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

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.

