# 2-connectivity of dual of a minimal cut in a bounded genus graph

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.