Hamiltonian Tetrahedralizations with Steiner Points

A tetrahedralization of a point set in 3-dimensional space is Hamiltonian if its dual graph has a Hamiltonian cycle. Let S be a set of n points in general position in 3-dimensional space. We prove that by adding to S at most ⌊ 2 ⌋ Steiner points in the interior of the convex hull of S, we obtain a point set that admits a Hamiltonian tetrahedralization. We also obtain an O(m 3 2 ) + O(n log n) algorithm to solve this problem, where m is the number of elements of S on its convex hull. We also prove that point sets with at most 20 convex hull points have a Hamiltonian tetrahedralization without the addition of any Steiner points.