Finding MST for connected graph with $|E|=|V|+87$
The "Cycle Property" helps (If you don't know it before, try to prove it first):


  Let $C$ be any cycle in $G$ and $e$ a heaviest (not necessarily unique) edge in $C$. Then there exists some MST which does not contain $e$.


You are encouraged to answer the following questions:


How to identify a cycle and then a heaviest edge $e$ in it?
What to do with the edge $e$?
How many times should you apply the cycle property?
When to stop?


Then analyze the time complexity.