  In 1996, Matheson and Tarjan proved that every near planar triangulation on
$n$ vertices contains a dominating set of size at most $n/3$, and conjectured
that this upper bound can be reduced to $n/4$ for planar triangulations when
$n$ is sufficiently large. In this paper, we consider the analogous problem for
independent dominating sets: What is the minimum $\epsilon$ for which every
near planar triangulation on $n$ vertices contains an independent dominating
set of size at most $\epsilon n$? We prove that $2/7 \leq \epsilon \leq 5/12$.
Moreover, this upper bound can be improved to $3/8$ for planar triangulations,
and to $1/3$ for planar triangulations with minimum degree 5.
