Ideal Tree-drawings of Approximately Optimal Width (And Small Height)

保持高度不变的平面图形图形转换

Transforming planar graph drawings while maintaining height

平面图形绘制有许多样式，特别是直线绘制、折线绘制、正交图形绘制和可见性表示。在这篇文章中，我们展示了许多这样的绘图可以从一种风格转换为另一种风格，而不需要改变绘图的高度。然后我们给出了这些变换的一些应用。

Drawing Halin-graphs with small height

In this paper, we study how to draw Halin-graphs, i.e., planar graphs that consist of a tree $T$ and a cycle among the leaves of that tree. Based on tree-drawing algorithms and the pathwidth $ pw(T) $, a well-known graph parameter, we find poly-line drawings of height at most $6pw(T)+3\in O(\log n)$. We also give an algorithm for straight-line drawings, and achieve height at most $12pw(T)+1$ for Halin-graphs, and smaller if the Halin-graph is cubic. We show that the height achieved by our algorithms is optimal in the worst case (i.e. for some Halin-graphs).

Order-preserving drawings of trees with approximately optimal height (and small width)

In this paper, we study how to draw trees so that they are planar, straight-line and respect a given order of edges around each node. We focus on minimizing the height, and show that we can always achieve a height of at most 2pw(T)+1, where pw(T) (the so-called pathwidth) is a known lower bound on the height. Hence we give an asymptotic 2-approximation algorithm. We also create a drawing whose height is at most 3pw(T ), but where the width can be bounded by the number of nodes. Finally we construct trees that require height 2pw(T)+1 in all planar order-preserving straight-line drawings.

Minimum Height Drawings of Ordered Trees in Polynomial Time: Homotopy Height of Tree Duals

We consider drawings of graphs in the plane in which vertices are assigned distinct points in the plane and edges are drawn as simple curves connecting the vertices and such that the edges intersect only at their common endpoints. There is an intuitive quality measure for drawings of a graph that measures the height of a drawing $ϕ: G \rightarrow \mathbb{R}^2$ as follows. For a vertical line $\ell$ in $\mathbb{R}^2$, let the height of $\ell$ be the cardinality of the set $\ell \cap ϕ(G)$. The height of a drawing of $G$ is the maximum height over all vertical lines. In this paper, instead of ab…

Homotopy height, grid-major height and graph-drawing height

It is well-known that both the pathwidth and the outer-planarity of a graph can be used to obtain lower bounds on the height of a planar straight-line drawing of a graph. But both bounds fall short for some graphs. In this paper, we consider two other parameters, the (simple) homotopy height and the (simple) grid-major height. We discuss the relationship between them and to the other parameters, and argue that they give lower bounds on the straight-line drawing height that are never worse than the ones obtained from pathwidth and outer-planarity.

A 2-Approximation for the Height of Maximal Outerplanar Graph Drawings

In this paper, we study planar drawings of maximal outerplanar graphs with the objective of achieving small height. A recent paper gave an algorithm for such drawings that is within a factor of 4 of the optimum height. In this paper, we substantially improve the approximation factor to become 2. The main ingredient is to define a new parameter of outerplanar graphs (the so-called umbrella depth, obtained by recursively splitting the graph into graphs called umbrellas). We argue that the height of any poly-line drawing must be at least the umbrella depth, and then devise an algorithm that achie…