Pole Dancing: 3D Morphs for Tree Drawings

Abstract

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...

On Morphing 1-Planar Drawings

“…The main challenge is to design morphing algorithms that maintain some additional geometric properties of the input drawings throughout the transformation, such as planarity with straight-line edges (see, e.g., [1,16,24]), convexity [6,33], orthogonality [12,25,25], and upwardness [20]. We point the interested reader to [4,8,10,11] for additional related work.…”

Morphing tree drawings in a small 3D grid

“…For general 3D graph drawings the problem seems challenging: it is tightly connected to unknot recognition problem, that is in NP ∩ co-NP [16,18], and its containment in P is widely open. If the given graph is a tree, the worst-case tight bound of Θ(n) steps holds for 3D crossing-free linear morph [5] (and the lower-bound example is again a path). If both the initial and the final drawing are planar, then O(log n) steps suffice [5].…”

“…If the given graph is a tree, the worst-case tight bound of Θ(n) steps holds for 3D crossing-free linear morph [5] (and the lower-bound example is again a path). If both the initial and the final drawing are planar, then O(log n) steps suffice [5].…”

“…A morph consists of morphing steps, where vertices move simultaneously along straight-line trajectories at constant speed. A crossingfree morph is known between two drawings of an n-vertex planar graph G with O(n) morphing steps and sometimes Ω(n) steps is necessary even when G is a path [Alamdari et al 2017], while using 3D can reduce the number of steps to O(log n) for an n-vertex tree [Arseneva et al 2019]. However, these morphs do not bound one practical parameter, the resolution.…”

How to Morph a Tree on a Small Grid

“…We remark that, although tree drawing algorithms are well investigated in Graph Drawing, morphs of tree drawings have not been the subject of research until now, with the exception of the recent work by Arseneva et al [6], who showed how to construct a three-dimensional crossing-free morph between two straight-line planar drawings of an n-node tree in O(log n) morphing steps.…”