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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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Huaming Zhang
    • 1
  • Xin He
    • 1
  1. 1.Department of Computer Science and EngineeringSUNY at BuffaloBuffaloUSA

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