We show that every planar triangulation on $n>10$ vertices has a dominating set of size $n/7=n/3.5$. This approaches the $n/4$ bound conjectured by Matheson and Tarjan [MT'96], and improves significantly on the previous best bound of $17n/53\approx n/3.117$ by \v{S}pacapan [\v{S}'20]. From our proof it follows that every 3-connected $n$-vertex near-triangulation (except for 3 sporadic examples) has a dominating set of size $n/3.5$. On the other hand, for 3-connected near-triangulations, we show a lower bound of $3(n-1)/11\approx n/3.666$, demonstrating that the conjecture by Matheson and Tarjan [MT'96] cannot be strengthened to 3-connected near-triangulations. Our proof uses a penalty function that, aside from the number of vertices, penalises vertices of degree 2 and specific constellations of neighbours of degree 3 along the boundary of the outer face. To facilitate induction, we not only consider near-triangulations, but a wider class of graphs (skeletal triangulations), allowing us to delete vertices more freely. Our main technical contribution is a set of attachments, that are small graphs we inductively attach to our graph, in order both to remember whether existing vertices are already dominated, and that serve as a tool in a divide and conquer approach. Along with a well-chosen potential function, we thus both remove and add vertices during the induction proof. We complement our proof with a constructive algorithm that returns a dominating set of size $\le 2n/7$. Our algorithm has a quadratic running time.

翻译：我们证明：每个顶点数 $n>10$ 的平面三角剖分图包含一个大小为 $n/7=n/3.5$ 的控制集。这一结果逼近 Matheson 与 Tarjan [MT'96] 猜想的 $n/4$ 界，并显著改进了 \v{S}pacapan [\v{S}'20] 此前的最优界 $17n/53\approx n/3.117$。由证明可知，除3个零星反例外，每个3连通 $n$ 顶点近三角剖分图均存在大小为 $n/3.5$ 的控制集。另一方面，对于3连通近三角剖分图，我们给出下界 $3(n-1)/11\approx n/3.666$，表明 Matheson 与 Tarjan [MT'96] 的猜想无法推广至3连通近三角剖分图。本证明采用惩罚函数，该函数除顶点数量外，还对2度顶点及外边界上特定3度邻域构型施加惩罚。为便于归纳，我们不仅考虑近三角剖分图，还引入更广泛的图类（骨架三角剖分图），从而更自由地删除顶点。我们的主要技术贡献是一组附着结构：在归纳过程中逐步附着到图上的小规模子图，其作用既包括记录现有顶点是否已被控制，又作为分治策略的工具。结合精心选取的势函数，我们在归纳证明中同时进行顶点删减与添加。我们为证明配备了构造性算法，该算法可返回大小不超过 $2n/7$ 的控制集，且运行时间为二次复杂度。