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Chapter 28 Minimum Spanning Trees
Exercises Practice Problems
1. Prim's Algorithm.
Prim's Algorithm "grows a minimum edge into a MST." Given weighted graph \(G=(V,E)\text{:}\)
Initialize \(T\) to be a vertex \(v\)
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At each step, grow the tree:
Stop when you can't add more edges.
Use Prim's Algorithm to find the MST of the following weighted graph.
2. Kruskal's Algorithm.
Kruskal's Algorithm grows a minimum forest until it is a MST. Given weighted graph \(G=(V,E)\text{:}\)
Initialize \(T\) to contain all the vertices of \(G\text{,}\) but no edges.
-
At each step, grow the forest:
Stop when you can't add more edges.
3. MST Examples.
Give an example of a weighted graph on 6 vertices that \(\ldots\)
has a unique minimum spanning tree.
has exactly 2 minimum spanning trees.
has more than 2 minimum spanning trees.
4. A Minimum Edge.
Let \(G=(V,E)\) be a weighted graph.
Suppose that \(G\) contains a unique edge \(e\) of minimum weight in \(G\text{.}\) Prove that \(e\) is contained in every minimum spanning tree of \(G\text{.}\)
Suppose that \(G\) contains an edge \(e\) such that for every cycle \(C\) containing \(e\text{,}\) the weight \(w(e)\) is smaller than all the other edge weights in \(C-e\text{.}\) Prove that \(e\) is contained in every minimum spanning tree of \(G\text{.}\)
5. Minimum Spanning Forest.
Extend the concept of a minimum spanning tree to a disconnected, weighted graph. Define a "minimum spanning forest" and describe how you would find it.
