On the NP-hardness of GRacSim drawing and k-SEFE Problems

Abstract

We study the complexity of two problems on simultaneous graph drawing. The first problem, $\rm{GR{\small AC} S{\small IM~DRAWING}}$, asks for finding a simultaneous geometric embedding of two planar graphs, sharing a common subgraph, such that only crossings at right angles are allowed, and every crossing must involve a private edge of one graph and a private edge of the other graph. The second problem, $k-\rm{SEFE}$, is a restricted version of the topological simultaneous embedding with fixed edges ($\rm{SEFE}$) 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 $\rm{GR{\small AC} S{\small IM~DRAWING}}$ is $\mathcal{NP}$-hard and that $k-\rm{SEFE}$ is $\mathcal{NP}$-complete. The $\mathcal{NP}$-hardness of both problems is proved using two similar reductions from $\rm{3-P\small{ARTITION}}$.