Algorithms Analysis Practice Test

In a minimum spanning tree (MST) of a graph, it is not guaranteed that the shortest path between any two vertices \( s \) and \( t \) will always be along the edges of the tree \( T \). The MST is constructed to minimize the total edge weight while connecting all vertices, but this does not necessarily imply that the paths within the MST represent the shortest paths between every pair of vertices in the original graph.

The shortest path between two vertices in the original graph may involve edges that are not included in the minimum spanning tree, especially if those edges provide a more direct or less costly route than what is provided by the substructure of the MST. In fact, the MST can have longer paths compared to the shortest paths that might be available in the complete graph.

Thus, the assertion that the shortest path from any vertices \( s \) to \( t \) is along \( T \) is incorrect, affirming that the answer is indeed false.