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Optimal st-Orientations for Plane Triangulations

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Algorithmic Aspects in Information and Management (AAIM 2007)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 4508))

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Abstract

For a plane triangulation G with n vertices, it has been proved that there exists a plane triangulation G with n vertices such that for any st-orientation of G, the length of the longest directed paths of G from s to t is \(\geq \lfloor \frac{2n}{3}\rfloor\) [18] . In this paper, we prove the bound \(\frac{2n}{3}\) is optimal by showing that every plane triangulation G with n-vertices admits an st-orientation with the length of its longest directed paths bounded by \(\frac {2n}{3}+O(1)\). In addition, this st-orientation is constructible in linear time. A by-product of this result is that every plane graph G with n vertices admits a visibility representation with height \(\le \frac{2n}{3}+O(1)\), constructible in linear time, which is also optimal.

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References

  1. di Battista, G., Eades, P., Tammassia, R., Tollis, I.: Graph Drawing: Algorithms for the Visualization of Graphs. Princeton Hall, Princeton (1998)

    Google Scholar 

  2. Chiang, Y.-T., Lin, C.-C., Lu, H.-I.: Orderly spanning trees with applications to graph encoding and graph drawing. In: Proc. of the 12th Annual ACM-SIAM SODA, pp. 506–515. ACM Press, New York (2001)

    Google Scholar 

  3. Ebert, J.: st-ordering the vertices of biconnected graphs. Computing 30(1), 19–33 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  4. Fraysseix, H.D., de Mendez, P.O., Rosenstiehl, P.: Bipolar orientations revisited. Discrete Applied Mathematics 56, 157–179 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  5. Kant, G.: A more compact visibility representation. International Journal of Computational Geometry and Applications 7, 197–210 (1997)

    Article  MathSciNet  Google Scholar 

  6. Kant, G., He, X.: Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems. Theoretical Computer Science 172, 175–193 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. Lempel, A., Even, S., Cederbaum, I.: An algorithm for planarity testing of graphs. In: Theory of Graphs, Proc. of an International Symposium, Rome, July 1966, pp. 215–232 (1967)

    Google Scholar 

  8. Even, S., Tarjan, R.: Computing an st-numbering. Theoretical Computer Science 2, 339–344 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  9. Lin, C.-C., Lu, H.-I., Sun, I.-F.: Improved compact visibility representation of planar graph via Schnyder’s realizer. SIAM Journal on Discrete Mathematics 18, 19–29 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Maon, Y., Schieber, B., Vishkin, U.: Parallel ear decomposition search (eds) and st-numbering in graphs. Theoretical Computer Science 47, 277–298 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ossona de Mendez, P.: Orientations bipolaires. PhD thesis, Ecole des Hautes Etudes en Sciences Sociales, Paris (1994)

    Google Scholar 

  12. Papamanthou, C., Tollis, I.G.: Applications of parameterized st-orientations in graph drawing algorithms. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 355–367. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  13. Papakostas, A., Tollis, I.G.: Algorithms for area-efficient orthogonal drawings. Computational Geometry: Theory and Applications 9, 83–110 (1998)

    MATH  MathSciNet  Google Scholar 

  14. Rosenstiehl, P., Tarjan, R.E.: Rectilinear planar layouts and bipolar orientations of planar graphs. Discrete Comput. Geom. 1, 343–353 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  15. Schnyder, W.: Planar graphs and poset dimension. Order 5, 323–343 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  16. Tamassia, R., Tollis, I.G.: An unified approach to visibility representations of planar graphs. Discrete Comput. Geom. 1, 321–341 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  17. Tarjan, R.: Two streamlined depth-first search algorithms. Fundamentae Informatica 9, 85–94 (1986)

    MATH  MathSciNet  Google Scholar 

  18. Zhang, H., He, X.: New Theoretical Bounds of Visibility Representation of Plane Graphs. In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 425–430. Springer, Heidelberg (2005)

    Google Scholar 

  19. He, X., Zhang, H.: Nearly Optimal Visibility Representations of Plane Graphs. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 407–418. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Computer Science Department, University of Alabama in Huntsville, Huntsville, AL, 35899, USA

    Huaming Zhang

  2. Department of Computer Science and Engineering, SUNY at Buffalo, Buffalo, NY, 14260, USA

    Xin He

Authors
  1. Huaming Zhang
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  2. Xin He
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Ming-Yang Kao Xiang-Yang Li

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Zhang, H., He, X. (2007). Optimal st-Orientations for Plane Triangulations. In: Kao, MY., Li, XY. (eds) Algorithmic Aspects in Information and Management. AAIM 2007. Lecture Notes in Computer Science, vol 4508. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72870-2_28

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  • DOI: https://doi.org/10.1007/978-3-540-72870-2_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-72868-9

  • Online ISBN: 978-3-540-72870-2

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