On the Complexity of Some Geometric Problems With Fixed Parameters
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 simulta...
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| Published in: | Journal of graph algorithms and applications Vol. 25; no. 1; pp. 195 - 218 |
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| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
01.01.2021
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| ISSN: | 1526-1719, 1526-1719 |
| Online Access: | Get full text |
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| Summary: | 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 $k$. |
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| ISSN: | 1526-1719 1526-1719 |
| DOI: | 10.7155/jgaa.00557 |