Suppose that T is a minimum spanning tree for a connected, weighted graph G and that G contains an edge e (not a loop) that is not in T. Let v and w be the endpoints of e. By exercise 18 there is a unique path in T from v to w. Let e’ be any edge of this path. Prove that .
Suppose, there is a minimum spanning tree for a connected, weighted graph .
The connected graph contains an edge which is not in spanning tree .
Correctness of Prim’s Algorithm:
When a connected, weighted graph is input to prim’s algorithm, the output is a minimum spanning tree for .
Consider the end points of the edge are .
The objective is to prove that .
Assume that .
Form a new graph by adding and deleting .
The addition of an edge to a spanning tree creates a circuit, and the deletion of an edge from a circuit does not disconnect a graph.
Consequently is also a spanning tree for
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