Saturated Partial Embeddings of Maximal Planar Graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929620440317952 |
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| author | Clifton, Alexander Simon, Dániel G. |
| author_facet | Clifton, Alexander Simon, Dániel G. |
| contents | 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 $psr(G) \geq \frac{\frac{3}{2}n - 3}{3n - 6} = \frac{1}{2}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_06068 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Saturated Partial Embeddings of Maximal Planar Graphs Clifton, Alexander Simon, Dániel G. Combinatorics 52C99 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 $psr(G) \geq \frac{\frac{3}{2}n - 3}{3n - 6} = \frac{1}{2}$. |
| title | Saturated Partial Embeddings of Maximal Planar Graphs |
| topic | Combinatorics 52C99 |
| url | https://arxiv.org/abs/2412.06068 |