  In this work, we study how far one can deviate from optimal behavior when
embedding a planar graph. For a planar graph $G$, we say that a plane subgraph
$H\subseteq G$ is a \textit{plane-saturated subgraph} if adding any edge
(possibly with new vertices) to $H$ would either violate planarity or make the
resulting graph no longer a subgraph of $G$. For a planar graph $G$, we define
the \textit{plane-saturation ratio}, $\psr(G)$, as the minimum value of
$\frac{e(H)}{e(G)}$ for a plane-saturated subgraph $H$ of $G$ and investigate
how small $\psr(G)$ can be. While there exist planar graphs where $\psr(G)$ is
arbitrarily close to $0$, we show that for all twin-free planar graphs,
$\psr(G)>1/16$, and that there exist twin-free planar graphs where $\psr(G)$ is
arbitrarily close to $1/16$. In fact, we study a broader category of planar
graphs, focusing on classes characterized by a bounded number of degree $1$ and
degree $2$ twin vertices. We offer solutions for some instances of bounds while
positing conjectures for the remaining ones.
