How is a minimum bottleneck spanning tree different from a minimum spanning tree?

Question

A minimum bottleneck spanning tree of a weighted graph G is a spanning tree of G such that minimizes the maximum weight of any edge in the spanning tree. A MBST is not necessarily a MST (minimum spanning tree).

Please give an example where these statements make sense.

Answer 1

Look at the MST example on Wikipedia for reference:

A bottleneck in a spanning tree is a maximum-weight edge in that tree. There may be several bottlenecks (all of the same weight of course) in a spanning tree. In the Wikipedia MST there are two bottlenecks of weight 8.

Now, take a minimum spanning tree of a given graph (there may be several MSTs, all with the same total edge weight of course) and call the maximum edge weight B. In our example B = 8.

Any spanning tree that also has a bottleneck of B = 8 is an MBST. But it may not be an MST (because the total edge weight is bigger than the best possible).

So, take the Wikipedia MST and modify it (add / remove some edges) so that

1. it remains a spanning tree, and
2. it still doesn't use any weight > 8, yet
3. you increase the total edge weight

For example change just the sub-tree "on the left" of the Wikipedia MST (consisting of weights {2, 2, 3}) to {2, 3, 6}, thus increasing total edge weight by 4 without changing the bottleneck of 8. Bingo, you've created an MBST which is not an MST.