Given a collection of planar graphs G1,…,Gk on the same set V of
n vertices, the simultaneous geometric embedding (with mapping) problem, or
simply k-SGE, is to find a set P of n points in the plane and a bijection
ϕ:V→P such that the induced straight-line drawings of G1,…,Gk
under ϕ are all plane.
This problem is polynomial-time equivalent to weak rectilinear realizability
of abstract topological graphs, which Kyn\v{c}l (doi:10.1007/s00454-010-9320-x)
proved to be complete for ∃R, the existential theory of the
reals. Hence the problem k-SGE is polynomial-time equivalent to several other
problems in computational geometry, such as recognizing intersection graphs of
line segments or finding the rectilinear crossing number of a graph.
We give an elementary reduction from the pseudoline stretchability problem to
k-SGE, with the property that both numbers k and n are linear in the
number of pseudolines. This implies not only the ∃R-hardness
result, but also a 22Ω(n) lower bound on the minimum size of a
grid on which any such simultaneous embedding can be drawn. This bound is
tight. Hence there exists such collections of graphs that can be simultaneously
embedded, but every simultaneous drawing requires an exponential number of bits
per coordinates. The best value that can be extracted from Kyn\v{c}l's proof is
only 22Ω(n)