Euclidean Constructibility in Graph-Minimization Problems

1. Let b1, * * *, bN be any set of distinct points in the plane. By a tree U on the vertices b1, , bv we mean any set consisting of some of the (2) closed straight segments bibj with the property that any two vertices can be joined by a sequence of segments belonging to U in one and only one way. A segment bibj is called a branch of U, the length L(U) of U is the sum of the lengths of its branches, {bi} is the set of all vertices sending branches to the vertex bi and w(b,) is their number. We now formulate the problem: