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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, a minimum spanning tree cannot contain a cycle, because by definition a tree, including a minimum spanning tree, is acyclic.

Step by step solution

01

Understanding Graphs and Trees

In graph theory, a graph consists of vertices (or nodes), and edges (or arcs) connecting these vertices. A tree is a special type of a graph which is minimally connected (meaning there exists a path from any node to any other node) and has no cycles (which would imply that there is more than one path from at least one node to another).
02

Understand Minimum Spanning Trees

A minimum spanning tree (MST) of a graph is a subset of the edges of the graph that forms a tree and includes every vertex, with the total weight of all the edges in the tree being the smallest possible. Given that an MST is a tree, it is minimally connected and it inherently does not contain a cycle.
03

Conclude the answer

By the definition of a tree in graph theory, and the assumption that a minimum spanning tree, being a tree, therefore contains no cycles, you can conclude that, no, a minimum spanning tree cannot have a cycle.

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Most popular questions from this chapter

Assume that in a network of computers any two computers can be linked. Given a cost estimate for each possible link, should Algorithm 4.1 (Prim's Algorithm) or Algorithm 4.2 (Kruskal's Algorithm) be used? Justify your answer.

Use induction to prove the correctness of Dijkstra's Algorithm (Algorithm 4.3).

Suppose we assign \(n\) persons to \(n\) jobs. Let \(C_{U}\) be the cost of assigning the ith person to the jth job. Use a greedy approach to write an algorithm that finds an assignment that minimizes the total cost of assigning all \(n\) persons to all \(n\) jobs. Analyze your algorithm, and show the results using order notation.

Prove that a complete graph (a graph in which there is an edge between every pair of vertices has \(n^{n-2}\) spanning trees, Here \(n\) is the number of vertices in the graph.

Implement Prim's Algorithm (Algorithm 4,1) on your system, and study its performance using different graphs.
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