On the number of edges in saturated partial embeddings of maximal planar graphs
János Barát, Zoltán L. Blázsik, Balázs Keszegh, Zeyu Zheng
TL;DR
This work investigates the plane-saturation problem for maximal planar graphs by studying the minimum size of a plane-saturated subgraph embedded in a maximal planar graph. It establishes a universal upper bound: for all maximal planar graphs with $v(G)\ge16$, $\mathrm{sat}_{\mathcal{P}}(G)<(3-\epsilon)\,v(G)$ for some $\epsilon>0$, and it derives a simple lower bound $\mathrm{sat}_{\mathcal{P}}(G)\ge\frac{v(G)+4}{6}$ along with an explicit infinite family achieving $\mathrm{sat}_{\mathcal{P}}(G)\le\frac{v(G)+82}{3}$, which implies $\mathrm{sat}_{\mathcal{P}}(G)/e(G)\le\frac{1}{9}+o(1)$ since $e(G)=3v(G)-6$. These results address Clifton and Salia's question and narrow the asymptotic range of the plane-saturation ratio in maximal planar graphs, advancing understanding of extremal plane embeddings.
Abstract
We investigate the extremal properties of saturated partial plane embeddings of maximal planar graphs. For a planar graph $G$, the plane-saturation number $\mathrm{sat}_{\mathcal{P}}(G)$ denotes the minimum number of edges in a plane subgraph of $G$ such that the addition of any edge either violates planarity or results in a graph that is not a subgraph of $G$. We focus on maximal planar graphs and establish an upper bound on $\mathrm{sat}_{\mathcal{P}}(G)$ by showing there exists a universal constant $ε> 0$ such that $\mathrm{sat}_{\mathcal{P}}(G) < (3-ε)v(G)$ for any maximal planar graph $G$ with $v(G) \geq 16$. This answers a question posed by Clifton and Simon. Additionally, we derive lower bound results and demonstrate that for maximal planar graphs with sufficiently large number of vertices, the minimum ratio $\mathrm{sat}_{\mathcal{P}}(G)/e(G)$ lies within the interval $(1/16, 1/9 + o(1)]$.