Re-embedding a 1-Plane Graph into a Straight-line Drawing in Linear Time

TL;DR

This paper presents a linear-time algorithm to determine if a 1-plane graph embedding can be re-embedded into a straight-line drawing without forbidden configurations, extending Thomassen's characterization.

Contribution

It introduces a new characterization of 1-plane embeddings for straight-line re-embedding and provides an efficient linear-time algorithm for the process.

Findings

Linear-time algorithm for re-embedding 1-plane graphs

Characterization of forbidden configurations for straight-line drawings

Preservation of crossing edge pairs during re-embedding

Abstract

Thomassen characterized some 1-plane embedding as the forbidden configuration such that a given 1-plane embedding of a graph is drawable in straight-lines if and only if it does not contain the configuration [C. Thomassen, Rectilinear drawings of graphs, J. Graph Theory, 10(3), 335-341, 1988]. In this paper, we characterize some 1-plane embedding as the forbidden configuration such that a given 1-plane embedding of a graph can be re-embedded into a straight-line drawable 1-plane embedding of the same graph if and only if it does not contain the configuration. Re-embedding of a 1-plane embedding preserves the same set of pairs of crossing edges. We give a linear-time algorithm for finding a straight-line drawable 1-plane re-embedding or the forbidden configuration.