[3] In each of these cases, the Kleetope is formed by attaching pyramids onto each face of the original polyhedron.
One method of forming the Kleetope of a polytope P is to place a new vertex outside P, near the centroid of each facet.
[7] If a polyhedron with n vertices is formed by repeating the Kleetope construction some number of times, starting from a tetrahedron, then its longest path has length O(nlog3 2); that is, the shortness exponent of these graphs is log3 2, approximately 0.630930.
The same technique shows that in any higher dimension d, there exist simplicial polytopes with shortness exponent logd 2.
[8] Similarly, Plummer (1992) used the Kleetope construction to provide an infinite family of examples of simplicial polyhedra with an even number of vertices that have no perfect matching.