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GATE | GATE CS Mock 2018 | Question 37

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Let G = (V, E) be any connected undirected edge-weighted graph. The weights of the edges in E are positive. Consider the following statements:
  1. The path between a pair of vertices in a minimum spanning tree of an undirected graph is necessarily the shortest (minimum weight) path.
  2. Minimum Spanning Tree of G is always unique and shortest path between a pair of vertices may not be unique.
Which of the above statements is/are necessarily true? (A) (1) only (B) (2) only (C) both (1) and (2) (D) neither (1) nor (2)

Answer: (D)

Explanation: Since, edge weights are not defined. If edge weights are distinct then MST is always unique but may not be shortest paths.
(A)---1----(B)----2---(C)
 \                    /
  ---------3----------
And, if edge weights are not distinct then both MST and shorted paths are not unique.
(A)---2----(B)----2---(C)
 \                    /
  ---------2----------
So, both statements are false. Option (D) is correct.

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Last Updated : 03 Jan, 2018
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