On the $\mathcal{NP}$-hardness of GRacSim Drawing and k-SEFE Problems

TL;DR

This paper proves that finding right-angle crossings in simultaneous graph drawings and a restricted version of SEFE are both computationally hard problems, with GRacSim Drawing being NP-hard and k-SEFE being NP-complete.

Contribution

It establishes the NP-hardness and NP-completeness of two specific problems in simultaneous graph drawing, expanding understanding of their computational complexity.

Findings

GRacSim Drawing is NP-hard.

k-SEFE is NP-complete.

Reductions from 3-Partition demonstrate complexity.

Abstract

We study the complexity of two problems in simultaneous graph drawing. The first problem, GRacSim Drawing, asks for finding a simultaneous geometric embedding of two graphs such that only crossings at right angles are allowed. The second problem, k-SEFE, is a restricted version of the topological simultaneous embedding with fixed edges (SEFE) problem, for two planar graphs, in which every private edge may receive at most $k$ crossings, where $k$ is a prescribed positive integer. We show that GRacSim Drawing is $\mathcal{NP}$-hard and that k-SEFE is $\mathcal{NP}$-complete. The $\mathcal{NP}$-hardness of both problems is proved using two similar reductions from 3-Partition.