Algorithms for updating minimum spanning trees is kina grannis dating

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A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted (un)directed graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight.

Proof: Assume that there is a MST T that does not contain e.

Adding e to T will produce a cycle, that crosses the cut once at e and crosses back at another edge e' .

Proof: if e was not included in the MST, removing any of the (larger cost) edges in the cycle formed after adding e to the MST, would yield a spanning tree of smaller weight.

If T is a tree of MST edges, then we can contract T into a single vertex while maintaining the invariant that the MST of the contracted graph plus T gives the MST for the graph before contraction.

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Deleting e' we get a spanning tree T\U of strictly smaller weight than T. By a similar argument, if more than one edge is of minimum weight across a cut, then each such edge is contained in some minimum spanning tree.

If the minimum cost edge e of a graph is unique, then this edge is included in any MST.

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and want to visit every other node, then what is the most efficient path to do that?

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