GD 2004: Graph Drawing pp 425-430

# New Theoretical Bounds of Visibility Representation of Plane Graphs

• Huaming Zhang
• Xin He
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3383)

## Abstract

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.

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