Independent Dominating Sets in Planar Triangulations

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 \ (\) for which every near planar triangulation on \ (n\) vertices contains an independent dominating set of size at most \ (n\)? We prove that \ (2/7 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.