Upward Planar Morphs

TL;DR

This paper proves the existence of upward planar morphs between topologically-equivalent upward planar drawings of directed graphs, with bounds on the number of steps needed depending on the graph class, and shows some cases require linear steps.

Contribution

It establishes bounds on the number of morphing steps needed for upward planar drawings, including optimal bounds for specific graph classes and a lower bound example.

Findings

O(1) steps for reduced planar st-graphs

O(n) steps for planar st-graphs and reduced upward planar graphs

O(n^2) steps for general upward planar graphs

Abstract

We prove that, given two topologically-equivalent upward planar straight-line drawings of an $n$-vertex directed graph $G$, there always exists a morph between them such that all the intermediate drawings of the morph are upward planar and straight-line. Such a morph consists of $O(1)$ morphing steps if $G$ is a reduced planar $st$-graph, $O(n)$ morphing steps if $G$ is a planar $st$-graph, $O(n)$ morphing steps if $G$ is a reduced upward planar graph, and $O(n^2)$ morphing steps if $G$ is a general upward planar graph. Further, we show that $\Omega(n)$ morphing steps might be necessary for an upward planar morph between two topologically-equivalent upward planar straight-line drawings of an $n$-vertex path.