. Anal. 16:346–358, 1979).

Our divide and conquer algorithm explicitly exploits the fundamental properties of symmetric drawing, which consists of congruent drawings of isomorphic subgraphs. We first find an isomorphic subgraph of a given symmetric plane graph, and compute an angle-constrained convex drawing of the subgraph. Finally, a symmetric convex drawing of the given graph is constructed by merging repetitive copies of the congruent drawings of isomorphic subgraphs. For this purpose, we define a new problem of angle-constrained convex drawing of plane graphs, where some of outer vertices have angle constraints.

Our results also imply that there is a linear-time algorithm that constructs maximally symmetric convex drawings of triconnected planar graphs. Previous algorithm (Hong et al. in Discrete Comput. Geom. 36:283–311, 2006) constructs symmetric drawings of triconnected planar graphs with straight-lines.

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References

  1. Abelson, D., Hong, S., Taylor, D.E.: A Group-theoretic method for drawing graphs symmetrically. In: Graph Drawing (Proc. of GD 2002). Lecture Notes in Computer Science, vol. 2265, pp. 86–97. Springer, Berlin (2003)

    Google Scholar 

  2. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs (1993)

    Google Scholar 

  3. Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. North Holland, Amsterdam (1976)

    Google Scholar 

  4. Bonichon, N., Felsner, S., Mosbah, M.: Convex drawings of 3-connected plane graphs. In: Graph Drawing (Proc. of GD 2004). Lecture Notes in Computer Science, vol. 3383, pp. 60–70. Springer, Berlin (2005)

    Google Scholar 

  5. Buchheim, C., Junger, M.: Detecting symmetries by branch and cut. In: Graph Drawing (Proc. of GD 2001). Lecture Notes in Computer Science, pp. 178–188. Springer, Berlin (2001)

    Google Scholar 

  6. Chiba, N., Yamanouchi, T., Nishizeki, T.: Linear algorithms for convex drawings of planar graphs. In: Progress in Graph Theory, pp. 153–173. Academic Press, San Diego (1984)

    Google Scholar 

  7. Chrobak, M., Kant, G.: Convex grid drawings of 3-connected planar graphs. Int. J. Comput. Geom. Appl. 7, 211–223 (1997)

    Article  MathSciNet  Google Scholar 

  8. Chrobak, M., Goodrich, M.T., Tamassia, R.: Convex drawings of graphs in two and three dimensions. In: Proc. of the 12th Annual Symposium on Computational Geometry (SoCG 1996), pp. 319–328. ACM, New York (1996)

    Chapter  Google Scholar 

  9. Cook, S.A., Reckhow, R.A.: Time bounded random access machines. J. Comput. Syst. Sci. 7, 354–375 (1976)

    Article  MathSciNet  Google Scholar 

  10. Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, New York (1999)

    MATH  Google Scholar 

  11. de Fraysseix, H.: An heuristic for graph symmetry detection. In: Graph Drawing (Proc. of GD 1999). Lecture Notes in Computer Science, vol. 1731, pp. 276–285. Springer, Berlin (2000)

    Google Scholar 

  12. Eades, P., Lin, X.: Spring algorithms and symmetry. Theor. Comput. Sci. 240(2), 379–405 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fáry, I.: On straight line representations of planar graphs. Acta Sci. Math. Szeged 11, 229–233 (1948)

    Google Scholar 

  14. Hong, S., Eades, P.: Symmetric layout of disconnected graphs. In: Algorithms and Computation (Proc. of ISAAC 2003). Lecture Notes in Computer Science, vol. 2906, pp. 405–414. Springer, Berlin (2003)

    Google Scholar 

  15. Hong, S., Eades, P.: Drawing planar graphs symmetrically II: biconnected planar graphs. Algorithmica 42(2), 159–197 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hong, S., Eades, P.: Drawing planar graphs symmetrically III: oneconnected planar graphs. Algorithmica 44(1), 67–100 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hong, S., Nagamochi, H.: Convex drawings of hierarchical plane graphs. In: Proc. of the 17th Australasian Workshop on Combinatorial Algorithms (AWOCA 2006), Uluru, NT, 13–16 July 2006

