# The dual graph of the triangulation

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.

Thanks!

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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.

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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