For any cycle, must there exist a planar graph embedding with it as the face boundary?

Question

For a plane graph $G$, if $C$ is a cycle in $G$, can $G$ be embedded in the plane so that $C$ is the face boundary of the outer face of $G$?

Answer 1

Not necessarily.

Given any embedding of connected $G$ in the plane, if I can find a cycle $C_1$ that is outsides of $C$ and another cycle $C_2$ that is inside of $C$, then $C$ will always have to separate these 2 other cycles, hence can never be the outer face boundary.

Answer 2

For a concrete example, let $G$ be the complete graph on five vertices with a single edge removed. Let $C$ be the $3$-cycle in this graph around the remaining degree $4$ vertices. If you try to embed $G$ in the plane with $C$ as the outer face, then the remaining two vertices must be embedded inside of $C$, and each must be connected to each of outer three vertices. Drawing this out, you'll notice it's impossible.