On Balanced Open image in new window-Contact Representations

  • Stephane Durocher
  • Debajyoti Mondal
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8242)

Abstract

In a Open image in new window-contact representation of a planar graph G, each vertex is represented as an axis-aligned plus shape consisting of two intersecting line segments (or equivalently, four axis-aligned line segments that share a common endpoint), and two plus shapes touch if and only if their corresponding vertices are adjacent in G. Let the four line segments of a plus shape be its arms. In a c-balanced representation, c ≤ 1, every arm can touch at most \(\lceil c\varDelta\rceil\) other arms, where \(\varDelta\) is the maximum degree of G. The widely studied T- and L-contact representations are c-balanced representations, where c could be as large as 1. In contrast, the goal in a c-balanced representation is to minimize c. Let ck, where k ∈ {2,3}, be the smallest c such that every planar k-tree has a c-balanced representation. In this paper we show that 1/4 ≤ c2 ≤ 1/3 ( = b2) and 1/3 < c3 ≤ 1/2 ( = b3). Our result has several consequences. Firstly, planar k-trees admit 1-bend box-orthogonal drawings with boxes of size \(\lceil b_k\varDelta\rceil \times \lceil b_k\varDelta\rceil\), which generalizes a result of Tayu, Nomura, and Ueno. Secondly, they admit 1-bend polyline drawings with \(2\lceil b_k\varDelta\rceil\) slopes, which is significantly smaller than the \(2\varDelta\) upper bound established by Keszegh, Pach, and Pálvölgyi for arbitrary planar graphs.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    de Fraysseix, H., de Mendez, P.O., Rosenstiehl, P.: On triangle contact graphs. Combinatorics, Probability and Computing 3(2), 233–246 (1994)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Czyzowicz, J., Kranakis, E., Urrutia, J.: A simple proof of the representation of bipartite planar graphs as the contact graphs of orthogonal straight line segments. Information Processing Letters 66(3), 125–126 (1998)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Kobourov, S.G., Ueckerdt, T., Verbeek, K.: Combinatorial and geometric properties of planar Laman graphs. In: Khanna, S. (ed.) Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1668–1678. SIAM (2013)Google Scholar
  4. 4.
    Biedl, T.C., Kaufmann, M.: Area-efficient static and incremental graph drawings. In: Burkard, R.E., Woeginger, G.J. (eds.) ESA 1997. LNCS, vol. 1284, pp. 37–52. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  5. 5.
    Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16(1), 4–32 (1996)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Zhou, X., Nishizeki, T.: Orthogonal drawings of series-parallel graphs with minimum bends. SIAM Journal on Discrete Mathematics 22(4), 1570–1604 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Tayu, S., Nomura, K., Ueno, S.: On the two-dimensional orthogonal drawing of series-parallel graphs. Discrete Applied Mathematics 157(8), 1885–1895 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dujmović, V., Suderman, M., Wood, D.R.: Graph drawings with few slopes. Computational Geometry 38(3), 181–193 (2007)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Lenhart, W., Liotta, G., Mondal, D., Nishat, R.I.: Planar and plane slope number of partial 2-trees. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 412–423. Springer, Heidelberg (2013)Google Scholar
  10. 10.
    Jelínek, V., Jelínková, E., Kratochvíl, J., Lidický, B., Tesar, M., Vyskocil, T.: The planar slope number of planar partial 3-trees of bounded degree. Graphs and Combinatorics 29(4), 981–1005 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Keszegh, B., Pach, J., Pálvölgyi, D.: Drawing planar graphs of bounded degree with few slopes. SIAM Journal on Discrete Mathematics 27(2), 1171–1183 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Stephane Durocher
    • 1
  • Debajyoti Mondal
    • 1
  1. 1.Department of Computer ScienceUniversity of ManitobaCanada

Personalised recommendations