GATE | GATE-CS-2017 (Set 1) | Question 37

Let G = (V, E) be any connected undirected edge-weighted graph. The weights of the edges in E are positive any distinct. Consider the following statements:

I. Minimum Spanning Tree of G is always unique.
II. Shortest path between any two vertices of G is always unique.

Which of the above statements is/are necessarily true?

(A)

I only

(B)

II only

(C)

both I and II

(D)

neither I and II

Answer: (A)

Explanation:

I. Minimum Spanning Tree of G is always unique – MST will always be distinct if the edges are unique so Correct II. Shortest path between any two vertices of G is always unique – Shortest path between any two vertices can be same so incorrect Therefore, option A is correct 

Alternate solution: We know that minimum spanning tree of a graph is always unique if all the weight are distinct, so statement 1 is correct. Now statement 2 , this might not be true in all cases. Consider the graph.
There are two shortest paths from a to b (one is direct and other via node c) So statement 2 is false Hence option a is correct.

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