GD 2012: Graph Drawing pp 291-302

# Point-Set Embeddability of 2-Colored Trees

• Fabrizio Frati
• Marc Glisse
• William J. Lenhart
• Giuseppe Liotta
• Tamara Mchedlidze
• Rahnuma Islam Nishat
Conference paper
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.

## Keywords

Planar Graph Blue Point Consecutive Point Outerplanar Graph Proof Sketch
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.

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## 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