Question $2$: PROVE WITHOUT ANY POSSIBILITY OF REFUTATION the computational complexity $($worst case asymptotic lower bound$)$ and computational complexity class of the problem of determining, if a given set $S$ of cycles $($defined as sequences of alternating vertices and edges that start and end at the same vertex$)$ within a graph $G=(V,E)$ is minimal, in the sense that it is not possible to obtain the same set of generated $($as combinations of$)$ cycles $($considering that $2$ cycles that differ only from the number of times the same sequence of alternating vertices and edges is traversed are identical$),$ from either a strict subset of $S,$ or a set $S^{\text{'}}$ where the only differences with $S$ correspond to always shorter cycles in $S^{\text{'}},$ and design and PROVE WITHOUT ANY POSSIBILITY OF REFUTATION the computational complexity $($worst case asymptotic upper bound$)$ of an optimal algorithm solving that problem $.$ hint if the problem is undecidable then there is no posible solution.