Let $G = (V,E)$ be a plane graph. A face $f$ of $G$ is guarded by an edge $vw \in E$ if at least one vertex from $\{v,w\}$ is on the boundary of $f$. For a planar graph class $\mathcal{G}$ we ask for the minimal number of edges needed to guard all faces of any $n$-vertex graph in $\mathcal{G}$. We prove that $\lfloor n/3 \rfloor$ edges are always sufficient for quadrangulations and give a construction where $\lfloor (n-2)/4 \rfloor$ edges are necessary. For $2$-degenerate quadrangulations we improve this to a tight upper bound of $\lfloor n/4 \rfloor$ edges. We further prove that $\lfloor 2n/7 \rfloor$ edges are always sufficient for stacked triangulations (that are the $3$-degenerate triangulations) and show that this is best possible up to a small additive constant.

中文翻译：

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用边保护四边形和堆叠三角

令 $G = (V,E)$ 为平面图。如果 $\{v,w\}$ 中的至少一个顶点在 $f$ 的边界上，则 $G$ 的面 $f$ 由边 $vw \in E$ 保护。对于平面图类 $\mathcal{G}$，我们要求保护 $\mathcal{G}$ 中任何 $n$-顶点图的所有面所需的最少边数。我们证明 $\lfloor n/3 \rfloor$ 边对于四边形总是足够的，并给出了 $\lfloor (n-2)/4 \rfloor$ 边是必要的构造。对于 $2$-退化四边形，我们将其改进为 $\lfloor n/4 \rfloor$ 边缘的紧上界。我们进一步证明 $\lfloor 2n/7 \rfloor$ 边对于堆叠三角剖分（即 $3$-退化三角剖分）总是足够的，并表明这在一个小的附加常数下是最好的。