On Morphing 1-Planar Drawings

TL;DR

This paper investigates the existence of topology-preserving morphs between graph drawings, establishing that such morphs always exist for a significant family of 1-planar graphs, advancing understanding in non-planar graph morphing.

Contribution

It proves that topology-preserving morphs always exist for a meaningful family of 1-planar graphs, extending morphing theory beyond planar graphs.

Findings

Topology-preserving morphs exist for certain 1-planar graphs.

Constructive proof with potentially complex vertex trajectories.

Advances understanding of morphing in non-planar graph drawings.

Abstract

Computing a morph between two drawings of a graph is a classical problem in computational geometry and graph drawing. While this problem has been widely studied in the context of planar graphs, very little is known about the existence of topology-preserving morphs for pairs of non-planar graph drawings. We make a step towards this problem by showing that a topology-preserving morph always exists for drawings of a meaningful family of $1$-planar graphs. While our proof is constructive, the vertices may follow trajectories of unbounded complexity.