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Open Rectangle-of-Influence Drawings of Inner Triangulated Plane Graphs

  • Kazuyuki Miura
  • Tetsuya Matsuno
  • Takao Nishizeki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4372)

Abstract

A straight-line drawing of a plane graph is called an open rectangle-of-influence drawing if there is no vertex in the proper inside of the axis-parallel rectangle defined by the two ends of every edge. In an inner triangulated plane graph, every inner face is a triangle although the outer face is not always a triangle. In this paper, we first obtain a sufficient condition for an inner triangulated plane graph G to have an open rectangle-of-influence drawing; the condition is expressed in terms of a labeling of angles of a subgraph of G. We then present an O(n 1.5/logn)-time algorithm to examine whether G satisfies the condition and, if so, construct an open rectangle-of-influence drawing of G on an (n − 1) ×(n − 1) integer grid, where n is the number of vertices in G.

Keywords

Plane Graph Outer Face Good Label Outer Vertex Facial Cycle 
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.
    Biedl, T., Bretscher, A., Meijer, H.: Rectangle of influence drawings of graphs without filled 3-cycles. In: Kratochvíl, J. (ed.) GD 1999. LNCS, vol. 1731, pp. 359–368. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  2. 2.
    Biedl, T., Kant, G., Kaufmann, M.: On triangulating planar graphs under the four-connectivity constraint. Algorithmica 19(4), 427–446 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chrobak, M., Kant, G.: Convex grid darwings of 3-connected planar graphs. Int. J. Comput. Geom Appl. 7(3), 211–223 (1997)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, Upper Saddle River (1999)zbMATHGoogle Scholar
  5. 5.
    Di Battista, G., Lenhart, W., Liotta, G.: Proximity drawability: A survey. In: Tamassia, R., Tollis, I(Y.) G. (eds.) GD 1994. LNCS, vol. 894, pp. 328–339. Springer, Heidelberg (1995)Google Scholar
  6. 6.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a graph on a grid. Combinatorica 10, 41–51 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)Google Scholar
  8. 8.
    Hochbaum, D.S.: Faster pseudoflow-based algorithms for the bipartite matching and the closure problems (Abstract). In: CORS/SCRO-INFORMS Joint Int. Meeting, Banff, Canada, May 16-19, p. 46 (2004)Google Scholar
  9. 9.
    Hochbaum, D.S., Chandran, B.G.: Further below the flow decomposition barrier of maximum flow for bipartite matching and maximum closure. Working paper (2004)Google Scholar
  10. 10.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Liotta, G., Lubiw, A., Meijer, H., Whitesides, S.H.: The rectangle of influence drawability problem. Comput. Geom. Theory and Applications 10(1), 1–22 (1998)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Miura, K., Nishizeki, T.: Rectangle-of-influence drawings of four-connected plane graphs. In: Proc. of Information Visualisation 2005, Asia-Pacific Symposium on Information Visualisation (APVIS2005). ACS, vol. 45, pp. 71–76 (2005)Google Scholar
  13. 13.
    Miura, K., Nakano, S., Nishizeki, T.: Grid drawings of four-connected plane graphs. Discrete & Computational Geometry 26(1), 73–87 (2001)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Nishizeki, T., Rahman, M.S.: Planar Graph Drawing. World Scientific, Singapore (2004)zbMATHGoogle Scholar
  15. 15.
    Schnyder, W.: Embedding planar graphs in the grid. In: Proc. 1st Annual ACM-SIAM Symp. on Discrete Algorithms, San Francisco, pp. 138–147. ACM Press, New York (1990)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Kazuyuki Miura
    • 1
  • Tetsuya Matsuno
    • 2
  • Takao Nishizeki
    • 2
  1. 1.Faculty of Symbiotic Systems Science, Fukushima University, Fukushima 960-1296Japan
  2. 2.Graduate School of Information Sciences, Tohoku University, Sendai 980-8579Japan

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