Point-Set Embeddability of 2-Colored Trees

  • Fabrizio Frati
  • Marc Glisse
  • William J. Lenhart
  • Giuseppe Liotta
  • Tamara Mchedlidze
  • Rahnuma Islam Nishat
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)

Abstract

In this paper we study bichromatic point-set embeddings of 2-colored trees on 2-colored point sets, i.e., point-set embeddings of trees (whose vertices are colored red and blue) on point sets (whose points are colored red and blue) such that each red (blue) vertex is mapped to a red (resp. blue) point. We prove that deciding whether a given 2-colored tree admits a bichromatic point-set embedding on a given convex point set is an \(\cal NP\)-complete problem; we also show that the same problem is linear-time solvable if the convex point set does not contain two consecutive points with the same color. Furthermore, we prove a 3n/2 − O(1) lower bound and a 2n upper bound (a 7n/6 − O(logn) lower bound and a 4n/3 upper bound) on the minimum size of a universal point set for straight-line bichromatic embeddings of 2-colored trees (resp. 2-colored binary trees). Finally, we show that universal convex point sets with n points exist for 1-bend bichromatic point-set embeddings of 2-colored trees.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Fabrizio Frati
    • 1
  • Marc Glisse
    • 2
  • William J. Lenhart
    • 3
  • Giuseppe Liotta
    • 4
  • Tamara Mchedlidze
    • 5
  • Rahnuma Islam Nishat
    • 6
  1. 1.School of Information TechnologiesThe University of SydneyAustralia
  2. 2.INRIA Saclay – Ile-de-FranceFrance
  3. 3.Computer Science DepartmentWilliams CollegeU.S.A.
  4. 4.Dipartimento Ingegneria Elettronica e dell’InformazioneUniversitá di PerugiaItaly
  5. 5.Institute of Theoretical InformaticsKarlsruhe Institute of Technology (KIT)Germany
  6. 6.Department of Computer ScienceUniversity of VictoriaCanada

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