Non-trivial runs of Prim's algorithm
Prim's algorithm greedily chooses the smallest edge leaving the set of already connected nodes (until all nodes are connected).

It is not clear what "non-trivial run" means here; I propose this: a run of Prim is trivial if in every step, it can choose the smallest available edge. This would happen on linear chains, for instance.

With this "definition", a non-trivial run is one during which the algorithm can not choose a cheap edge because it would create a cycle, but has to take a more expensive one. Here is an example graph:


[source]

If you start the algorithm with $A$, edge $\{B,C\}$ would be the cheapest after having chosen $\{A,B\}$ and $\{A,C\}$, but it can not be chosen.