Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I study Polygon Triangulation and have an execise.

Prove or disprove: The dual graph of the triangulation of a monotone polygon is always a chain, that is, any node in this graph has degree at most two.

It seems like the assumption that the dual graph of the triangulation of a monotone polygon is always a chain is false. But how to prove this.

Let's say if it was true, at least one edge of every triangle would be a part of the boundary of the polygon.

I don't have any idea how approach the proof. Please help me out.


share|cite|improve this question
up vote 3 down vote accepted

Does the following serve as a contradiction to the claim? I think a more interesting question would be to prove or disprove that for all monotone polygons there exists a triangulation such that its dual graph is a chain. I'm afraid I do not have anything non-trivial to say about this question. Contradiction for assertion

share|cite|improve this answer
Thanks, I thought maybe there are any rigorous math proof why this is not true – fog Apr 14 '12 at 6:23
And by "this" do you mean the original problem you had asked (proving or disproving that dual of any triangulation of a monotone polygon is a chain) or the the one I suggested (proving or disproving that there always exists a triangulation of a monotone polygon such that it's dual is a chain)? – Shitikanth Apr 14 '12 at 6:28
both of them are interesting ;) – fog Apr 14 '12 at 10:13
Well I dont see how showing a producing counterexample is not sufficiently rigorous. – Shitikanth Apr 14 '12 at 10:25

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.