## Abstract

In three-dimensional space an embedded network is called gradient-constrained if the absolute gradient of any differentiable point on the edges in the network is no more than a given value *m*. A *gradient-constrained minimum Steiner tree T* is a minimum gradient-constrained network interconnecting a given set of points. In this paper we investigate some of the fundamental properties of these minimum networks. We first introduce a new metric, the *gradient metric*, which incorporates a new definition of distance for edges with gradient greater than *m*. We then discuss the variational argument in the gradient metric, and use it to prove that the degree of Steiner points in *T* is either three or four. If the edges in *T* are labelled to indicate whether the gradients between their endpoints are greater than, less than, or equal to *m*, then we show that, up to symmetry, there are only five possible labellings for degree 3 Steiner points in *T*. Moreover, we prove that all four edges incident with a degree 4 Steiner point in *T* must have gradient *m* if *m* is less than 0.38. Finally, we use the variational argument to locate the Steiner points in T in terms of the positions of the neighbouring vertices.

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