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.
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Hossain, M.I., Mondal, D., Rahman, M.S., Salma, S.A. (2012). Universal Line-Sets for Drawing Planar 3-Trees. In: Rahman, M.S., Nakano, Si. (eds) WALCOM: Algorithms and Computation. WALCOM 2012. Lecture Notes in Computer Science, vol 7157. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28076-4_15
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DOI: https://doi.org/10.1007/978-3-642-28076-4_15
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