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The edge-length ratio of a straight-line drawing of a graph is the ratio between the lengths of the longest and of the shortest edge in the drawing. The planar edge-length ratio of a planar graph is the minimum edge-length ratio of any planar straight-line drawing of the graph.In this paper, we study the planar edge-length ratio of planar graphs. We prove that there exist n-vertex planar graphs whose planar edge-length ratio is in Ω(n); this bound is tight. We also prove upper bounds on the planar edge-length ratio of several families of planar graphs, including series-parallel graphs and bipartite planar graphs.

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We study the problem of drawing a dynamic graph, where each vertex appears in the graph at a certain time and remains in the graph for a fixed amount of time, called the window size. This defines a graph story, i.e., a sequence of subgraphs, each induced by the vertices that are in the graph at the same time. The drawing of a graph story is a sequence of drawings of such subgraphs. To support readability, we require that each drawing is straight-line and planar and that each vertex maintains its placement in all the drawings. Ideally, the area of the drawing of each subgraph should be a function of the window size, rather than a function of the size of the entire graph, which could be too large. We show that the graph stories of paths and trees can be drawn on a $2W \times 2W$ and on an $(8W+1) \times (8W+1)$ grid, respectively, where $W$ is the window size. These results are constructive and yield linear-time algorithms. Further, we show that there exist graph stories of planar graphs whose subgraphs cannot be drawn within an area that is only a function of $W$.

In this paper we study planar morphs between straight-line planar grid drawings of trees. A morph consists of a sequence of morphing steps, where in a morphing step vertices move along straight-line trajectories at constant speed. We show how to construct planar morphs that simultaneously achieve a reduced number of morphing steps and a polynomially-bounded resolution. We assume that both the initial and final drawings lie on the grid and we ensure that each morphing step produces a grid drawing; further, we consider both upward drawings of rooted trees and drawings of arbitrary trees.Upward planar morphs between strictly-upward drawings of rooted ordered trees maintain the drawing order-preserving at all times, as proved in the following.Lemma 2. Let Γ 0 and Γ 1 be two order-preserving strictly-upward straight-line planar drawings of a rooted ordered tree T . Let M be any upward planar morph between Γ 0 and Γ 1 . Then any intermediate drawing of M is order-preserving.Proof. Assume that the morph M happens between the time instants t = 0 and t = 1. For any t ∈ [0, 1], denote by Γ t the drawing of T in M at time t. Since M is upward, the drawing Γ t is strictly-upward, for any t ∈ [0, 1]. Hence, it suffices to prove that, for any internal node v of T , the edges from v to its children enter v in Γ t in the left-to-right order associated with v; this is indeed true for t = 0 and t = 1. Suppose that, for some t ∈ (0, 1), the left-to-right order in which two edges (u 1 , v) and (u 2 , v) enter v in Γ t is different than in Γ 0 . Since such edges are represented by curves monotonically increasing in the y-direction from u 1 and u 2 to v throughout M, it follows that there is a time t * ∈ (0, t) such that the edges (u 1 , v) and (u 2 , v) overlap in Γ t * . However, this cannot happen due to the planarity of M. 4