Drawing Graphs on Few Lines and Few Planes

Self Cite

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.

Section: Complexity Of Computing Weak Line Covers In 2dmentioning

confidence: 99%

“…Theorem 1 (Collapse of the Affine Hierarchy [3]) For any integers 1 ≤ l < 3 ≤ d and for any graph G, it holds that π l d (G) = π l 3 (G) and ρ l d (G) = ρ l 3 (G).…”

Section: Introductionmentioning

confidence: 99%

Line and Plane Cover Numbers Revisited

Biedl

Felsner

Meijer

et al. 2019

Self Cite

“…We will use the following corollary of the theory: Corollary 1 Let P = (X, <) be a partial order on n elements, then there is an orthogonal pair A, C where A is a k-antichain and C an -chain and k + Fig. 4 together with two orthogonal pairs of L corresponding to the boundary points (1,3) and (3,1) of G(L); chains of C are blue, antichains of A are red, green, and yellow.…”

Section: Orthogonal Partitions Of Posetsmentioning

confidence: 99%

4-Connected Triangulations on Few Lines

Felsner

2019

We show that 4-connected plane triangulations can be redrawn such that edges are represented by straight segments and the vertices are covered by a set of at most p 2n lines each of them horizontal or vertical. The same holds for all subgraphs of such triangulations.The proof is based on a corresponding result for diagrams of planar lattices which makes use of orthogonal chain and antichain families.

“…These trivial lower bounds are the same as for the slope number of graphs [25], that is, the minimum number of slopes required to draw all edges, and the slope number is upper bounded by the number of segments required. 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

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.