New Theoretical Bounds of Visibility Representation of Plane Graphs

Zhang, Huaming; He, Xin

doi:10.1007/978-3-540-31843-9_43

Cited by 4 publications

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References 8 publications

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“…On the other hand, Zhang and He considered the problem in another domain about how to reduce the height of VR of plane graphs by applying a new strategy to obtain better st-numberings from canonical ordering trees of plane graphs [11,12].…”

Section: Introductionmentioning

confidence: 99%

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Improved visibility representation of plane graphs

Zhang

2005

Computational Geometry

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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 [Discrete Comput. Geom. 1 (1986) 343-353] and Tamassia and Tollis [Discrete Comput. Geom. 1 (1986) 321-341] independently gave linear time VR algorithms for 2-connected plane graph. Using this approach, the height of the VR is bounded by (n − 1), the width is bounded by (2n − 5). After that, some work has been done to find a more compact VR. Kant and He [Theoret. Comput. Sci. 172 (1997) 175-193] proved that a 4-connected plane graph has a VR with width bounded by (n − 1). Kant [Internat. J. Comput. Geom. Appl. 7 (1997) 197-210] reduced the width bound to 3n−6 2 for general plane graphs. Recently, using a sophisticated greedy algorithm, Lin et al. reduced the width bound to 22n−42 15 [In this paper, we prove that any plane graph G has a VR with width at most 13n−24 9 , which can be constructed by using the simple standard VR algorithm in [P. Rosenstiehl, R.E. Tarjan, Discrete Comput.

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Section: Introductionmentioning

confidence: 99%

“…Rosenstiehl and Tarjan [6] conjectured that it is NP-hard. In [12], Zhang and He gave a plane triangulation G where any VR of G requires a grid size of ( 2n 3 ) × ( 4n 3 − 3). How to reduce the gap remains open.…”

Section: Introductionmentioning

confidence: 99%

Improved visibility representation of plane graphs

Zhang

2005

Computational Geometry

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An Application of Well-Orderly Trees in Graph Drawing

Zhang

2006

Graph Drawing

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

Chen

Hung

2009

Graph Drawing

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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 n 2 + 2 n−2 2. 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 n 2 .

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New Theoretical Bounds of Visibility Representation of Plane Graphs

Cited by 4 publications

References 8 publications

Improved visibility representation of plane graphs

Improved visibility representation of plane graphs

An Application of Well-Orderly Trees in Graph Drawing

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

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