Here I begin an investigation into the related question: how few cells can you remove from the plane to exclude the shape of a given polyhex?
The dihex is hard to exclude! You must remove at least 2/3 of the cells:
To exclude the bent trihex you can remove half the cells. If you have a better exclusion, please let me know.
These trihexes and tetrahexes are excluded minimally.
These tetrahex exclusions are minimal:
These are not necessarily minimal:
A straight polyhex of odd order n can be excluded with 1/n holes. Here is an example for n=5:
