Track Layouts of Graphs

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.

Section: Placingmentioning

confidence: 92%

Section: B the Parallel Affine Cover Numbersmentioning

confidence: 99%

Drawing Graphs on Few Lines and Few Planes

Chaplick

Fleszar

Lipp

et al. 2016

“…Proof. (i)⇒(ii) Pemmaraju proves in his thesis [13] (see also [4]) that if G is a graph, π is a vertex ordering of G with no (k + 1)-rainbow, V 1 , . .…”

Section: Planar Posets Of Bounded Heightmentioning

confidence: 99%

The Queue-Number of Posets of Bounded Width or Height

Knauer

Micek

Ueckerdt

2018

Heath and Pemmaraju [9] conjectured that the queuenumber of a poset is bounded by its width and if the poset is planar then also by its height. We show that there are planar posets whose queue-number is larger than their height, refuting the second conjecture. On the other hand, we show that any poset of width 2 has queue-number at most 2, thus confirming the first conjecture in the first non-trivial case. Moreover, we improve the previously best known bounds and show that planar posets of width w have queue-number at most 3w − 2 while any planar poset with 0 and 1 has queue-number at most its width.

“…Chaplick et al [4] also investigated the complexity of computing the affine cover numbers. Among others, they showed that in 3D, for l ∈ {1, 2}, it is NP-complete to decide whether π l 3 (G) ≤ 2 for a given graph G. In 2D, the question has still been open, but a related question was raised by Dujmović et al [7] already in 2004. They investigated so-called track layouts which are defined as follows.…”

Section: Introductionmentioning

confidence: 99%

Line and Plane Cover Numbers Revisited

Biedl

Felsner

Meijer

et al. 2019

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.