Let $G$ be a finite graph embedded in $\mathbb{C}$. Let $F$ denote denote its unbounded face. Is there notation or a name for $F^c$ without referring directly to $F$. Of course this is equivalent to the union of $G$ with all its bounded faces.
More generally, if $G$ is a bounded set in $\mathbb{C}$, and $F$ is the unbounded component of $G^c$, I would like notation for $F^c$ which refers to $G$ and not $F$.
If there is no notation already I will probably use "$\text{fill}(G)$", because I am "filling" $G$. Any better suggestion?
I remember doing this a while ago: "given set $A$, let $B$ be the complement of $\infty$ in $\mathbb C\setminus A$". I did not find any notation or term for this thing,
But I know that the term filled-in Julia set is in wide use, and it means precisely the operation you describe, applied to the Julia set of something. So, to generalize this to an arbitrary set $G$ and call the result $\operatorname{fill}(G)$ makes perfect sense.