Abstract
This paper investigates the combinatorial property of ultrafilters that Mathias forcing relativized to them does not add dominating reals. We prove that the characterization due to Hrušák and Minami is equivalent to the strong P-point property.
We also consistently construct a P-point that has no rapid Rudin-Keisler predecessor but that is not a strong P-point. These results answer questions of Canjar and Laflamme.