We show that every $n$-vertex triangulation has a connected dominating set of size at most $10n/21$. Equivalently, every $n$ vertex triangulation has a spanning tree with at least $11n/21$ leaves. Prior to the current work, the best known bounds were $n/2$, which follows from work of Albertson, Berman, Hutchinson, and Thomassen (J. Graph Theory \textbf{14}(2):247--258). One immediate consequence of this result is an improved bound for the SEFENOMAP graph drawing problem of Angelini, Evans, Frati, and Gudmundsson (J. Graph Theory \textbf{82}(1):45--64). As a second application, we show that every $n$-vertex planar graph has a one-bend non-crossing drawing in which some set of at least $11n/21$ vertices is drawn on the $x$-axis.

翻译：我们证明每个 $n$ 个顶点的三角剖分图存在一个大小不超过 $10n/21$ 的连通支配集。等价地，每个 $n$ 个顶点的三角剖分图存在一棵至少有 $11n/21$ 个叶子的生成树。此前已知的最佳界为 $n/2$，该结果源自 Albertson、Berman、Hutchinson 和 Thomassen 的工作（J. Graph Theory \textbf{14}(2):247--258）。该结果的一个直接推论是改进了 Angelini、Evans、Frati 和 Gudmundsson（J. Graph Theory \textbf{82}(1):45--64）提出的 SEFENOMAP 图绘制问题的界。作为第二个应用，我们证明每个 $n$ 个顶点的平面图存在一个单弯非交叉绘制，其中至少 $11n/21$ 个顶点被绘制在 $x$ 轴上。