Universal Line-Sets for Drawing Planar 3-Trees
نویسندگان
چکیده
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 ⌊ 2(n−1) 3 ⌋ parallel lines are universal for any planar graph with n vertices. In this paper we show that a set of ⌊ 2 ⌋ parallel lines or a set of ⌈ 4 ⌉ 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 ⌊ 2 ⌋ parallel lines. Submitted: April 2012 Reviewed: August 2012 Revised: September 2012 Accepted: December 2012 Final: December 2012 Published: January 2013 Article type: Regular paper Communicated by: S.-i. Nakano E-mail addresses: [email protected] (Md. Iqbal Hossain) [email protected] (Debajyoti Mondal) [email protected] (Md. Saidur Rahman) [email protected] (Sammi Abida Salma) 60 Hossain et al. Universal Line-Sets for Drawing Planar 3-Trees
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