Drawing Graphs on Few Circles and Few Spheres

Kryven, Myroslav; Ravsky, Alexander; Wolff, Alexander

doi:10.1007/978-3-319-74180-2_14

Cited by 4 publications

(2 citation statements)

References 19 publications

(42 reference statements)

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“…Chaplick et al [4,5] consider a similar problem where all edges are to be covered by few lines (or planes); the difference to our problem is that collinear segments are counted only once in their model. In the same fashion, Kryven et al [18] aim to cover all edges by few circles (or spheres).…”

Section: Classmentioning

confidence: 99%

Drawing Planar Graphs with Few Geometric Primitives

Hültenschmidt¹,

Kindermann²,

Meulemans³

et al. 2018

JGAA

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We define the visual complexity of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw a path with an arbitrary number of edges). Let n denote the number of vertices of a graph. We show that trees can be drawn with 3n/4 straight-line segments on a polynomial grid, and with n/2 straight-line segments on a quasi-polynomial grid. Further, we present an algorithm for drawing planar 3-trees with (8n − 17)/3 segments on an O(n) × O(n 2 ) grid. This algorithm can also be used with a small modification to draw maximal outerplanar graphs with 3n/2 edges on an O(n) × O(n 2 ) grid. We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only (5n − 11)/3 arcs. This is significantly smaller than the lower bound of 2n for line segments for a nontrivial graph class. * A preliminary version of this work appeared in: Proc. 43rd Int. Workshop Graph-Theor. Concepts Comput. Sci. (WG'17) [14]. The work of P. Kindermann and A.

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Section: Classmentioning

confidence: 99%

Drawing Planar Graphs with Few Geometric Primitives

Hültenschmidt¹,

Kindermann²,

Meulemans³

et al. 2018

JGAA

View full text Add to dashboard Cite

show abstract

“…Relevant to segment complexity are the studies by Chaplick et al [3,4] who consider drawings where all edges are to be covered by few lines (or planes); the difference to our problem is that collinear segments are counted only once in their model. In the same fashion, Kryven et al [18] aim to cover all edges by few circles (or spheres).…”

Section: Introductionmentioning

confidence: 99%

Drawing Planar Graphs with Few Segments on a Polynomial Grid

Kindermann

Mchedlidze

Schneck

et al. 2019

Lecture Notes in Computer Science

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The visual complexity of a graph drawing can be measured by the number of geometric objects used for the representation of its elements. In this paper, we study planar graph drawings where edges are represented by few segments. In such a drawing, one segment may represent multiple edges forming a path. Drawings of planar graphs with few segments were intensively studied in the past years. However, the area requirements were only considered for limited subclasses of planar graphs. In this paper, we show that trees have drawings with 3n/4 − 1 segments and n 2 area, improving the previous result of O(n 3.58 ). We also show that 3-connected planar graphs and biconnected outerplanar graphs have a drawing with 8n/3 − O(1) and 3n/2 − O(1) segments, respectively, and O(n 3 ) area.

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Drawing Graphs on Few Circles and Few Spheres

Kryven

Ravsky

Wolff

2018

Algorithms and Discrete Applied Mathematics

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Abstract. Given a drawing of a graph, its visual complexity is defined as the number of geometrical entities in the drawing, for example, the number of segments in a straight-line drawing or the number of arcs in a circular-arc drawing (in 2D). Recently, Chaplick et al. [4] introduced a different measure for the visual complexity, the affine cover number, which is the minimum number of lines (or planes) that together cover a crossing-free straight-line drawing of a graph G in 2D (3D). In this paper, we introduce the spherical cover number, which is the minimum number of circles (or spheres) that together cover a crossing-free circulararc drawing in 2D (or 3D). It turns out that spherical covers are sometimes significantly smaller than affine covers. Moreover, there are highly symmetric graphs that have symmetric optimum spherical covers but apparently no symmetric optimum affine cover. For complete, complete bipartite, and platonic graphs, we analyze their spherical cover numbers and compare them to their affine cover numbers as well as their segment and arc numbers. We also link the spherical cover number to other graph parameters such as chromatic number, treewidth, and linear arboricity.

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Drawing Graphs on Few Circles and Few Spheres

Cited by 4 publications

References 19 publications

Drawing Planar Graphs with Few Geometric Primitives

Drawing Planar Graphs with Few Geometric Primitives

Drawing Planar Graphs with Few Segments on a Polynomial Grid

Drawing Graphs on Few Circles and Few Spheres

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