Universal Line-Sets for Drawing Planar 3-Trees

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.