Morphing Planar Graph Drawings via Orthogonal Box Drawings

TL;DR

This paper presents an efficient algorithm for morphing planar graph drawings into orthogonal box drawings, enabling planarity-preserving transformations with small grid size and a constant number of bends per edge.

Contribution

It introduces a novel morphing algorithm that transforms planar graph drawings into orthogonal box drawings with controlled grid size and bends, running in quadratic time.

Findings

Achieves planarity-preserving morphs with small grid size.

Uses only a constant number of bends per edge.

Runs in O(n^2) time on a standard word RAM model.

Abstract

We give an algorithm to morph planar graph drawings that achieves small grid size at the expense of allowing a constant number of bends on each edge. The input is an $n$-vertex planar graph and two planar straight-line drawings of the graph on an $O(n) \times O(n)$ grid. The planarity-preserving morph is composed of $O(n)$ linear morphs between successive pairs of drawings, each on an $O(n) \times O(n)$ grid with a constant number of bends per edge. The algorithm to compute the morph runs in $O(n^2)$ time on a word RAM model with standard arithmetic operations -- in particular no square roots or cube roots are required. The first step of the algorithm is to morph each input drawing to a planar orthogonal box drawing where vertices are represented by boxes and each edge is drawn as a horizontal or vertical segment. The second step is to morph between planar orthogonal box drawings. This is done by extending known techniques for morphing planar orthogonal drawings with point vertices.