Convexity-increasing morphs of planar graphs

Abstract

We study the problem of convexifying drawings of planar graphs. Given any planar straightline drawing of an internally 3-connected graph, we show how to morph the drawing to one with strictly convex faces while maintaining planarity at all times. Our morph is convexity-increasing, meaning that once an angle is convex, it remains convex. We give an efficient algorithm that constructs such a morph as a composition of a linear number of steps where each step either moves vertices along horizontal lines or moves vertices along vertical lines. Moreover, we show that a linear number of steps is worst-case optimal. To obtain our result, we use a well-known technique by Hong and Nagamochi for finding redrawings with convex faces while preserving y-coordinates. Using a variant of Tutte's graph drawing algorithm, we obtain a new proof of Hong and Nagamochi's result which comes with a better running time. This is of independent interest, as Hong and Nagamochi's technique serves as a building block in existing morphing algorithms.

Planar and Toroidal Morphs Made Easier

“…First, we describe a much simpler algorithm to construct piecewise-linear morphs between planar straight-line graphs. Specifically, given two isomorphic planar straight-line graphs Γ 0 and Γ 1 with strictly convex faces and the same outer face, we construct a morph from Γ 0 to Γ 1 that consists of O(n) unidirectional morphing steps, in O(n 1+ω/2 ) time, matching the existing state of the art [32]; this also improves a prior result of Angelini, Da Lozzo, Frati, Lubiw, Patrignani, and Roselli [5] for computing convexity-preserving morphs. Our morphing algorithm replaces Cairns' edge collapses with an application of Floater and Gotsman's barycentric interpolation.…”

“…Our first result is a "best-of-both-worlds" result that simultaneously obtains nice properties enjoyed by two different approaches for planar morphing: Floater and Gotsman's barycentric interpolation method [26,28,[46][47][48] results in morphs that are natural and visually appealing but are represented implicitly; variations on Cairns' edge-collapse method [1,9,10,32,49] result in efficient explicit representations of morphs that are, unfortunately, not useful for visualization. In particular, we describe a very simple algorithm that achieves an efficient explicit representation of a morph that should still be useful for visualization.…”

“…A long series of later works, culminating in papers by Alamdari, Angelini, Barrera-Cruz, Chan, Da Lozzo, Di Battista, Frati, Haxell, Lubiw, Patrignani, Roselli, Singla, and Wilkinson [1], and Kleist, Klemz, Lubiw, Schlipf, Staals, and Strash [32], describe efficient algorithms to construct planar morphs with explicit piecewise-linear vertex trajectories, all ultimately based on Cairns' inductive edge-collapsing strategy. (Alamdari et al [1] and Roselli [42] provide more detailed history of these results.)…”

The Segment Number: Algorithms and Universal Lower Bounds for Some Classes of Planar Graphs

“…There are multiple ways to define this property and it will be convenient to refer to all of them. Therefore, we use the following characterization; for proofs see, e.g., Kleist et al [9]. In the context of our recursive strategy, we face a special case of the following problem: given an internally 3-connected plane graph G and a convex drawing Γ o of the boundary of its outer face, extend Γ o to a convex drawing of G. It is known that such an extension exists if and only if each segment of Γ o corresponds to an archfree path of G [3,4,16,17].…”

Dynamic Graph Map Animation

“…Although linear interpolation is smooth and easy to implement, it is not good for preserving the mental map [3]. Newer methods concentrate on poly-linear interpolation, that is, contiguous sequences of linear interpolations [6].…”