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Chapter 4: Problem 8

Do you think it is possible for a minimum spanning tree to have a cycle? Justify your answer.

Short Answer

Expert verified
No, it's impossible for a minimum spanning tree to contain a cycle, as by definition a tree is a connected graph without cycles, and an MST is a tree that minimizes the total edge weights.

Step by step solution

01

Analyzing the Definition of MST

A Minimum Spanning Tree (MST) is a subset of the graph that forms a tree that includes every vertex and has the minimum possible total edge weight. By definition, a tree is a connected graph that has no cycles. Therefore, an MST, being a tree, should not contain any cycles.
02

Explaining the Cycle Creation Paradox

Suppose we add an edge to an MST to form a cycle. Then one of the edges in the cycle must have a weight greater than or equal to the weight of all other edges in the cycle (since we are calling it an MST, it should have minimum weight). So, we can remove that edge to break the cycle while decreasing or keeping the total weight the same, contradicting the assumption that it was forming an MST.
03

Conclusion

So, based on the definition of MST and the cycle creation paradox, it is confirmed that an MST can't have a cycle. Adding a new edge to an MST would either increase its weight or create a cycle, which contradicts the fact that the MST has the minimum total weight and doesn't contain any cycle. Therefore, it is impossible for an MST to have a cycle.

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