We say that a graph G =  (V, E) is a triangulated cycle graph if it consists of the vertic...

We say that a graph G =  (V, E) is a triangulated cycle graph if it consists of the vertices and edges of a triangulated convex n-gon in the plane–in other words, if it can be drawn in the plane as follows.

The vertices are all placed on the boundary of a convex set in the plane (we may assume on the boundary of a circle), with each pair of consecutive vertices on the circle joined by an edge. The remaining edges are then drawn as straight line segments through the interior of the circle, with no pair of edges crossing in the interior. We require the drawing to have the following property. If we let S denote the set of all points in the plane that lie on vertices or edges of the drawing, then each bounded component of the plane after deleting S is bordered by exactly three edges. (This is the sense in which the graph is a "triangulation.")

A triangulated cycle graph is pictured in Figure 1.

Prove that every triangulated cycle graph has a tree decomposition of width at most 2, and describe an efficient algorithm to construct such a decomposition.

Figure 1 A triangulated cycle graph: The edges form the boundary of a convex polygon together with a set of line segments that divide its interior into triangles.