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may be several minimum spanning trees (e.g.

### If a adjacent has not been.

when tree removal devonport edges have weight 1, all spanning trees are minimum spanning trees). The cut theorem: For any set Sof vertices in a graph G= (V;E), let e= (u;v) be the lightest edge with exactly one endpoint in S.

Then emust be in the MST. Jul 17, Find the acyclic subset \(T \subseteq E\) connecting all vertices with the minimum weight \[w(T) = \sum_{(u,v) \in T} w(u,v)\] \(T\) is acyclic and connects all vertices \(T\) is known as a spanning tree; The problem of finding this tree is known as the minimum spanning tree problem.

Def. A spanning tree of a graph G is a subgraph T that is connected and acyclic. Property. MST of G is always a spanning tree. 15 Greedy Algorithms Simplifying assumption. All edge costs ce are distinct. Cycle property. Let C be any cycle, and let f be the max cost edge belonging to C. Then the MST does not contain f. Cut property. A minimum weight edge in an undirected graph belongs to some minimum spanning tree.

The inverse of the Theorem about safe edges is not true. In other words, if (u,v) is a safe edge for A crossing (S, V−S) then it is not necessarily a light edge. Consider the set: {(u,v): there exists a cut (S, V−S) such that (u,v) is a light edge crossing it}.

Minimum Spanning Trees 1 Minimum Spanning Tree Let G=(V,E)be a connected, weighted graph. Recall that a weighted graph is a graph where we associate with each edge a real number, called the weight.

graph. Recall that a spanning tree of Gis a subgraph T of Gwhich is a tree that spans G. In other words, it contains all of the vertices of G.