4-Connected Triangulations on Few Lines

Felsner, Stefan

doi:10.1007/978-3-030-35802-0_30

Cited by 2 publications

(2 citation statements)

References 19 publications

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“…Xin He [34] used transversal structures to give an algorithm for realizing these triangulations as the contact graphs of rectangles (equivalently, drawing their dual in such a way that every face is a rectangle), and together with Goos Kant showed that the transversal structure and drawing can be computed in linear time [35]. Since then, several other graph drawing algorithms based on transversal structures have been obtained [9,18,22,23,30].…”

Section: Introductionmentioning

confidence: 99%

Grand Schnyder Woods

Bernardi¹,

Fusy²,

Liang³

2023

Preprint

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We define a far-reaching generalization of Schnyder woods which encompasses many classical combinatorial structures on planar graphs.Schnyder woods are defined for planar triangulations as certain triples of spanning trees covering the triangulation and crossing each other in an orderly fashion. They are of theoretical and practical importance, as they are central to the proof that the order dimension of any planar graph is at most 3, and they are also underlying an elegant drawing algorithm. In this article we extend the concept of Schnyder wood well beyond its original setting: for any integer d we define a "grand-Schnyder" structure for (embedded) planar graphs which have faces of degree at most d and non-facial cycles of length at least d. We prove the existence of grand-Schnyder structures, provide a linear construction algorithm, describe 4 different incarnations (in terms of tuples of trees, corner labelings, weighted orientations, and marked orientations), and define a lattice for the set of grand Schnyder structures of a given planar graph. We show that the grand-Schnyder framework unifies and extends several classical constructions: Schnyder woods and Schnyder decompositions, regular edge-labelings (a.k.a. transversal structures), and Felsner woods.

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Section: Introductionmentioning

confidence: 99%

Grand Schnyder Woods

Bernardi¹,

Fusy²,

Liang³

2023

Preprint

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“…Answering a question of Chaplick et al [3] he also constructed a family of subcubic series-parallel graphs with unbounded π 1 2 -value. Felsner [11] proved that, for every 4-connected plane triangulation G on n vertices, it holds that π 1 2 (G) ≤ √ 2n. Chaplick et al [4] also investigated the complexity of computing the affine cover numbers.…”

Section: Introductionmentioning

confidence: 99%

Line and Plane Cover Numbers Revisited

Biedl

Felsner

Meijer

et al. 2019

Lecture Notes in Computer Science

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A measure for the visual complexity of a straight-line crossingfree drawing of a graph is the minimum number of lines needed to cover all vertices. For a given graph G, the minimum such number (over all drawings in dimension d ∈ {2, 3}) is called the d-dimensional weak line cover number and denoted by π 1 d (G). In 3D, the minimum number of planes needed to cover all vertices of G is denoted by π 2 3 (G). When edges are also required to be covered, the corresponding numbers ρ 1 d (G) and ρ 2 3 (G) are called the (strong) line cover number and the (strong) plane cover number. Computing any of these cover numbers-except π 1 2 (G)-is known to be NP-hard. The complexity of computing π 1 2 (G) was posed as an open problem by Chaplick et al. [WADS 2017]. We show that it is NP-hard to decide, for a given planar graph G, whether π 1 2 (G) = 2. We further show that the universal stacked triangulation of depth d, G d , has π 1 2 (G d ) = d+1. Concerning 3D, we show that any n-vertex graph G with ρ 2 3 (G) = 2 has at most 5n − 19 edges, which is tight.

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4-Connected Triangulations on Few Lines

Cited by 2 publications

References 19 publications

Grand Schnyder Woods

Grand Schnyder Woods

Line and Plane Cover Numbers Revisited

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