A dominating set of a graph $G$ is a subset $S$ of its vertices such that each vertex of $G$ not in $S$ has a neighbor in $S$. A face-hitting set of a plane graph $G$ is a set $T$ of vertices in $G$ such that every face of $G$ contains at least one vertex of $T$. We show that the vertex-set of every plane (multi-)graph without isolated vertices, self-loops or $2$-faces can be partitioned into two disjoint sets so that both the sets are dominating and face-hitting. We also show that all the three assumptions above are necessary for the conclusion. As a corollary, we show that every $n$-vertex simple plane triangulation has a dominating set of size at most $(1 - \alpha)n/2$, where $\alpha n$ is the maximum size of an independent set in the triangulation. Matheson and Tarjan [European J. Combin., 1996] conjectured that every plane triangulation with a sufficiently large number of vertices $n$ has a dominating set of size at most $n / 4$. Currently, the best known general bound for this is by Christiansen, Rotenberg and Rutschmann [SODA, 2024] who showed that every plane triangulation on $n > 10$ vertices has a dominating set of size at most $2n/7$. Our corollary improves their bound for $n$-vertex plane triangulations which contain a maximal independent set of size either less than $2n/7$ or more than $3n/7$.

翻译：图$G$的支配集是顶点子集$S$，使得$G$中不在$S$内的每个顶点在$S$中至少有一个邻居。平面图$G$的面覆盖集是顶点集$T$，使得$G$的每个面至少包含$T$中的一个顶点。我们证明，每个无孤立顶点、无自环或无$2$面的平面（多重）图的顶点集可以划分为两个不相交的集合，使得这两个集合既是支配集又是面覆盖集。我们还证明了上述三个假设对于结论的成立都是必要的。作为推论，我们证明每个$n$顶点简单平面三角剖分图存在一个大小至多为$(1 - \alpha)n/2$的支配集，其中$\alpha n$是该三角剖分图中最大独立集的大小。Matheson和Tarjan [European J. Combin., 1996] 猜想，每个具有足够大顶点数$n$的平面三角剖分图存在一个大小至多为$n / 4$的支配集。目前，对此的最佳已知一般界是由Christiansen、Rotenberg和Rutschmann [SODA, 2024] 给出的，他们证明每个顶点数$n > 10$的平面三角剖分图存在一个大小至多为$2n/7$的支配集。我们的推论改进了他们对包含最大独立集大小小于$2n/7$或大于$3n/7$的$n$顶点平面三角剖分图的界。