Description: Suppose n= (V,{e}) is a connected network, and U is a non-empty subset of the vertex set V. if (u,v) is an edge with a minimum weight (cost), where u∈u,v∈v-u, there must be a minimum spanning tree that contains an edge (U,V).
Prove:
Assume that any one of the smallest spanning trees of net n is not contained (U,V). Set T is a minimal spanning tree connected to the Web, when the Edge (U,V) is added to T, by the definition of the spanning tree, there must be a one-day loop containing (u,v) in T. On the other hand, since T is a spanning tree, there must be another edge (U ', V ') on T, where u ' ∈u,v ' ∈v-u, and between U and U ', V and V ' have a path that communicates. By deleting the Edge (U ', V '), the circuit can be eliminated and another spanning tree T ' is obtained. Because (U,V) is not higher than (U ', V '), T's standby is no higher than t,t ' is a minimal spanning tree containing (u,v). This contradicts the hypothesis.
MST properties (for constructing the smallest spanning tree)