Let $\cal T$ be a monohedral, edge-to-edge tiling of the plane, with
prototile $T$ a simple polygon, and with one edge $e^*$ of $T$ distinguished.
Associate a graph $G=G_{\cal T}$ with $\cal T$ as follows.
Each copy $T_i$ of $T$ in $\cal T$ is a node of $G$, and
$T_1$ and $T_2$ are connected by an arc in $G$ if they share a distinguished
edge of either tile.
$G$ is a planar graph without isolated nodes, often disconnected.
Note that $G$ depends on both $T$ and $e^*$:
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[![PentagonTiling][1]][1]
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<sup>
Distinguished edges marked red.</sup>
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<sup>
Top: $G =$ collection of $4$-cycles. Bottom: $G =$ collection of segments.
</sup>
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My question is: Which graphs $G_{\cal T}$ can be realized by some tiling $\cal T$?
Because there seems to be considerable freedom to "design" $G$
when $T$ is a regular $\{3,4,6\}$-gon, or tiles with several equal-length edges
(such as polyominoes<sup>1</sup>), perhaps this is an easier question:
> ***Q***. What are some graphs $G$ that cannot be realized by some tiling $\cal T$?
*Added*. An example where choosing the "base" of the horn-shape as the
distinguished edge, seems to produce an infinite chain.
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[![SpiralTiling][2]][2]
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<sup>
Image: [An introduction to tilings](http://pi.math.cornell.edu/~mec/2008-2009/KathrynLindsey/PROJECT/homepage.html). J.O'Rourke mods.</sup>
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<sup>
Original MO: [Radial tilings with variable area ratios
](https://mathoverflow.net/a/83148/6094).
Grünbaum & Shephard.
</sup>
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<sup>1</sup>
Rhoads, Glenn C. "Planar tilings by polyominoes, polyhexes, and polyiamonds." *Journal of Computational and Applied Mathematics* 174, no. 2 (2005): 329-353.
[Journal link](https://www.sciencedirect.com/science/article/pii/S0377042704002195).
[1]: https://i.sstatic.net/oaknB.jpg
[2]: https://i.sstatic.net/2xpjG.jpg