GATE | GATE-CS-2016 (Set 1) | Question 50

G = (V, E) is an undirected simple graph in which each edge has a distinct weight, and e is a particular edge of G. Which of the following statements about the minimum spanning trees (MSTs) of G is/are TRUE

I.  If e is the lightest edge of some cycle in G, 
    then every MST of G includes e
II. If e is the heaviest edge of some cycle in G, 
    then every MST of G excludes e

(A) I only
(B) II only
(C) both I and II
(D) neither I nor II

Answer: (B)

Explanation:
I is NOT true.
Let G=(V, E) be a rectangular graph where V = {a, b, c, d} and E = {ab, bc, cd, da, ac}.
Let the edges have weights: ab = 1, bc = 2, cd = 4, da = 5, ac = 3. Then, clearly, ac is the lightest edge of the cycle cdac, however, the MST abcd with cost 7 (= ab + bc + cd) does not include it.
Let the edges have weights: ab = 6, bc ā€“ 7, cd = 4, da = 5, ac = 3. Then, again, ac is the lightest edge of the cycle cdac, and, the MST bacd with cost 13 (= ba + ac + cd) includes it.
So, the MSTs of G may or may not include the lightest edge.

II is true
Let the heavies edge be e. Suppose the minimum spanning tree which contains e. If we add one more edge to the spanning tree we will create a cycle. Suppose we add edge eā€™ to the spanning tree which generated cycle C. We can reduce the cost of the minimum spanning tree if we choose an edge other than e from C for removal which implies that e must not be in minimum spanning tree and we get a contradiction.

Source: http://www.ece.northwestern.edu/~dda902/336/hw5-sol.pdf

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