  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 for every
set $P$ of $\lceil 11n/21\rceil$ points in $\R^2$ every $n$-vertex planar graph
has a one-bend non-crossing drawing in which some set of $11n/21$ vertices is
drawn on the points of $P$. The main result extends to $n$-vertex
triangulations of genus-$g$ surfaces, and implies that these have connected
dominating sets of size at most $10n/21+O(\sqrt{gn})$.
