Let $G=(V,E)$ be undirected, and let $s,t\in V$ and $C\subseteq E$ be a circle that contains $s$ and $t$. Assuming $s$ and $t$ are on the circle $C$, we are given a set of edges $F\subseteq E$ which are to be removed. After removing these edges, we want to verify whether the remaining edges from the circle $C(F) = C \backslash F$ have a subcircle $C^{'} \subseteq C(F)$ that contains both $s$ and $t$.
If the circle $C$ is known to be simple, the answer is quick. If $C\cap F \neq \emptyset$ the answer is false, there is no subcircle, let alone one that contains $s$ and $t$.
I'm having trouble to see how one can verify whether such a subcircle exists in the case where $C$ is not a simple circle. I tried locating subcircles that have been hit by $F$ and thus cannot be used, but its seems every rule I try to make has a counterexample.
Edit: Running an $s,t$-flow is too expensive, perhaps there is a faster way, as this was a circle.
