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Let $G$ be a graph of genus $g$ embedded on a surface of genus $g$. Let $s,t \in V(G)$. Consider a minimal $s,t$-cut $C$ in $G$. Let $H$ consist of the union of faces adjacent to $E(C)$. Notice that $H$ is also embedded on a surface. Is the dual $H^*$ of $H$ always $2$-connected?

Any pointer, reference or proof will be appreciated.

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