Steiner minimal trees on sets of four points

Abstract

Let S = {A, B, C, D} consist of the four corner points of a convex quadrilateral where diagonals [A, C] and [B, D] intersect at the point O. There are two possible full Steiner trees for S, the AB-CD tree has A and B adjacent to one Steiner point, and C and D to another; the AD-BC tree has A and D adjacent to one Steiner point, and B and C to another. Pollak proved that if both full Steiner trees exist, then the AB-CD (AD-BC) tree is the Steiner minimal tree if[Figure not available: see fulltext.]AOD>3 ( <90° but the AD-BC tree does not exist, then the AB-CD tree cannot be ruled out as a Steiner minimal tree, though under certain broad conditions it can. © 1987 Springer-Verlag New York Inc.