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Bichromatic Point-Set Embeddings of Trees with Fewer Bends

(Extended Abstract)
  • Khaled Mahmud Shahriar
  • Md. Saidur Rahman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8344)

Abstract

Let G be a planar graph such that each vertex of G is colored by either red or blue color. Assume that there are nr red vertices and n b blue vertices in G. Let S be a set of fixed points in the plane such that |S| = n r  + n b where nr points in S are colored by red color and nb points in S are colored by blue color. A bichromatic point-set embedding of G on S is a crossing free drawing of G such that each red vertex of G is mapped to a red point in S, each blue vertex of G is mapped to a blue point in S, and each edge is drawn as a polygonal curve. In this paper, we study the problem of computing bichromatic point-set embeddings of trees on two restricted point-sets which we call “ordered point-set” and “properly-colored point-set”. We show that trees have bichromatic point-set embeddings on these two special types of point-sets with at most one bend per edge and such embeddings can be found in linear time.

Keywords

Trees Bichromatic point-set embedding Bend Ordered point-set Properly-colored point-set 

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References

  1. 1.
    Cabello, S.: Planar embeddability of the vertices of a graph using a fixed point set is np-hard. Journal of Graph Algorithms and Applications 10(2), 353–363 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H., Trotta, F., Wismath, S.K.: k-colored point-set embeddability of outerplanar graphs. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 318–329. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. 3.
    Di Giacomo, E., Liotta, G., Trotta, F.: On embedding a graph on two sets of points. International Journal of Foundations of Computer Science 17(05), 1071–1094 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Di Giacomo, E., Liotta, G., Trotta, F.: Drawing colored graphs with constrained vertex positions and few bends per edge. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 315–326. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Frati, F., Glisse, M., Lenhart, W.J., Liotta, G., Mchedlidze, T., Nishat, R.I.: Point-set embeddability of 2-colored trees. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 291–302. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  6. 6.
    Kaufmann, M., Wiese, R.: Embedding vertices at points: Few bends suffice for planar graphs. Journal of Graph Algorithms and Applications 6(1), 115–129 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graphs and Combinatorics 17(4), 717–728 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Shahriar, K.M.: Bichromatic point-set embeddings of trees with fewer bends. M. Sc. Engg. Thesis, Department of CSE, BUET (2008), http://www.buet.ac.bd/library/Web/showBookDetail.asp?reqBookID=66772&reqPageTopBookId=66772

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Khaled Mahmud Shahriar
    • 1
  • Md. Saidur Rahman
    • 1
  1. 1.Department of Computer Science and EngineeringBangladesh University of Engineering and Technology (BUET)DhakaBangladesh

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