Geometric Graphs and Arrangements

Felsner, Stefan

doi:10.1007/978-3-322-80303-0

Cited by 82 publications

(68 citation statements)

References 3 publications

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“…Laman graphs are of interest in rigidity-theory, see e.g. [GSS93], [Fel04a]. Laman graphs admit a Henneberg construction, i.e., an ordering v 1 , .…”

Section: Segment Contact Graphsmentioning

confidence: 99%

Rectangle and Square Representations of Planar Graphs

Felsner

2012

Thirty Essays on Geometric Graph Theory

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In the first part of this survey we consider planar graphs that can be represented by a dissections of a rectangle into rectangles. In rectangular drawings the corners of the rectangles represent the vertices. The graph obtained by taking the rectangles as vertices and contacts as edges is the rectangular dual. In visibility graphs and segment contact graphs the vertices correspond to horizontal or to horizontal and vertical segments of the dissection. Special orientations of graphs turn out to be helpful when dealing with characterization and representation questions. Therefore, we look at orientations with prescribed degrees, bipolar orientations, separating decompositions, and transversal structures.In the second part we ask for representations by a dissections of a rectangle into squares. We review results by Brooks et al. (1940), Kenyon (1998) and Schramm (1993) and discuss a technique of computing squarings via solutions of systems of linear equations.Mathematics Subject Classifications (2010) 05C10, 05C62, 52C15.

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“…Laman graphs are of interest in rigidity-theory, see e.g. [GSS93], [Fel04a]. Laman graphs admit a Henneberg construction, i.e., an ordering v 1 , .…”

Section: Segment Contact Graphsmentioning

confidence: 99%

Rectangle and Square Representations of Planar Graphs

Felsner

2012

Thirty Essays on Geometric Graph Theory

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“…In computational and discrete geometry (and other fields) line arrangements are interesting objects of study and their properties have therefore been studied for decades [4].…”

Section: Motivation and More Related Workmentioning

confidence: 99%

“…The complexity of a wall is bounded by O(n 2 ), thus there are at most Cn 4 pairs of points on walls for some constant C. So the probability that a path within the wall is blocked is bounded by P r(A i ) ≤ (x,y) P r(S x,y is monochromatic) ≤ Cn 4 2 w−3 .…”

Section: Lemma 17 ([1])mentioning

confidence: 99%

Upper and Lower Bounds on Long Dual Paths in Line Arrangements

Hoffmann

Kleist

Miltzow

2015

Mathematical Foundations of Computer Science 2015

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Given a line arrangement A with n lines, we show that there exists a path of length n 2 /3 − O(n) in the dual graph of A formed by its faces. This bound is tight up to lower order terms. For the bicolored version, we describe an example of a line arrangement with 3k blue and 2k red lines with no alternating path longer than 14k. Further, we show that any line arrangement with n lines has a coloring such that it has an alternating path of length Ω(n 2 / log n). Our results also hold for pseudoline arrangements.arXiv:1506.03728v1 [math.CO]

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“…We should mention that various alternative notions of a combinatorial description of a pseudoline arrangement are available in the literature such as local sequences, allowable sequences and oriented matroids of rank 3 (see for instance [9]). …”

Section: Preliminariesmentioning

confidence: 99%

Arrangements of Pseudocircles and Circles

Kang

Müller

2014

Discrete Comput Geom

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An arrangement of pseudocircles is a finite collection of Jordan curves in the plane with the additional properties that (i) every two curves meet in at most two points; and (ii) if two curves meet in a point p, then they cross at p.We say that two arrangements C = (c 1 , . . . , c n ) and D = (d 1 , . . . , d n ) are equivalent if there is a homeomorphism ϕ of the plane onto itself such that ϕ[c i ] = d i for all i ∈ {1, . . . , n}. Linhart and Ortner (2005) gave an example of an arrangement of five pseudocircles that is not equivalent to an arrangement of circles, and they conjectured that every arrangement of at most four pseudocircles is equivalent to an arrangement of circles. Here we prove their conjecture.We consider two related recognition problems. The first is the problem of deciding, given a (combinatorial description of a) pseudocircle arrangement, whether it is equivalent to an arrangement of circles. The second is deciding whether it is equivalent to an arrangement of convex pseudocircles. We prove that both problems are NP-hard, answering questions of Bultena, Grünbaum and Ruskey (1998) and of Linhart and Ortner (2008).We also give an example of an arrangement of convex pseudocircles with the property that its intersection graph (i.e. the graph with one vertex for each pseudocircle and an edge between two vertices if and only if the corresponding pseudocircles intersect) cannot be realised as the intersection graph of a family of circles. This disproves a folklore conjecture communicated to us by Pyatkin.

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Geometric Graphs and Arrangements

Cited by 82 publications

References 3 publications

Rectangle and Square Representations of Planar Graphs

Rectangle and Square Representations of Planar Graphs

Upper and Lower Bounds on Long Dual Paths in Line Arrangements

Arrangements of Pseudocircles and Circles

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