Visibility representation of plane graphs via canonical ordering tree
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 [5n/6]. This improves the previously known bound [-15n/16]. We also construct a plane graph G with n vertices where any VR of G requires a size of ([2n/3]) x ([4n/3] - 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 [G. Kant, A more compact visibility representation, Internal. J. Comput. Geom. Appl. 7 (1997) 197]. (c) 2005 Elsevier B.V. All rights reserved.
visibility representation graph algorithms