  18. Hong, S., Nagamochi, H.: Convex drawings of graphs with non-convex boundary constraints. Discrete Appl. Math. 156, 2368–2380 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Hong, S., Eades, P., Lee, S.: Drawing series parallel digraphs symmetrically. Comput. Geom. Theory Appl. 17(3–4), 165–188 (2000)

    MATH  MathSciNet  Google Scholar 

  20. Hong, S., McKay, B., Eades, P.: A linear time algorithm for constructing maximally symmetric straight line drawings of triconnected planar graphs. Discrete Comput. Geom. 36, 283–311 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kamada, A., Miura, K., Nishizeki, T.: Convex grid drawings of plane graphs with rectangular contours. In: Proc. of ISAAC 2006. Lecture Notes in Computer Science, pp. 131–140. Springer, Berlin (2006)

    Google Scholar 

  22. Lin, X.: Analysis of algorithms for drawing graphs. Ph.D. Thesis, University of Queensland (1992)

  23. Lipton, R.J., Rose, D.J., Tarjan, R.E.: Generalized nested dissection. SIAM J. Numer. Anal. 16, 346–358 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lipton, R.J., North, S.C., Sandberg, J.S.: A method for drawing graphs. In: Proc. of Annual Symposium on Computational Geometry (SoCG), pp. 153–160. ACM, New York (1985)

    Chapter  Google Scholar 

  25. Liskovets, V.A.: A reductive technique for enumerating non-isomorphic planar maps. Discrete Math. 156, 197–217 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  26. Lubiw, A.: Some NP-complete problems similar to graph isomorphism. SIAM J. Comput. 10(1), 11–21 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  27. Mani, P.: Automorphismen von Polyedrischen Graphen. Math. Ann. 192, 279–303 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  28. Manning, J.: Geometric symmetry in graphs. Ph.D. Thesis, Purdue Univ. (1990)

  29. Manning, J., Atallah, M.J.: Fast detection and display of symmetry in trees. Congressus Numerantium 64, 159–169 (1988)

    MathSciNet  Google Scholar 

  30. Manning, J., Atallah, M.J.: Fast detection and display of symmetry in quterplanar graphs. Discrete Appl. Math. 39, 13–35 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  31. Miura, K., Azuma, M., Nishizeki, T.: Convex drawings of plane graphs of minimum outer apices. Int. J. Found. Comput. Sci. 17, 1115–1128 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  32. Miura, K., Nakano, S., Nishizeki, T.: Convex grid drawings of four-connected plane graphs. Int. J. Found. Comput. Sci. 17(5), 1031–1060 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  33. Purchase, H.: Which aesthetic has the greatest effect on human understanding? In: Graph Drawing (Proc. of GD 1997). Lecture Notes in Computer Science, vol. 1353, pp. 248–259. Springer, Berlin (1998)

    Google Scholar 

  34. Rote, G.: Strictly convex drawings of planar graphs. In: Proc. of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2005), pp. 728–734. SIAM, Philadelphia (2005)

    Google Scholar 

  35. Tarjan, R.E.: Data Structures and Network Algorithms. SIAM, Philadelphia (1983)

    Google Scholar 

  36. Thomassen, C.: Plane representations of graphs. In: Bondy, J.A., Murty, U.S.R. (eds.) Progress in Graph Theory, pp. 43–69. Academic Press, San Diego (1984)

    Google Scholar 

  37. Tutte, W.T.: Convex representations of graphs. Proc. Lond. Math. Soc. 10(3), 304–320 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  38. Tutte, W.T.: How to draw a graph. Proc. Lond. Math. Soc. 13, 743–768 (1963)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Information Technologies, University of Sydney, Sydney, Australia

    Seok-Hee Hong

  2. Dept. of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto, Japan

    Hiroshi Nagamochi

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  1. Seok-Hee Hong
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  2. Hiroshi Nagamochi
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Correspondence to Seok-Hee Hong.

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Hong, SH., Nagamochi, H. A Linear-Time Algorithm for Symmetric Convex Drawings of Internally Triconnected Plane Graphs. Algorithmica 58, 433–460 (2010). https://doi.org/10.1007/s00453-008-9275-y

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