On the Complexity of Some Geometric Problems With Fixed Parameters
Vol. 25, no. 1, pp. 195-218, 2021. Regular paper.
Abstract 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 $k$.

 This work is licensed under the terms of the CC-BY license.
Submitted: July 2020.
Reviewed: December 2020.
Revised: January 2021.
Accepted: January 2021.
Final: January 2021.
Published: January 2021.
Communicated by Martin Nöllenburg
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