We introduce a class of plane graphs called weak near-triangulations, and prove that this class is closed under certain graph operations. Then we use the properties of weak near-triangulations to prove that every plane triangulation on $n>6$ vertices has a dominating set of size at most $17n/53$. This improves the bound $n/3$ obtained by Matheson and Tarjan.

中文翻译：

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平面三角的支配数

我们引入一类称为弱近三角的平面图，并证明该类在某些图操作下是封闭的。然后，我们使用弱近三角剖分的性质来证明每个平面三角剖分$ñ>6$ 顶点最多具有一组主要的大小 $17ñ/53$。这改善了界限$ñ/3$ 由Matheson和Tarjan获得。