Let G = (V,E) be a planar graph. An arrangement of circular arcs is called a composite arc-drawing of G, if its 1-skeleton is isomorphic to G. Similarly, a composite segment-drawing is described by an arrangement of straight-line segments. We ask for the smallest ground set of arcs/segments for a composite arc/segment-drawing. We present algorithms for constructing composite arc-drawings for trees, series-parallel graphs, planar 3-trees and general planar graphs. In the case where G is a tree, we also introduce an algorithm that realizes the vertices of the composite drawing on a O(n 1.81) ×n grid. For each of the graph classes we provide a lower bound for the maximal size of the arrangement’s ground set.

methods

5. Linking and Cutting Trees

Drawing Graphs with Few Arcs

/pdf/drawing-graphs-with-few-arcs-mcvbfdbwh8.pdf

A stacking operation adds a $d$-simplex on top of a facet of a simplicial $d$-polytope while maintaining the convexity of the polytope. A stacked $d$-polytope is a polytope that is obtained from a $d$-simplex and a series of stacking operations. We show that for a fixed $d$ every stacked $d$-polytope with $n$ vertices can be realized with nonnegative integer coordinates. The coordinates are bounded by $O(n^{2\log(2d)})$, except for one axis, where the coordinates are bounded by $O(n^{3\log(2d)})$. The described realization can be computed with an easy algorithm. 
The realization of the polytopes is obtained with a lifting technique which produces an embedding on a large grid. We establish a rounding scheme that places the vertices on a sparser grid, while maintaining the convexity of the embedding.

Embedding Stacked Polytopes on a Polynomial-Size Grid

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 two collinear edges of the same vertex). 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 $O(n)\,\times \,O(n^2)$ grid. This algorithm can also be used with a small modification to draw maximal outerplanar graphs with 3n / 2 edges on an $O(n)\,\times \,O(n^2)$ 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 provides a significant improvement over the lower bound of 2n for line segments for a nontrivial graph class.

/pdf/drawing-planar-graphs-with-few-geometric-primitives-p9ch0qzdif.pdf

Drawing planar graphs with few geometric primitives

We study the problem of arranging a set of n disks with prescribed radii on n rays emanating from the origin such that two neighboring rays are separated by an angle of 2π/n. The center of the disks have to lie on the rays, and no two disk centers are allowed to lie on the same ray. We require that the disks have disjoint interiors, and that for every ray the segment between the origin and the boundary of its associated disk avoids the interior of the disks. Let $\widetilde r$ be the sum of the disk radii. We introduce a greedy strategy that constructs such a disk arrangement that can be covered with a disk centered at the origin whose radius is at most $2\widetilde r$ , which is best possible. The greedy strategy needs O(n) arithmetic operations.

As an application of our result we present an algorithm for embedding unordered trees with straight lines and perfect angular resolution such that it can be covered with a disk of radius n3.0367, while having no edge of length smaller than 1. The tree drawing algorithm is an enhancement of a recent result by Duncan et al. [Symp. of Graph Drawing, 2010] that exploits the heavy-edge tree decomposition technique to construct a drawing of the tree that can be covered with a disk of radius 2 n4.

/pdf/pinning-balloons-with-perfect-angles-and-optimal-area-3aesafnm4p.pdf

5. Linking and Cutting Trees

Citations

Drawing Graphs with Few Arcs

Drawing Graphs with Few Arcs

Embedding Stacked Polytopes on a Polynomial-Size Grid

Drawing planar graphs with few geometric primitives

Pinning balloons with perfect angles and optimal area

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