The interval polynomial set proves to be an effective tool to deal with systems stability under bounded parameter perturbations. Based on the value set idea and vector operations in the complex plane, some early results on robust 𝒟 -slability of the interval polynomial set are improved. Extreme point results for some typical stability regions, including the relative stability region, the δ -operator stability region and the damping requirements stability region, are presented. Critical vertex polynomials with respect to each stability region are constructed, and some connections between these critical vertex polynomials have been established. The number of critical vertex polynomials is shown to be a cubic function of the order of the interval polynomial set. Explicit formulae for such numbers associated with each stability region have been derived. These results are then used to obtain extreme point results for robust 𝒟 -stabilization of the interval rational function set. Illustrative examples are also presented