Morphing Planar Graph Drawings with a Polynomial Number of Steps

Abstract

In 1944, Cairns proved the following theorem: given any two straight-line planar drawings of a triangulation with the same outer face, there exists a morph (i.e., a continuous transformation) between the two drawings so that the drawing remains straight-line planar at all times. Cairns's original proof required exponentially many morphing steps. We prove that there is a morph that consists of O(n 2 ) steps, where each step is a linear morph that moves each vertex at constant speed along a straight line. Using a known result on compatible triangulations this implies that for a general planar graph G and any two straight-line planar drawings of G with the same embedding, there is a morph between the two drawings that preserves straight-line planarity and consists of O(n 4 ) steps.

Free Edge Lengths in Plane Graphs

“…(c) and (1), we conclude that for a large enough L 2 ⌦(L 0 + 0 ) we can guarantee the same two properties we needed in (1), that is, ↵ maps every small rectangle R e to a parallelogram ↵(R e ) whose diameter is at least 0 , and every nonhorizontal edge to a segment of length at least L 0 + 2 0 . Hence, every remaining contracted vertex v e in b D 0 can be split within the parallelogram ↵(R e ) as in (1).…”

“…Hence, every remaining contracted vertex v e in b D 0 can be split within the parallelogram ↵(R e ) as in (1). To finish the construction, it remains to apply the inductive hypothesis to fill in the missing parts in the maximal separating triangles or 4-cycles.…”

“…An argument analogous to Lemma 7 (1) shows that H has an embedding such that the edges of the triangle abc are mapped to a given triangle and in the exterior of that triangle edge e has prescribed length. (Recall that in Lemma 7(1), the outer face was mapped to a given triangle such that disjoint edges in the interior of the triangle had prescribed lengths.)…”

Compatible Connectivity Augmentation of Planar Disconnected Graphs

“…Two planar drawings, ψ 1 and ψ 2 , of G are isomorphic if there exists a continuous family of planar drawings {ψ (t) : 0 ≤ t ≤ 1} of G such that ψ (0) = ψ 1 and ψ (1) = ψ 2 . 2 We call a graph, G = (V , E) a geometric planar graph if it is the image of some planar drawing of a graph G = (V , E). That is, V (G) = {ψ(v) : v ∈ V (G)}, E(G) = {(ψ(u), ψ(w)) : (u, w) ∈ E(G)}, and ψ is a planar drawing of G. When clear from context, we will sometimes treat a geometric planar graph interchangeably with the set of points and line segments defined by its vertices and edges, respectively.…”

“…Since then, a sequence of results has shown that morphs can be done Figure 2: Computer-assisted animation frequently involves morphing between a sequence of drawings of the same planar graph. Zooming in on a section of the image reveals that the artist's strokes are approximated by polygonal paths efficiently, so that the motion can be described concisely [2,3,12,19]. The most recent such result [3] shows that any planar drawing of an n-vertex connected planar graph can be morphed into any isomorphic drawing using a sequence of O(n) linear morphs, in which vertices move along linear trajectories at constant speed.…”