Geodesic
On the NP-hardness of GRacSim drawing and k-SEFE Problems
Journal of Graph Algorithms and Applications, Special Issue on Graph Drawing Beyond Planarity , Tome 22 (2018) no. 1, pp. 101-116.

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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 $\rm{3-P\small{ARTITION}}$. 
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     author = {Luca Grilli},
     title = {On the {NP-hardness} of {GRacSim} drawing and {k-SEFE} {Problems}},
     journal = {Journal of Graph Algorithms and Applications},
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Luca Grilli. On the NP-hardness of GRacSim drawing and k-SEFE Problems. Journal of Graph Algorithms and Applications, 
							Special Issue on Graph Drawing Beyond Planarity
					, Tome 22 (2018) no. 1, pp. 101-116. doi : 10.7155/jgaa.00456. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00456/

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