## Graham’s problem on shortest networks for points on a circle.(English)Zbl 0748.05051

Given a finite set of points $$x_ 1,x_ 2,\dots,x_ n$$ in the Euclidean plane. The problem of finding a network of smallest total length which connects up the points is called the Steiner problem and a solution is a Steiner tree or a Steiner minimum tree. When the points are located on a circle of radius $$r$$ with $$x_ i$$ adjacent to $$x_{i+1}$$, $$1\leq i\leq n-1$$ and at most one edge $$x_ ix_{i+1}$$ and $$x_ nx_ 1$$ has length greater than $$r$$, the authors prove that a Steiner tree consists of all edges $$x_ ix_{i+1}$$ plus $$x_ nx_ 1$$, $$1\leq i\leq n-1$$ with the longest edge removed.

### MSC:

 05C12 Distance in graphs 05C05 Trees

### Keywords:

Steiner problem; Steiner tree; tree; Graham’s conjecture
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### References:

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