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Consider the following undirected connected graph $G$ with weights on its edges as given in the figure below. A minimum spanning tree is a spanning tree of least weight and a maximum spanning tree is one with largest weight. A second best minimum spanning tree whose weight is the smallest among all spanning trees that are not minimum spanning trees in $G$.

Which of the following statements is TRUE in the above graph? (Note that all the edge weights are distinct in the above graph)

1. There is more than one minimum spanning tree and similarly, there is more than one maximum spanning tree here.
2. There is a unique minimum spanning tree, however there is more than one maximum spanning tree here.
3. There is more than one minimum spanning tree, however there is a unique maximum spanning tree here.
4. There is more than one minimum spanning tree and similarly, there is more than one second-best minimum spanning tree here.
5. There is unique minimum spanning tree, however there is more than one second-best minimum spanning tree here.

edited | 889 views
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option e ?
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I think Algorithms tag should be added in this question.

+1

To see that the second-best minimum spanning tree need not be unique, here is a weighted, undirected graph with a $unique\ MST$ of weight 7 and $two\ second-best\ MST's$ of weight 8:

$Ans: E$

In the graph we have all edge weights are distinct so we will get unique minimum and maximum spanning tree.
Each Cycle must exclude maximum weight edge in minimum spanning tree.
Here, we have two cycle of $3$ edges, $a-d-e$ and $c-g-k.$
For second best minimum spanning tree, exclude $a-e$ edge and include $d-e$ edge

Other way for second best minimum spanning tree: exclude $c-g$ edge and include $g-k$ edge.

So, e should be the answer.

by Boss (25.6k points)
edited by
+2
nice ans
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nice explanation papesh
Since, all the edge weights are distinct, we have a unique minimum weight spanning tree and maximum weight spanning tree.
And we can have more than one second best minimum spanning tree because different combinations of edge may give second largest value.

You can also try finding MinST and MaxST from the given graph.

Hence, option 'e' is only possible.
by Loyal (9.5k points)
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And we can have more than one second best minimum spanning tree because different combinations of edge may give second largest value.

Can u give some examples

the second minimum spanning tree wil contain the value +1 of the minimum spanning tree choosen

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second best minimum spanning tree is that whose weight is smallest among all spanning tree that are not minimum spanning trees in G it means to have second largest value we can get many combination of edges here so it will not be unique
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