The Complexity of Drawing Graphs on Few Lines and Few Planes

Chaplick, Steven; Fleszar, Krzysztof; Lipp, Fabian; Ravsky, Alexander; Verbitsky, Oleg; Wolff, Alexander

doi:10.1007/978-3-319-62127-2_23

Cited by 17 publications

(21 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We prove the last inequality. Since a graph K 9 has 36 = 6 · 6 edges, each cover of K 6 by six copies of K 4 generates a Steiner system S (2,4,9). The absence of such a system follows from the result of Hanani mentioned earlier, but we give a direct proof.…”

Section: The Affine Cover Numbers Of General Graphs: Proofsmentioning

confidence: 68%

Drawing Graphs on Few Lines and Few Planes

Chaplick

Fleszar

Lipp

et al. 2016

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

Abstract. We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight-line edges and crossing-free drawings. This problem has many connections to other challenging graph-drawing problems such as small-area or small-volume drawings, layered or track drawings, and drawing graphs with low visual complexity. While some facts about our problem are implicit in previous work, this is the first treatment of the problem in its full generality. Our contribution is as follows.-We show lower and upper bounds for the numbers of lines and planes needed for covering drawings of graphs in certain graph classes. In some cases our bounds are asymptotically tight; in some cases we are able to determine exact values. -We relate our parameters to standard combinatorial characteristics of graphs (such as the chromatic number, treewidth, maximum degree, or arboricity) and to parameters that have been studied in graph drawing (such as the track number or the number of segments appearing in a drawing). -We pay special attention to planar graphs. For example, we show that there are planar graphs that can be drawn in 3-space on a lot fewer lines than in the plane.

show abstract

Section: The Affine Cover Numbers Of General Graphs: Proofsmentioning

confidence: 68%

Drawing Graphs on Few Lines and Few Planes

Chaplick

Fleszar

Lipp

et al. 2016

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

show abstract

“…This point must be a tripod or a bend. Hence, in any drawing δ of I k , t(δ) + b(δ) ≥ k and, by Lemma 2, seg 2 (I k ) = seg 3…”

Section: Proofmentioning

confidence: 95%

“…Proposition 2. There is an infinite family (G k ) k≥1 of connected cubic graphs such that G k has n k = 6k − 2 vertices and seg 2 (G k ) = seg 3…”

Section: Singly-connected Cubic Graphsmentioning

confidence: 99%

Variants of the Segment Number of a Graph

Okamoto

Ravsky

Wolff

2019

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

The segment number of a planar graph is the smallest number of line segments whose union represents a crossing-free straight-line drawing of the given graph in the plane. The segment number is a measure for the visual complexity of a drawing; it has been studied extensively.In this paper, we study three variants of the segment number: for planar graphs, we consider crossing-free polyline drawings in 2D; for arbitrary graphs, we consider crossing-free straight-line drawings in 3D and straight-line drawings with crossings in 2D. We first construct an infinite family of planar graphs where the classical segment number is asymptotically twice as large as each of the new variants of the segment number. Then we establish the ∃R-completeness (which implies the NPhardness) of all variants. Finally, for cubic graphs, we prove lower and upper bounds on the new variants of the segment number, depending on the connectivity of the given graph.

show abstract

“…Among others, Chaplick et al showed that the affine cover number can be asymptotically smaller than the segment number, constructing an infinite family of triangulations (T n ) n>1 such that T n has n vertices and ρ 1 2 (T n ) = O( √ n), but seg(T n ) = Ω(n). On the other hand, they showed that seg(G) = O(ρ 1 2 (G) 2 ) for any connected planar graph G. In a companion paper [5], Chaplick et al show that most variants of the affine cover number are NP-hard to compute.…”

Section: Introductionmentioning

confidence: 99%

Drawing Graphs on Few Circles and Few Spheres

Kryven

Ravsky

Wolff

2018

Algorithms and Discrete Applied Mathematics

Self Cite

View full text Add to dashboard Cite

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.

show abstract

The Complexity of Drawing Graphs on Few Lines and Few Planes

Cited by 17 publications

References 28 publications

Drawing Graphs on Few Lines and Few Planes

Drawing Graphs on Few Lines and Few Planes

Variants of the Segment Number of a Graph

Drawing Graphs on Few Circles and Few Spheres

Contact Info

Product

Resources

About