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# Title: Rectangular Spiral Galaxies are Still Hard

Abstract: Spiral Galaxies is a pencil-and-paper puzzle played on a grid of unit squares: given a set of points called centers, the goal is to partition the grid into polyominoes such that each polyomino contains exactly one center and is $180^\circ$ rotationally symmetric about its center. We show that this puzzle is NP-complete even if the polyominoes are restricted to be (a) rectangles of arbitrary size or (b) 1$\times$1, 1$\times$3, and 3$\times$1 rectangles. The proof for the latter variant also implies NP-completeness of finding a non-crossing matching in modified grid graphs where edges connect vertices of distance $2$. Moreover, we prove NP-completeness of the design problem of minimizing the number of centers such that there exist a set of Spiral Galaxies that exactly cover a given shape.
 Comments: 20 pages, 20 figures Subjects: Computational Geometry (cs.CG) Cite as: arXiv:2110.00058 [cs.CG] (or arXiv:2110.00058v1 [cs.CG] for this version)

## Submission history

From: Christiane Schmidt [view email]
[v1] Thu, 30 Sep 2021 19:33:32 GMT (2644kb,D)

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