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 $e^*$: <hr /> [![PentagonTiling][1]][1] <br /> <sup> Distinguished edges marked red.</sup> <br /> <sup> Top: $G =$ collection of $4$-cycles. Bottom: $G =$ collection of segments. </sup> <hr /> 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. <hr /> [![SpiralTiling][2]][2] <br /> <sup> Image: [An introduction to tilings](http://pi.math.cornell.edu/~mec/2008-2009/KathrynLindsey/PROJECT/homepage.html).</sup> <br /> <sup> Original MO: [Radial tilings with variable area ratios ](https://mathoverflow.net/a/83148/6094). </sup> <hr /> <hr /> <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/cg4Nd.jpg