Point-Set embeddability of 2-colored trees

Authors:
Fabrizio Frati;Marc Glisse;William J. Lenhart;Giuseppe Liotta;Tamara Mchedlidze;Rahnuma Islam Nishat
Affiliations:
School of Information Technologies, The University of Sydney, Australia;INRIA Saclay --- Ile-de-France, France;Computer Science Department, Williams College;Dipartimento Ingegneria Elettronica e dell'Informazione, Universitá di Perugia, Italy;Institute of Theoretical Informatics, Karlsruhe Institute of Technology (KIT), Germany;Department of Computer Science, University of Victoria, Canada
Venue:
GD'12 Proceedings of the 20th international conference on Graph Drawing
Year:
2012

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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.