Suppose you are given a connected graph G = (V, E), with a cost ce on each edge e. In an earlier problem, we saw that when all edge costs are distinct, G has a unique minimum spanning tree. However, G may have many minimum spanning trees when the edge costs are not all distinct. Here we formulate the question: Can Kruskal’s Algorithm be made to find all the minimum spanning trees of G?
Recall that Kruskal’s Algorithm sorted the edges in order of increasing cost, then greedily processed edges one by one, adding an edge e as long as it did not form a cycle. When some edges have the same cost, the phrase "in order of increasing cost" has to be specified a little more carefully: we’ll say that an ordering of the edges is valid if the corresponding sequence of edge costs is nondecreasing. We’ll say that a valid execution of Kruskal’s Algorithm is one that begins with a valid ordering of the edges of G.
For any graph G, and any minimum spanning tree T of G, is there a valid execution of Kruskal’s Algorithm on G that produces T as output? Give a proof or a counterexample.
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