Improved Compact Visibility Representation of Planar Graph via Schnyder's Realizer

Lin, Ching-Chi; Lu, Hsueh-I; Sun, I-Fan

doi:10.1137/s0895480103420744

Cited by 25 publications

(10 citation statements)

References 28 publications

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“…These structures, nowadays known as Schnyder woods (respectively Schnyder realizers) and Schnyder angle labelings have found many applications. For illustration we cite some applications to graph drawing models [27], [13], [6], [2] and to the enumeration and encoding of planar maps [8], and [24]. Regarding the limits of possible extensions of Schnyder's results there are two observations:…”

Section: Order Dimension and Planaritymentioning

confidence: 99%

The Order Dimension of Planar Maps Revisited

Felsner¹

2014

SIAM J. Discrete Math.

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Abstract. Schnyder characterized planar graphs in terms of order dimension. This seminal result found several extensions. A particularly far reaching extension is the Brightwell-Trotter Theorem about planar maps. It states that the order dimension of the incidence poset P M of vertices, edges and faces of a planar map M has dimension at most 4. The original proof generalizes the machinery of Schnyder-paths and Schnyder-regions. In this note we use a simple result about the order dimension of grid intersection graphs to show a slightly stronger result: dim(split(P M )) 4.This may be the rst result in the area that is obtained without using the tools introduced by Schnyder.Mathematics Subject Classications (2010) 05C10, 06A07, 68R10

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Section: Order Dimension and Planaritymentioning

confidence: 99%

The Order Dimension of Planar Maps Revisited

Felsner¹

2014

SIAM J. Discrete Math.

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“…Schnyder woods continue to find applications in graph drawing, see e.g. [8,23]. A new line of applications of Schnyder woods was recently found in the area of bijective enumeration of planar structures [26].…”

Section: Rule Of Facesmentioning

confidence: 99%

Orthogonal Surfaces and Their CP-Orders

Felsner

Kappes

2008

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Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial already the three dimensional case has a rich structure with connections to Schnyder woods, planar graphs and three-polytopes. Our objective is to detect more of the structure of orthogonal surfaces in four and higher dimensions. In particular we are driven by the question which non-generic orthogonal surfaces have a polytopal structure. We review the state of knowledge of the three-dimensional situation. On that basis we introduce terminology for higher dimensional orthogonal surfaces and continue with the study of characteristic points and the cp-orders of orthogonal surfaces, i.e., the dominance orders on the characteristic points. In the generic case these orders are (almost) face lattices of polytopes. Examples show that in general cp-orders can lack key properties of face lattices. We investigate extra requirements which may help to have cp-orders which are face lattices. Finally, we turn the focus and ask for the realizability of polytopes on orthogonal surfaces. There are criteria which prevent large classes of simplicial polytopes from being realizable. On the other hand we identify some families of polytopes which can be realized on orthogonal surfaces.

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“…They also proved that the problem of finding a minimum-area visibility drawing of a planar graph is NP-hard in general. Many other interesting results on visibility drawings of plane graphs with small area are already present in the literature, which are the outcome of extensive study and research on visibility drawings [8,9,12]. Let G be a plane graph with n vertices.…”

Section: Introductionmentioning

confidence: 97%

Visibility Drawings of Plane 3-Trees with Minimum Area

2011

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A visibility drawing of a plane graph G is a drawing of G where each vertex is drawn as a horizontal line segment and each edge is drawn as a vertical line segment such that the line segments use only grid points as their endpoints. The area of a visibility drawing is the area of the smallest rectangle on the grid which encloses the drawing. A minimum-area visibility drawing of a plane graph G is a visibility drawing of G where the area is the minimum among all possible visibility drawings of G. The area minimization for grid visibility representation of planar graphs is NP-hard. However, the problem can be solved for a fixed planar embedding of a hierarchically planar graph in quadratic time. In this paper, we give a polynomial-time algorithm to obtain minimum-area visibility drawings of plane 3-trees.

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Improved Compact Visibility Representation of Planar Graph via Schnyder's Realizer

Cited by 25 publications

References 28 publications

The Order Dimension of Planar Maps Revisited

The Order Dimension of Planar Maps Revisited

Orthogonal Surfaces and Their CP-Orders

Visibility Drawings of Plane 3-Trees with Minimum Area

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