Layouts with Wires of Balanced Length

For any graph G with fixed boundary there exists a layout in the plane, which minimizes the maximum Euclidean distance of any node to its neighbors. This layout balances the length of the graph edges and is therefore called a (length-) balanced layout of G. Furthermore the existence of a unique optimal balanced layout L with the following properties is proved: (i) L is the minimal element of an order defined on the set of layouts of a graph with fixed boundary. (ii) L may be constructed as the limit of the /,-optimal layouts L,, of G. (iii) If G is a planar graph with fixed boundary, then the optimal balanced layout L of G is quasi-planar.