Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems
Theoretical Computer Science
Improved Compact Visibility Representation of Planar Graph via Schnyder's Realizer
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
Optimal st-Orientations for Plane Triangulations
AAIM '07 Proceedings of the 3rd 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
An application of well-orderly trees in graph drawing
GD'05 Proceedings of the 13th international conference on Graph Drawing
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In a visibility representation (VR for short) of a plane graph G, each vertex of G is represented by a horizontal line segment such that the line segments representing any two adjacent vertices of G are joined by a vertical line segment. Rosenstiehl and Tarjan [6], Tamassia and Tollis [7] independently gave linear time VR algorithms for 2-connected plane graph. Afterwards, one of the main concerns for VR is the size of VR. In this paper, we prove that any plane graph G has a VR with height bounded by $\lfloor \frac{5n}{6} \rfloor$. This improves the previously known bound $\lceil \frac{15n}{16} \rceil$. We also construct a plane graph G with n vertices where any VR of G require a size of $(\lfloor \frac{2n}{3} \rfloor) \times (\lfloor \frac{4n}{3} \rfloor-3)$. Our result provides an answer to Kant's open question about whether there exists a plane graph G such that all of its VR require width greater that cn, where c 1.