Question

Let G be any undirected graph with positive edge weights, and T be a minimum spanning tree of G. For any two vertices, u and v, let \( d_1(u, v) \) and \( d_2(u, v) \) be the shortest distances between u and v in G and T, respectively. Which ONE of the following options is CORRECT for all possible G, T, u, and v?

• A. \( d_1(u, v) = d_2(u, v) \)
• B. \( d_1(u, v) \leq d_2(u, v) \)
• C. \( d_1(u, v) \geq d_2(u, v) \)
• D. \( d_1(u, v) \neq d_2(u, v) \)

Hint:

In an MST, the shortest path between two vertices in the original graph \( G \) can be shorter or equal to the shortest path in the MST. The MST may exclude certain edges that could shorten the path.

Answer 1

The Correct Option is B

\( d_1(u, v) \) represents the shortest path between vertices \( u \) and \( v \) in the original graph \( G \), while \( d_2(u, v) \) represents the shortest path between the same vertices in the minimum spanning tree (MST) \( T \).

Key Concept: A minimum spanning tree (MST) is a subgraph of \( G \) that connects all the vertices together with the minimum possible total edge weight. However, it does not necessarily preserve the shortest paths between all pairs of vertices.

When we compute \( d_1(u, v) \), it represents the shortest path in the original graph \( G \), which can use any edges of \( G \), including some that are excluded from the MST. In contrast, \( d_2(u, v) \) represents the shortest path in the MST \( T \), where only edges included in the MST are used.

Important property of MST: Since the MST includes only a subset of the edges of \( G \), it is possible for the shortest path in \( G \) to be shorter than or equal to the shortest path in the MST. The MST may exclude edges that could reduce the distance between two vertices, so the shortest path in the MST is greater than or equal to the shortest path in the original graph.

Thus, the correct relationship is: \[ d_1(u, v) \leq d_2(u, v) \] This means the shortest path in the original graph \( G \) is always less than or equal to the shortest path in the minimum spanning tree \( T \).

Therefore, the correct answer is \( \boxed{B} \).