On the Complexity of Some Geometric Problems With Fixed Parameters - Schaefer, Marcus (Journal of graph algorithms and applications, 2021)
The following graph-drawing problems are known to be complete for the existential theory of the reals (${\exists \mathbb{R}}$-complete) as long as the parameter $k$ is unbounded. Do they remain ${\exists \mathbb{R}}$-complete for a fixed value $k$? Do $k$ graphs on a shared vertex set have a simultaneous geometric embedding? Is $G$ a segment intersection graph, where $G$ has maximum degree at most $k$? Given a graph $G$ with a rotation system and maximum degree at most $k$, does $G$ have a straight-line drawing which realizes the rotation system? We show that these, and some related, problems remain ${\exists \mathbb{R}}$-complete for constant $k$, where $k$ is in the double or triple digits. To obtain these results we establish a new variant of Mnëv's universality theorem, in which the gadgets are placed so as to interact minimally; this variant leads to fixed values for $k$, where the traditional variants of the universality theorem require unbounded values of $k$.