Compact Visibility Representation and Straight-Line Grid Embedding of Plane Graphs

Zhang, Huaming; He, Xin

doi:10.1007/978-3-540-45078-8_43

Cited by 23 publications

(17 citation statements)

References 15 publications

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“…An interesting special case of this bound (Corollary 1) was first obtained by Zhang and He [26] with a different method. Let again S Min be the minimum Schnyder wood in the lattice and recall that ∆ S Min = 0.…”

Section: The Number Of Mergesmentioning

confidence: 86%

“…In general Schnyder's result from [21] is still unbeaten. Lately, Zhang and He [26] used the minimum Schnyder wood of a triangulation to prove a bound of (n − 1 − ∆ ) × (n − 1 − ∆ ), where ∆ is the number of cyclic faces in the minimum Schnyder wood.…”

Section: Previous Workmentioning

confidence: 99%

“…In their recent paper about drawings of triangulations Zhang and He [26] use a similar idea of ignoring faces when counting faces in regions.…”

Section: Ignoring Facesmentioning

confidence: 99%

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Convex Drawings of 3-Connected Plane Graphs

2007

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We use Schnyder woods of 3-connected planar graphs to produce convex straight line drawings on a grid of size (n − 2 − ∆) × (n − 2 − ∆). The parameter ∆ ≥ 0 depends on the Schnyder wood used for the drawing. This parameter is in the range 0The algorithm is a refinement of the face-counting-algorithm, thus, in particular, the size of the grid is at most (f − 2) × (f − 2).The above bound on the grid size simultaneously matches or improves all previously known bounds for convex drawings, in particular Schnyder's and the recent Zhang and He bound for triangulations and the Chrobak and Kant bound for 3-connected planar graphs. The algorithm takes linear time.The drawing algorithm has been implemented and tested. The expected grid size for the drawing of a random triangulation is close to

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Section: The Number Of Mergesmentioning

confidence: 86%

Section: Previous Workmentioning

confidence: 99%

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Convex Drawings of 3-Connected Plane Graphs

2007

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“…[5] Width of VR ≤ 22n−42 15 [3] Width of VR ≤ (n − 1) [8] Height of VR ≤ 15n 16 [9] Width of VR ≤ 13n−24 9…”

Section: Referencesmentioning

confidence: 99%

New Theoretical Bounds of Visibility Representation of Plane Graphs

Zhang

2005

Graph Drawing

Self Cite

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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 [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 . This improves the previously known bound . We also construct a plane graph G with n vertices where any VR of G require a size of (

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“…In [27] a straight-line drawing algorithm base on minimal realizers is presented. This algorithm first computes the minimal realizer of a triangulation of the graph.…”

Section: Theoremmentioning

confidence: 99%

Planar Graphs, via Well-Orderly Maps and Trees

Gavoille

Hanusse

Poulalhon

et al. 2004

Graph-Theoretic Concepts in Computer Science

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Abstract. The family of well-orderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a well-orderly map. We show that the number of well-orderly maps with n nodes is at most 2 αn+O(log n) , where α ≈ 4.91. A direct consequence of this is a new upper bound on the number p(n) of unlabeled planar graphs with n nodes, log 2 p(n) 4.91n. The result is then used to show that asymptotically almost all (labeled or unlabeled), (connected or not) planar graphs with n nodes have between 1.85n and 2.44n edges. Finally we obtain as an outcome of our combinatorial analysis an explicit linear time encoding algorithm for unlabeled planar graphs using, in the worst-case, a rate of 4.91 bits per node and of 2.82 bits per edge.

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Compact Visibility Representation and Straight-Line Grid Embedding of Plane Graphs

Cited by 23 publications

References 15 publications

Convex Drawings of 3-Connected Plane Graphs

Convex Drawings of 3-Connected Plane Graphs

New Theoretical Bounds of Visibility Representation of Plane Graphs

Planar Graphs, via Well-Orderly Maps and Trees

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