Drawing Arrangement Graphs in Small Grids, or How to Play Planarity

Eppstein, David

doi:10.1007/978-3-319-03841-4_38

Cited by 3 publications

(3 citation statements)

References 20 publications

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“…Simply-nested planar graphs (graphs that can be decomposed into nested induced cycles) have universal point sets of size O n(log n/ log log n) 2 [1], and planar 3-trees have universal point sets of size O(n 3/2 log n) [16]. Based in part on the results in this paper, the graphs of simple line and pseudoline arrangements have been shown to have universal point sets of size O(n log n) [14].…”

Section: Introductionmentioning

confidence: 84%

Superpatterns and Universal Point Sets

Bannister¹,

Cheng²,

Devanny³

et al. 2014

JGAA

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An old open problem in graph drawing asks for the size of a universal point set, a set of points that can be used as vertices for straight-line drawings of all n-vertex planar graphs. We connect this problem to the theory of permutation patterns, where another open problem concerns the size of superpatterns, permutations that contain all patterns of a given size. We generalize superpatterns to classes of permutations determined by forbidden patterns, and we construct superpatterns of size n 2 /4 + Θ(n) for the 213-avoiding permutations, half the size of known superpatterns for unconstrained permutations. We use our superpatterns to construct universal point sets of size n 2 /4−Θ(n), smaller than the previous bound by a 9/16 factor. We prove that every proper subclass of the 213-avoiding permutations has superpatterns of size O(n log O(1) n), which we use to prove that the planar graphs of bounded pathwidth have near-linear universal point sets.

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Section: Introductionmentioning

confidence: 84%

Superpatterns and Universal Point Sets

Bannister¹,

Cheng²,

Devanny³

et al. 2014

JGAA

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“…Goodman [8] shows that every arrangement has a representation as a wiring diagram. More recently there have been results on drawing arrangements as convex polygonal chains with few bends [6] and on small grids [5]. Goodman and Pollack [11] consider arrangements whose pseudo-lines are the function-graphs of polynomial functions with bounded degree.…”

Section: Related Workmentioning

confidence: 99%

Arrangements of Approaching Pseudo-Lines

Felsner

Pilz

Schnider

2022

Discrete Comput Geom

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We consider arrangements of n pseudo-lines in the Euclidean plane where each pseudo-line $$\ell _i$$ ℓ i is represented by a bi-infinite connected x-monotone curve $$f_i(x)$$ f i ( x ) , $$x \in \mathbb {R}$$ x ∈ R , such that for any two pseudo-lines $$\ell _i$$ ℓ i and $$\ell _j$$ ℓ j with $$i \!<\! j$$ i < j , the function $$x \!\mapsto \! f_j(x) \!-\! f_i(x)$$ x ↦ f j ( x ) - f i ( x ) is monotonically decreasing and surjective (i.e., the pseudo-lines approach each other until they cross, and then move away from each other). We show that such arrangements of approaching pseudo-lines, under some aspects, behave similar to arrangements of lines, while for other aspects, they share the freedom of general pseudo-line arrangements. For the former, we prove: There are arrangements of pseudo-lines that are not realizable with approaching pseudo-lines. Every arrangement of approaching pseudo-lines has a dual generalized configuration of points with an underlying arrangement of approaching pseudo-lines. For the latter, we show: There are $$2^{\Theta (n^2)}$$ 2 Θ ( n 2 ) isomorphism classes of arrangements of approaching pseudo-lines (while there are only $$2^{\Theta (n \log n)}$$ 2 Θ ( n log n ) isomorphism classes of line arrangements). It can be decided in polynomial time whether an allowable sequence is realizable by an arrangement of approaching pseudo-lines. Furthermore, arrangements of approaching pseudo-lines can be transformed into each other by flipping triangular cells, i.e., they have a connected flip graph, and every bichromatic arrangement of this type contains a bichromatic triangular cell.

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“…Using a method based on compaction of wiring diagrams, the graphs of pseudoline arrangements may be given straight line drawings in small grids [9]. The planar dual graph of a weak pseudoline arrangement may be characterized as having drawings in which each bounded face is a centrally symmetric polygon [6]. The pseudoline arrangements in which each pseudoline is a translated quadrant can be used to visualize learning spaces, representing the possible states of knowledge of a human learner [8].…”

Section: Introductionmentioning

confidence: 99%

Convex-Arc Drawings of Pseudolines

Eppstein,

van Garderen,

Speckmann

et al. 2016

Preprint

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A weak pseudoline arrangement is a topological generalization of a line arrangement, consisting of curves topologically equivalent to lines that cross each other at most once. We consider arrangements that are outerplanar-each crossing is incident to an unbounded face-and simple-each crossing point is the crossing of only two curves. We show that these arrangements can be represented by chords of a circle, by convex polygonal chains with only two bends, or by hyperbolic lines. Simple but non-outerplanar arrangements (non-weak) can be represented by convex polygonal chains or convex smooth curves of linear complexity.

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Drawing Arrangement Graphs in Small Grids, or How to Play Planarity

Cited by 3 publications

References 20 publications

Superpatterns and Universal Point Sets

Superpatterns and Universal Point Sets

Arrangements of Approaching Pseudo-Lines

Convex-Arc Drawings of Pseudolines

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