On Visibility Representation of Plane Graphs

Zhang, Huaming; He, Xin

doi:10.1007/978-3-540-24749-4_42

Cited by 7 publications

(5 citation statements)

References 12 publications

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“…4(a) gives an st-numbering of G according to one of its canonical ordering tree T . The longest directed path passes through vertices 1, 2, 3, 8,9,10,11,12,13,14. Its length is 9.…”

Section: New Theoretical Bounds Of Vr Of Plane Graphsmentioning

confidence: 99%

“…In this paper, we will restrict ourselves to the VR where the exte-0020-0190/$ -see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2005.05.024 Table 1 References Plane graph G 4-connected plane graph G [8,11] Width of VR (2n − 5) Height of VR (n − 1) [4] Width of VR 3n−6 2 [7] Width of VR 22n−42 15 [5] Width of VR (n − 1) [12] Height of VR 15n 16 [13,14] Width of VR 13n−24 9…”

Section: Introductionmentioning

confidence: 99%

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Visibility representation of plane graphs via canonical ordering tree

Zhang

2005

Information Processing Letters

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“…4(a) gives an st-numbering of G according to one of its canonical ordering tree T . The longest directed path passes through vertices 1, 2, 3, 8,9,10,11,12,13,14. Its length is 9.…”

Section: New Theoretical Bounds Of Vr Of Plane Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Visibility representation of plane graphs via canonical ordering tree

Zhang

2005

Information Processing Letters

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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

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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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“…Algorithms for longest path parameterized st-orientations-namely, algorithms that apart from computing a correct st-orientation also manage to effectively control the length of the longest path of the produced st-orientation-, that, however, do not run in linear time and the application of which we study in this paper, firstly appeared in [19]. Towards this direction, but in a more theoretical context, Zhang and He have recently improved on various theoretical bounds on the longest path length of st-orientations for the case of plane triangulations only [11,26,27,28]. They also provided an algorithm to construct an st-orientation of a plane triangulation of minimum longest path length, equal to 2n/3 + O(1) [29], by essentially matching a lower bound presented in [28].…”

Section: Introductionmentioning

confidence: 97%

Applications of Parameterized st-Orientations

Papamanthou¹,

Tollis²

2010

JGAA

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An st-orientation of a biconnected undirected graph defines a directed graph with no cycles, a single source s and a single sink t. Given an undirected graph G as input, linear-time algorithms have been proposed for computing an st-orientation of G. Such an orientation is useful especially in graph drawing algorithms which use it at their first stage [23]. Namely, before they process the original undirected graph they receive as input, they transform it into an st-DAG, by computing an st-orientation of it. In this paper we observe that using st-orientations of different longest path lengths in various applications can result in different solutions, each one having its own merit. Guided by this intuition, we present results concerning applications of proposed algorithms for longest path parameterized st-orientations. Specifically, we show how to achieve considerable space savings (e.g., O(n)) for visibility representations of planar graphs by using st-orientations computed by algorithms that can control the length of the longest path. Also, we apply our results to the graph coloring problem, where we use an st-orientation as an intermediate step to compute a good coloring of a graph, and to other problems, such as computing space-efficient orthogonal drawings and longest paths.

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On Visibility Representation of Plane Graphs

Abstract: The visibility representation (VR for short) is a classical representation of plane graphs. It has various applications and has been extensively studied.

Cited by 7 publications

References 12 publications

Visibility representation of plane graphs via canonical ordering tree

Visibility representation of plane graphs via canonical ordering tree

New Theoretical Bounds of Visibility Representation of Plane Graphs

Applications of Parameterized st-Orientations

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