We study the question whether a crossing-free 3D morph between two straight-line drawings of an n-vertex tree can be constructed consisting of a small number of linear morphing steps. We look both at the case in which the two given drawings are two-dimensional and at the one in which they are three-dimensional. In the former setting we prove that a crossing-free 3D morph always exists with O(log n) steps, while for the latter Θ(n) steps are always sufficient and sometimes necessary. * We here refer to pole dancing as a fitness and competitive sport. The authors hope that many of our readers try this activity themselves, and will in return introduce many pole dancers to Graph Drawing, thereby alleviating the gender imbalance in both communities. The authors do not condone any pole activity used for sexual exploitation or abuse of women or men. between any two topologically-equivalent † planar straight-line ‡ drawings of the same planar graph always exists; this was proved for maximal planar graphs by Cairns [8] back in 1944, and then for all planar graphs by Thomassen [16] almost forty years later. Note that a planar morph between two planar graph drawings that are not topologically equivalent does not exist.It has lately been well investigated whether a planar morph between any two topologically-equivalent planar straight-line drawings of the same planar graph always exists such that the vertex trajectories have low complexity. This is usually formalized as follows. Let Γ and Γ be two topologically-equivalent planar straight-line drawings of the same planar graph G. Then a morph M is a sequence Γ 1 , Γ 2 , . . . , Γ k of planar straight-line drawings of G such that Γ 1 = Γ , Γ k = Γ , and Γ i , Γ i+1 is a planar linear morph, for each i = 1, . . . , k − 1. A linear morph Γ i , Γ i+1 is such that each vertex moves along a straight-line segment at uniform speed; that is, assuming that the morph happens between time t = 0 and time t = 1, the position of a vertex v at any time t ∈The complexity of a morph M is then measured by the number of its steps, i.e., by the number of linear morphs it consists of.A recent sequence of papers [3,4,5,6] culminated in a proof [2] that a planar morph between any two topologically-equivalent planar straight-line drawings of the same n-vertex planar graph can always be constructed consisting of Θ(n) steps. This bound is asymptotically optimal in the worst case, even for paths.The question we study in this paper is whether morphs with sub-linear complexity can be constructed if a third dimension is allowed to be used. That is: Given two topologically-equivalent planar straight-line drawings Γ and Γ of the same n-vertex planar graph G does a morph M = Γ = Γ 1 , Γ 2 , . . . , Γ k = Γ exist such that: (i) for i = 1, . . . , k, the drawing Γ i is a crossing-free straight-line 3D drawing of G, i.e., a straight-line drawing of G in R 3 such that no two edges cross; (ii) for i = 1, . . . , k − 1, the step Γ i , Γ i+1 is a crossing-free linear morph, i.e., no two edges cross throughout the tr...

No abstract

In this paper we study the area requirements of planar greedy drawings of triconnected planar graphs. Cao, Strelzoff, and Sun exhibited a family H of subdivisions of triconnected plane graphs and claimed that every planar greedy drawing of the graphs in H respecting the prescribed plane embedding requires exponential area. However, we show that every n-vertex graph in H actually has a planar greedy drawing respecting the prescribed plane embedding on an O(n)×O(n) grid. This reopens the question whether triconnected planar graphs admit planar greedy drawings on a polynomial-size grid. Further, we provide evidence for a positive answer to the above question by proving that every n-vertex Halin graph admits a planar greedy drawing on an O(n) × O(n) grid. Both such results are obtained by actually constructing drawings that are convex and anglemonotone. Finally, we consider α-Schnyder drawings, which are angle-monotone and hence greedy if α ≤ 30 • , and show that there exist planar triangulations for which every α-Schnyder drawing with a fixed α < 60 • requires exponential area for any resolution rule.

We study the question whether a crossing-free 3D morph between two straight-line drawings of an $n$-vertex tree $T$ can be constructed consisting of a small number of linear morphing steps. We look both at the case in which the two given drawings are two-dimensional and at the one in which they are three-dimensional. In the former setting we prove that a crossing-free 3D morph always exists with $O({rpw}(T))\subseteq O(\log n)$ steps, where ${rpw}(T)$ is the rooted pathwidth or Strahler number of $T$, while for the latter setting $\Theta(n)$ steps are always sufficient and sometimes necessary.

Given a set P of n points in R 2 and an input line γ in R 2 , we present an algorithm that runs in optimal Θ(n log n) time and Θ(n) space to solve a restricted version of the 1-Steiner tree problem. Our algorithm returns a minimum-weight tree interconnecting P using at most one Steiner point s ∈ γ, where edges are weighted by the Euclidean distance between their endpoints. We then extend the result to j input lines. Following this, we show how the algorithm of Brazil et al.[10] that solves the k-Steiner tree problem in R 2 in O(n 2k ) time can be adapted to our setting. For k > 1, restricting the (at most) k Steiner points to lie on an input line, the runtime becomes O(n k ). Next we show how the results of Brazil et al. [10] allow us to maintain the same time and space bounds while extending to some non-Euclidean norms and different tree cost functions. Lastly, we extend the result to j input curves.