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.

翻译：1996年，Matheson与Tarjan证明了每个包含$n$个顶点的几乎平面三角剖分均含有一个大小至多为$n/3$的支配集，并猜想当$n$充分大时，平面三角剖分的这一上界可降至$n/4$。本文考虑独立支配集的类似问题：对于包含$n$个顶点的几乎平面三角剖分，存在大小不超过$\epsilon n$的独立支配集的最小$\epsilon$是多少？我们证明$2/7 \leq \epsilon \leq 5/12$。此外，该上界在平面三角剖分中可改进至$3/8$，而在最小度为5的平面三角剖分中可改进至$1/3$。