GATE | GATE-CS-2000 | Question 41

Let G be an undirected connected graph with distinct edge weight. Let emax be the edge with maximum weight and emin the edge with minimum weight. Which of the following statements is false?
(A) Every minimum spanning tree of G must contain emin
(B) If emax is in a minimum spanning tree, then its removal must disconnect G
(C) No minimum spanning tree contains emax
(D) G has a unique minimum spanning tree


Answer: (C)

Explanation:

In kruskal’s algorithm, we pick the edges in ascending order and add them to the forest if no cycle is formed. Option A is True because first edge could never create a cycle.
The only reason for emax to be present in the minimum spanning tree could be that it is the only possible edge to cover a particular vertex in a tree since all vertices have to be present in a spanning tree by definition. Consider the image below

pranjul_41

See question 2 of https://www.geeksforgeeks.org/data-structures-and-algorithms-set-8/

This solution is contributed by Pranjul Ahuja


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