Steiner Reducing Sets of Minimum Weight Triangulations

This paper develops techniques for computing the minimum weight Steiner triangulation of a planar point set. We call a Steiner point P a Steiner reducing point of a planar point set X if the weight (sum of edge lengths) of a minimum weight triangulation of X ∪{P} is less than that of X. We define the Steiner reducing set St(X) to be the collection of all Steiner reducing points of X. We provide here necessary conditions for membership in the Steiner reducing set. We prove that St(X) can be topologically complex, containing multiple connected components or even holes. We construct families of sets X for which the number of connected components of St(X) grows linearly in the cardinality of X. We further prove that St(X) need not be simply connected, and the rank of H1(St(X)) (i.e. the number of holes) can also grow linearly in the cardinality of X.