Visibility Representations of Four-Connected Plane Graphs with Near Optimal Heights

  • Authors:

  • Chieh-Yu Chen;Ya-Fei Hung;Hsueh-I Lu

  • Affiliations:

  • Department of Computer Science and Information Engineering, National Taiwan University,;Graduate Institute of Networking and Multimedia, National Taiwan University, Taipei 106, Taiwan, ROC;Department of Computer Science and Information Engineering, National Taiwan University, and Graduate Institute of Networking and Multimedia, National Taiwan University, Taipei 106, Taiwan, ROC

  • Venue:
  • Graph Drawing
  • Year:
  • 2009

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Abstract

A visibility representation of a graph G is to represent the nodes of G with non-overlapping horizontal line segments such that the line segments representing any two distinct adjacent nodes are vertically visible to each other. If G is a plane graph, i.e., a planar graph equipped with a planar embedding, a visibility representation of G has the additional requirement of reflecting the given planar embedding of G . For the case that G is an n -node four-connected plane graph, we give an O (n )-time algorithm to produce a visibility representation of G with height at most $\left\lceil\frac{n}{2}\right\rceil+2\left\lceil\sqrt{\frac{n-2}{2}}\right\rceil$. To ensure that the first-order term of the upper bound is optimal, we also show an n -node four-connected plane graph G , for infinite number of n , whose visibility representations require heights at least $\frac{n}{2}$.