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

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 NP-hard and that K-SEFE is NP-complete. The NP-hardness of both problems is proved using two similar reductions from 3-PARTITION.