A Survey on Graph Drawing Beyond Planarity <s> Problem 19. Does every acyclic planar digraph admit a 1-bend upward RAC drawing? <s> Let G1 and G2 be two planar graphs having some vertices in common. A simultaneous embedding of G1 and G2 is a pair of crossing-free drawings of G1 and G2 such that each vertex in common is represented by the same point in both drawings. In this paper we show that an outerplanar graph and a simple path can be simultaneously embedded with fixed edges such that the edges in common are straight-line segments while the other edges of the outerplanar graph can have at most one bend per edge. We then exploit the technique for outerplanar graphs and paths to study simultaneous embeddings of other pairs of graphs. Namely, we study simultaneous embedding with fixed edges of: (i) two outerplanar graphs sharing a forest of paths and (ii) an outerplanar graph and a cycle. <s> BIB001 </s> A Survey on Graph Drawing Beyond Planarity <s> Problem 19. Does every acyclic planar digraph admit a 1-bend upward RAC drawing? <s> Given two planar graphs that are defined on the same set of vertices, a RAC simultaneous drawing is one in which each graph is drawn planar, there are no edge overlaps and the crossings between the two graphs form right angles. The geometric version restricts the problem to straight-line drawings. It is known, however, that there exists a wheel and a matching which do not admit a geometric RAC simultaneous drawing. In order to enlarge the class of graphs that admit RAC simultaneous drawings, we allow bends in the resulting drawings. We prove that two planar graphs always admit a RAC simultaneous drawing with six bends per edge each, in quadratic area. For more restricted classes of planar graphs (i.e., matchings, paths, cycles, outerplanar graphs and subhamiltonian graphs), we manage to significantly reduce the required number of bends per edge, while keeping the area quadratic. <s> BIB002
Every pair of planar graphs sharing their vertex set admits an SRE with at most six bends per edge BIB002 . On the other hand, if no restriction on the edge crossings is given, such a pair admits an SE with at most two bends per edge BIB001 .