Visibility representation of plane graphs via canonical ordering tree

  • Authors:

  • Huaming Zhang;Xin He

  • Affiliations:

  • Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA;Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA

  • Venue:
  • Information Processing Letters
  • Year:
  • 2005

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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 [Rectilinear planar layouts and bipolar orientations of planar graphs, Discrete Comput. Geom. 1 (1986) 343], Tamassia and Tollis [An unified approach to visibility representations of planar graphs, Discrete Comput. Geom. 1 (1986) 321] independently gave linear time VR algorithms for 2-connected plane graph. Afterwards, one of the main concerns for VR is the size of the representation. In this paper, we prove that any plane graph G has a VR with height bounded by @?5n6@?. This improves the previously known bound @?15n16@?. We also construct a plane graph G with n vertices where any VR of G requires a size of (@?2n3@?)x(@?4n3@?-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 c1 [G. Kant, A more compact visibility representation, Internat. J. Comput. Geom. Appl. 7 (1997) 197].