Complement is connected iff Connected components are Simply Connected
If the connected open set $H \subset \mathbb C$ is not simply connected, there is a simple closed curve $C$ in $H$ that is not homotopic to a point in $H$. Therefore there must be points inside $C$ that is not in $H$. Such a point and $\infty$ are in different connected components of $({\mathbb C} \cup \{\infty\} - G$.