WALCOM 2012: WALCOM: Algorithms and Computation pp 136-147

# Universal Line-Sets for Drawing Planar 3-Trees

• Md. Iqbal Hossain
• Debajyoti Mondal
• Md. Saidur Rahman
• Sammi Abida Salma
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7157)

## Abstract

A set S of lines is universal for drawing planar graphs with n vertices if every planar graph G with n vertices can be drawn on S such that each vertex of G is drawn as a point on a line of S and each edge is drawn as a straight-line segment without any edge crossing. It is known that $$\lfloor \frac{2(n-1)}{3}\rfloor$$ parallel lines are universal for any planar graph with n vertices. In this paper we show that a set of $$\lfloor \frac{n-3}{2} \rfloor +3$$ parallel lines or a set of $$\lceil \frac{n+3}{4} \rceil$$ concentric circles are universal for drawing planar 3-trees with n vertices. In both cases we give linear-time algorithms to find such drawings. A by-product of our algorithm is the generalization of the known bijection between plane 3-trees and rooted full ternary trees to the bijection between planar 3-trees and unrooted full ternary trees. We also identify some subclasses of planar 3-trees whose drawings are supported by fewer than $$\lfloor \frac{n-3}{2} \rfloor +3$$ parallel lines.

## Keywords

Planar Graph Parallel Line Concentric Circle Outer Face Planar Embedding
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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