The degree of a point in a triangulation is the number of edges incident to it. Give an example of a set of n points in the plane such that, no matter how the set is triangulated, there is always a point whose degree is n−1.

Explanation:

The vertices of a square is one such set of points. Either diagonal will bring the degree of the points involved to 3 = 4-1.

The vertices of a regular pentagon is another such set of points. After joining the points in a convex hull, any interior connection will create a triangle and a quadrilateral. The diagonal of the quadrilateral will bring the degree of at least one of the points to 4 = 5-1.

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