TJJCCGG 2012: Computational Geometry and Graphs pp 20-29

# Colored Quadrangulations with Steiner Points

• Victor Alvarez
• Atsuhiro Nakamoto
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8296)

## Abstract

Let P ⊂ ℝ2 be a k-colored set of n points in general position, where k ≥ 2. A k-colored quadrangulation on P is a properly colored straight-edge plane graph G with vertex set P such that the boundary of the unbounded face of G coincides with CH(P) and that each bounded face of G is quadrilateral, where CH(P) stands for the boundary of the convex hull of P. It is easily checked that in general not every k-colored P admits a k-colored quadrangulation, and hence we need the use of Steiner points, that is, auxiliary points which can be put in any position of the interior of the convex hull of P and can have any color among the k colors. In this paper, we show that if P satisfies some condition for colors of the points in the convex hull, then a k-colored quadrangulation of P can always be constructed using less than $$\frac{(16 k-2) n+7 k-2}{39 k-6}$$ Steiner points. Our upper bound improves the known upper bound for k = 3, and gives the first bounds for k ≥ 4.

## Keywords

Convex Hull Interior Point Convex Polygon Steiner Point Color Class
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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© Springer-Verlag Berlin Heidelberg 2013

## Authors and Affiliations

• Victor Alvarez
• 1
• Atsuhiro Nakamoto
• 2
1. 1.Fachrichtung InformatikUniversität des SaarlandesSaarbrückenGermany
2. 2.Department of MathematicsYokohama National UniversityYokohamaJapan