I am attempting a question on MST below, but I do not understand the answer of the question. I am trying to work backwards form the answer to find out how they got to the solution.
The question I am attempting is F.
Question:
Solution:
2026-02-23 16:52:00.1771865520
Minimum Spanning Tree - Finding X in an interval (adjustment of an arc)
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If a minimal spanning tree were to contain the edge $TU$, the two subtrees we'd get by removing $TU$ would be connected by at least one other edge, and we could replace $TU$ by that edge, reducing the total cost. Thus a minimal spanning tree doesn't contain $TU$. By the same argument, it doesn't contain $UW$. Thus it must contain $RU$ (to reach $U$). These arguments go through just as well if $RU$ has any weight less than $31$. Since increasing the weight of $RU$ up to that point doesn't change the membership of $RU$ in the unique minimal spanning tree, it doesn't change the minimal spanning tree.
On the other hand, if the weight of $RU$ is $31$ or more, we can replace it by $UW$ without increasing the total cost.