Geodesic Embeddings and Planar Graphs

Zickfeld

2007

Discrete Comput Geom

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In this paper we study connections between planar graphs, Schnyder woods, and orthogonal surfaces. Schnyder woods and the face counting approach have important applications in graph drawing and the dimension theory of orders. Orthogonal surfaces explain connections between these seemingly unrelated notions. We use these connections for an intuitive proof of the Brightwell-Trotter Theorem which says, that the face lattice of a 3-polytope minus one face has order dimension three. Our proof yields a linear time algorithm for the construction of the three linear orders that realize the face lattice.Coplanar orthogonal surfaces are in correspondence with a large class of convex straight line drawings of 3-connected planar graphs. We show that Schnyder's face counting approach with weighted faces can be used to construct all coplanar orthogonal surfaces and hence the corresponding drawings. Appropriate weights are computable in linear time.

“…Together with the following proposition from [16] (see also [10]) this implies the BrightwellTrotter Theorem (Theorem 2).…”

Section: Rigid Orthogonal Surfaces Via Flat Shiftingsupporting

confidence: 59%

“…All the proofs omitted in this section can be found in [11,10,9]. A planar map M is a simple planar graph G together with a fixed planar embedding of G in the plane.…”

Section: Basics On Schnyder Woods and Orthogonal Surfacesmentioning

confidence: 99%

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Schnyder Woods and Orthogonal Surfaces

Zickfeld

2007

Discrete Comput Geom

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“…In this section we give a simpler proof of this result using geodesic embeddings on orthogonal surfaces. The theory was again developed in the context of order dimension [18,5,8].…”

Section: Schnyder Woods and Primal-dual Contact Representationsmentioning

confidence: 99%

“…An orthogonal surface is rigid if all its bounded flats are rigid. It has been shown in [5] and [8] that every Schnyder wood has a geodesic embedding on some rigid orthogonal surface. From now on we assume that the given orthogonal surface is rigid, this assumption will be critical in the proof of Proposition 7.…”

Section: Schnyder Woods and Primal-dual Contact Representationsmentioning

confidence: 99%

Straight Line Triangle Representations

Aerts

2017

Discrete Comput Geom

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A straight line triangle representation (SLTR) of a planar graph is a straight line drawing such that all the faces including the outer face have triangular shape. Such a drawing can be viewed as a tiling of a triangle using triangles with the input graph as skeletal structure. In this paper we present a characterization of graphs that have an SLTR. The characterization is based on flat angle assignments, i.e., selections of angles of the graph that have size π in the representation. We also provide a second characterization in terms of contact systems of pseudosegments. With the aid of discrete harmonic functions we show that contact systems of pseudosegments that respect certain conditions are stretchable. The stretching procedure is then used to get straight line triangle representations. Since the discrete harmonic function approach is quite flexible it allows further applications, we mention some of them.The drawback of the characterization of SLTRs is that we are not able to effectively check whether a given graph admits a flat angle assignment that fulfills the conditions. Hence it is still open to decide whether the recognition of graphs that admit straight line triangle representation is polynomially tractable.

Posets and planar graphs

Journal of Graph Theory

Trotter

2005

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Abstract. Usually dimension should be an integer valued parameter, we introduce a refined version of dimension for graphs which can assume a value [ t − 1 t ] which is thought to be between t − 1 and t. We have the following two results:• A graph is outerplanar if and only if its dimension is at most [ 2 3 ].This characterization of outerplanar graphs is closely related to the celebrated result of W. Schnyder [16] who proved that a graph is planar if and only if its dimension is at most 3.• The largest n for which the dimension of the complete graph Kn is at most [ t − 1 t ] is the number of antichains in the lattice of all subsets of a set of size t − 2. Accordingly, the refined dimension problem for complete graphs is equivalent to the classical combinatorial problem known as Dedekind's problem. This result extends work of Hoşten and Morris [14].The main results are enriched by background material which links to a line of reserch in extremal graph theory which was stimulated by a problem posed by G. Agnarsson: Find the maximum number of edges in a graph on n nodes with dimension at most t.