Geometric conditions for Euclidean Steiner trees in ℜdℜd

We present geometric conditions that can be used to restrict or eliminate candidate topologies for Euclidean Steiner minimal trees in ℜdℜd, d⩾2d⩾2. Our emphasis is on conditions that are not restricted to the planar case (d=2d=2). For trees with a Steiner topology we give restrictions on terminal–Steiner connections that are based on the Voronoi diagram associated with the set of terminal nodes. We then describe more restrictive conditions for trees with a full Steiner topology and show how these conditions can be used to improve implicit enumeration algorithms for finding Euclidean Steiner minimal trees with d>2d>2.