On embedding an outer-planar graph in a point set
Computational Geometry: Theory and Applications
On the Parameterized Complexity of Layered Graph Drawing
Algorithmica - Parameterized and Exact Algorithms
Characterization of unlabeled level planar trees
Computational Geometry: Theory and Applications
A note on minimum-area straight-line drawings of planar graphs
GD'07 Proceedings of the 15th international conference on Graph drawing
On graphs supported by line sets
GD'10 Proceedings of the 18th international conference on Graph drawing
Minimum-layer drawings of trees
WALCOM'11 Proceedings of the 5th international conference on WALCOM: algorithms and computation
Point-set embeddings of plane 3-trees
Computational Geometry: Theory and Applications
Embedding stacked polytopes on a polynomial-size grid
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
On point-sets that support planar graphs
GD'11 Proceedings of the 19th international conference on Graph Drawing
Minimum-width grid drawings of plane graphs
Computational Geometry: Theory and Applications
Universal point sets for planar three-trees
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
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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.