Orderly spanning trees with applications to graph encoding and graph drawing
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
An Information-Theoretic Upper Bound of Planar Graphs Using Triangulation
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
Improved visibility representation of plane graphs
Computational Geometry: Theory and Applications
Visibility representation of plane graphs via canonical ordering tree
Information Processing Letters
An application of well-orderly trees in graph drawing
GD'05 Proceedings of the 13th international conference on Graph Drawing
Optimal st-Orientations for Plane Triangulations
AAIM '07 Proceedings of the 3rd international conference on Algorithmic Aspects in Information and Management
On Representation of Planar Graphs by Segments
AAIM '08 Proceedings of the 4th international conference on Algorithmic Aspects in Information and Management
Visibility representations of four-connected plane graphs with near optimal heights
Computational Geometry: Theory and Applications
Width-optimal visibility representations of plane graphs
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Compact visibility representation of 4-connected plane graphs
COCOA'10 Proceedings of the 4th international conference on Combinatorial optimization and applications - Volume Part I
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The visibility representation (VR for short) is a classical representation of plane graphs. VR has various applications and has been extensively studied in literature. One of the main focuses of the study is to minimize the size of VR. It is known that there exists a plane graph G with n vertices where any VR of G requires a size at least $(\lfloor \frac{2n}{3} \rfloor) \times (\lfloor \frac{4n}{3} \rfloor -3)$. In this paper, we prove that every plane graph has a VR with height at most $\frac{2n}{3}+2\lceil \sqrt{n/2}\rceil$, and a VR with width at most $\frac{4n}{3}+2\lceil \sqrt{n}\rceil$. These representations are nearly optimal in the sense that they differ from the lower bounds only by a lower order additive term. Both representations can be constructed in linear time. However, the problem of finding VR with optimal height and optimal width simultaneously remains open.