Polygons with a Unique Triangulation

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For each n > 3, find a polygon with n vertices that has a unique triangulation.

I want to say that you can somehow build these polygons by continuously adding triangles somehow, but I'm not sure.

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2 Answers

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Here is a nonagon with a unique triangulation. Not quite a full answer, but perhaps it could give you some ideas.

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You are right. You can achieve this by continuously adding triangles. You start with a polygon of size 4. You can easily see that it consists of some vertex $v_0$ and a reflex chain. Let's say w.l.o.g that $v_0$ is always right to the reflex chain. Then every vertex on that chain can only see $v_0$ and thus the diagonal to $v_0$ is the only diagonal possible for a specific vertex on that chain.

The above nonagon is way too complicated and with such a spike construction you must be very careful that you do not get a convex quadrilateral lying completely inside the polygon. This would destroy the uniqueness, because you will then have a flippable edge in your triangulation.

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