Saturated Partial Embeddings of Maximal Planar Graphs

We investigate two notions of saturation for partial planar embeddings of maximal planar graphs. Let $G = (V, E)$ be a vertex-labeled maximal planar graph on $n$ vertices, which by definition has $3n - 6$ edges. We say that a labeled plane graph $H = (V, E')$ with $E' \subseteq E$ is a \emph{labeled plane-saturated subgraph} of $G$ if no edge in $E \setminus E'$ can be added to $H$ in a manner that preserves vertex labels, without introducing a crossing. The \emph{labeled plane-saturation ratio} $lpsr(G)$ is defined as the minimum value of $\frac{e(H)}{e(G)}$ over all such $H$. We establish almost tight bounds for $lpsr(G)$, showing $lpsr(G) \leq \frac{n+7}{3n-6}$ for $n \geq 47$, and constructing a maximal planar graph $G$ with $lpsr(G) \geq \frac{n+2}{3n-6}$ for each $n\ge 5$. Dropping vertex labels, a \emph{plane-saturated subgraph} is defined as a plane subgraph $H\subseteq G$ where adding any additional edge to the drawing either introduces a crossing or causes the resulting graph to no longer be a subgraph of $G$. The \emph{plane-saturation ratio} $psr(G)$ is defined as the minimum value of $\frac{E(H)}{E(G)}$ over all such $H$. For all sufficiently large $n$, we demonstrate the existence of a maximal planar graph $G$ with $psr(G) \geq \frac{\frac{3}{2}n - 3}{3n - 6} = \frac{1}{2}$.